Afrikaans Mathspeak Steve Noble's samples. Locale: af, Style: Verbose.

0515623=negative 5 and one fifth minus 6 and two thirds equalsnegative 5 and een vyf de minus 6 and twee derdes is gelyk aan
1734(478)=negative 7 and three fourths minus left parenthesis negative 4 and seven eighths right parenthesis equalsnegative 7 and drie kwarts minus links hakkie negative 4 and sewe agstes regs hakkie is gelyk aan
224.15(13.7)=negative 24.15 minus left parenthesis 13.7 right parenthesis equalsnegative 24,15 minus links hakkie 13,7 regs hakkie is gelyk aan
3(4)×3=12left parenthesis negative 4 right parenthesis times 3 equals negative 12links hakkie negative 4 regs hakkie maal 3 is gelyk aan negative 12
412÷3=4negative 12 divided by 3 equals negative 4negative 12 divided by 3 is gelyk aan negative 4
512÷(4)=3negative 12 divided by left parenthesis negative 4 right parenthesis equals 3negative 12 divided by links hakkie negative 4 regs hakkie is gelyk aan 3
66×56 times 56 maal 5
76×(5)6 times left parenthesis negative 5 right parenthesis6 maal links hakkie negative 5 regs hakkie
86×5negative 6 times 5negative 6 maal 5
96×(5)negative 6 times left parenthesis negative 5 right parenthesisnegative 6 maal links hakkie negative 5 regs hakkie
108×7negative 8 times 7negative 8 maal 7
118×(7)negative 8 times left parenthesis negative 7 right parenthesisnegative 8 maal links hakkie negative 7 regs hakkie
128×(7)8 times left parenthesis negative 7 right parenthesis8 maal links hakkie negative 7 regs hakkie
138×78 times 78 maal 7
14m1=30°m angle 1 equals 30 degreem hoek 1 is gelyk aan 30 graad
15m2=60° m angle 2 equals 60 degreem hoek 2 is gelyk aan 60 graad
16m1+m2=90° m angle 1 plus m angle 2 equals 90 degreem hoek 1 plus m hoek 2 is gelyk aan 90 graad
17mM+mN=180° m angle upper M plus m angle upper N equals 180 degreem hoek großes M plus m hoek großes N is gelyk aan 180 graad
18A=12bhupper A equals one half b hgroßes A is gelyk aan een helfte b h
19area of trianglearea of square=1 unit216 units2StartFraction area of triangle Over area of square EndFraction equals StartFraction 1 unit squared Over 16 units squared EndFractionAnfang Bruch area of triangle durch area of square Ende Bruch is gelyk aan Anfang Bruch 1 unit squared durch 16 units squared Ende Bruch
200.620.6 squared0,6 squared
211.52 1.5 squared1,5 squared
224(2x+3x)4 left parenthesis 2 x plus 3 x right parenthesis4 links hakkie 2 x plus 3 x regs hakkie
2336+4y1y2+5y2236 plus 4 y minus 1 y squared plus 5 y squared minus 236 plus 4 y minus 1 y squared plus 5 y squared minus 2
24(5+9)4+3=left parenthesis 5 plus 9 right parenthesis minus 4 plus 3 equalslinks hakkie 5 plus 9 regs hakkie minus 4 plus 3 is gelyk aan
25BCModifyingAbove upper B upper C With left right arrowModifyingAbove großes B großes C With links regs pyltjie
26PQModifyingAbove upper P upper Q With right arrowModifyingAbove großes P großes Q With regs pyltjie
27GH¯ModifyingAbove upper G upper H With barModifyingAbove großes G großes H With makron
28WX¯YZ¯ModifyingAbove upper W upper X With bar approximately equals ModifyingAbove upper Y upper Z With barModifyingAbove großes W großes X With makron ongeveer gelyk aan ModifyingAbove großes Y großes Z With makron
29BEFangle upper B upper E upper Fhoek großes B großes E großes F
30BEDangle upper B upper E upper Dhoek großes B großes E großes D
31DEFangle upper D upper E upper Fhoek großes D großes E großes F
32x=b±b24ac2ax equals StartFraction negative b plus or minus StartRoot b squared minus 4 a c EndRoot Over 2 a EndFractionx is gelyk aan Anfang Bruch negative b plus of minus Anfang Wurzel b squared minus 4 a c Ende Wurzel durch 2 a Ende Bruch
33y=x2+8x+16y equals x squared plus 8 x plus 16y is gelyk aan x squared plus 8 x plus 16
34y=13(3x)y equals one third left parenthesis 3 Superscript x Baseline right parenthesisy is gelyk aan een derde links hakkie 3 hoch x Grundlinie regs hakkie
35y=102xy equals 10 minus 2 xy is gelyk aan 10 minus 2 x
36y=2x3+5y equals 2 x cubed plus 5y is gelyk aan 2 x cubed plus 5
37y=(x2+1)(x2+3)y equals left parenthesis x squared plus 1 right parenthesis left parenthesis x squared plus 3 right parenthesisy is gelyk aan links hakkie x squared plus 1 regs hakkie links hakkie x squared plus 3 regs hakkie
38y=0.5xy equals 0.5 Superscript xy is gelyk aan 0,5 hoch x
39y=222xy equals 22 minus 2 xy is gelyk aan 22 minus 2 x
40y=3xy equals StartFraction 3 Over x EndFractiony is gelyk aan Anfang Bruch 3 durch x Ende Bruch
41y=(x+4)(x+4)y equals left parenthesis x plus 4 right parenthesis left parenthesis x plus 4 right parenthesisy is gelyk aan links hakkie x plus 4 regs hakkie links hakkie x plus 4 regs hakkie
42y=(4x3)(x+1)y equals left parenthesis 4 x minus 3 right parenthesis left parenthesis x plus 1 right parenthesisy is gelyk aan links hakkie 4 x minus 3 regs hakkie links hakkie x plus 1 regs hakkie
43y=20x4x2y equals 20 x minus 4 x squaredy is gelyk aan 20 x minus 4 x squared
44y=x2y equals x squaredy is gelyk aan x squared
45y=3x1y equals 3 Superscript x minus 1y is gelyk aan 3 hoch x minus 1
46y=162(x+3)y equals 16 minus 2 left parenthesis x plus 3 right parenthesisy is gelyk aan 16 minus 2 links hakkie x plus 3 regs hakkie
47y=4x2x3y equals 4 x squared minus x minus 3y is gelyk aan 4 x squared minus x minus 3
48y=x+1xy equals x plus StartFraction 1 Over x EndFractiony is gelyk aan x plus Anfang Bruch 1 durch x Ende Bruch
49y=4x(5x)y equals 4 x left parenthesis 5 minus x right parenthesisy is gelyk aan 4 x links hakkie 5 minus x regs hakkie
50y=2(x3)+6(1x)y equals 2 left parenthesis x minus 3 right parenthesis plus 6 left parenthesis 1 minus x right parenthesisy is gelyk aan 2 links hakkie x minus 3 regs hakkie plus 6 links hakkie 1 minus x regs hakkie
510.25>516 0.25 greater than five sixteenths0,25 groter as vyf sestien des
5232(57)32 dot left parenthesis 5 dot 7 right parenthesis32 punt links hakkie 5 punt 7 regs hakkie
53(12×12×π×2)+(2×12×π×5)left parenthesis one half times one half times pi times 2 right parenthesis plus left parenthesis 2 times one half times pi times 5 right parenthesislinks hakkie een helfte maal een helfte maal pi maal 2 regs hakkie plus links hakkie 2 maal een helfte maal pi maal 5 regs hakkie
54liminfnEn=n1knEk,limsupnEn=n1knEk.liminf Underscript n right arrow infinity Endscripts upper E Subscript n Baseline equals union Underscript n greater than or equals 1 Endscripts intersection Underscript k greater than or equals n Endscripts upper E Subscript k Baseline comma limsup Underscript n right arrow infinity Endscripts upper E Subscript n Baseline equals intersection Underscript n greater than or equals 1 Endscripts union Underscript k greater than or equals n Endscripts upper E Subscript k Baseline periodliminf Unterschrift n regs pyltjie oneindigheid großes E Index n Grundlinie is gelyk aan union Unterschrift n groter of gelyk aan 1 intersection Unterschrift k groter of gelyk aan n großes E Index k Grundlinie komma limsup Unterschrift n regs pyltjie oneindigheid großes E Index n Grundlinie is gelyk aan intersection Unterschrift n groter of gelyk aan 1 union Unterschrift k groter of gelyk aan n großes E Index k Grundlinie punt
55(i)𝒮𝒜;(ii)ifE𝒜thenE𝒜;(iii)ifE1,E2𝒜thenE1E2𝒜.StartLayout 1st Row 1st Column left parenthesis i right parenthesis 2nd Column script upper S element of script upper A semicolon 2nd Row 1st Column left parenthesis ii right parenthesis 2nd Column if upper E element of script upper A then upper E overbar element of script upper A semicolon 3rd Row 1st Column left parenthesis iii right parenthesis 2nd Column if upper E 1 comma upper E 2 element of script upper A then upper E 1 union upper E 2 element of script upper A period EndLayoutStartLayout 1. Row 1. Column links hakkie i regs hakkie 2. Column Schreibschrift großes S element van Schreibschrift großes A kommapunt 2. Row 1. Column links hakkie ii regs hakkie 2. Column if großes E element van Schreibschrift großes A then großes E overbar element van Schreibschrift großes A kommapunt 3. Row 1. Column links hakkie iii regs hakkie 2. Column if großes E 1 komma großes E 2 element van Schreibschrift großes A then großes E 1 eenheid großes E 2 element van Schreibschrift großes A punt EndLayout
56(A.1)IfAthen0P{A}1.(1)(A.2)P{𝒮}=1.(2)(A.3)If{En,n1}is a sequence ofdisjoint(3)StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column left parenthesis normal upper A period 1 right parenthesis upper I f upper A element of script upper F t h e n 0 less than or equals upper P left brace upper A right brace less than or equals 1 period 4th Column left parenthesis 1 right parenthesis 2nd Row 1st Column Blank 2nd Column Blank 3rd Column left parenthesis normal upper A period 2 right parenthesis upper P left brace script upper S right brace equals 1 period 4th Column left parenthesis 2 right parenthesis 3rd Row 1st Column Blank 2nd Column Blank 3rd Column left parenthesis normal upper A period 3 right parenthesis upper I f left brace upper E Subscript n Baseline comma n greater than or equals 1 right brace element of script upper F is a sequence of disjoint 4th Column left parenthesis 3 right parenthesis EndLayoutStartLayout 1. Row 1. Column Blank 2. Column Blank 3. Column links hakkie normales großes A punt 1 regs hakkie großes I f großes A element van Schreibschrift großes F t h e n 0 kleiner of gelyk aan großes P links krulhakkie großes A regs krulhakkie kleiner of gelyk aan 1 punt 4. Column links hakkie 1 regs hakkie 2. Row 1. Column Blank 2. Column Blank 3. Column links hakkie normales großes A punt 2 regs hakkie großes P links krulhakkie Schreibschrift großes S regs krulhakkie is gelyk aan 1 punt 4. Column links hakkie 2 regs hakkie 3. Row 1. Column Blank 2. Column Blank 3. Column links hakkie normales großes A punt 3 regs hakkie großes I f links krulhakkie großes E Index n Grundlinie komma n groter of gelyk aan 1 regs krulhakkie element van Schreibschrift großes F is a sequence of disjoint 4. Column links hakkie 3 regs hakkie EndLayout
57P{Bj|A}=P{Bj}P{A|Bj}jJP{Bj}P{A|Bj}.upper P left brace upper B Subscript j Baseline vertical bar upper A right brace equals StartFraction upper P left brace upper B Subscript j Baseline right brace upper P left brace upper A vertical bar upper B Subscript j Baseline right brace Over sigma summation Underscript j prime element of upper J Endscripts upper P left brace upper B Subscript j prime Baseline right brace upper P left brace upper A vertical bar upper B Subscript j prime Baseline right brace EndFraction periodgroßes P links krulhakkie großes B Index j Grundlinie afstreep großes A regs krulhakkie is gelyk aan Anfang Bruch großes P links krulhakkie großes B Index j Grundlinie regs krulhakkie großes P links krulhakkie großes A afstreep großes B Index j Grundlinie regs krulhakkie durch sigma summation Unterschrift j priem element van großes J großes P links krulhakkie großes B Index j priem Grundlinie regs krulhakkie großes P links krulhakkie großes A afstreep großes B Index j priem Grundlinie regs krulhakkie Ende Bruch punt
58μ1(B)=Bf(x)dμ2(x)mu 1 left parenthesis upper B right parenthesis equals integral Underscript upper B Endscripts f left parenthesis x right parenthesis d mu 2 left parenthesis x right parenthesismy 1 links hakkie großes B regs hakkie is gelyk aan integraal Unterschrift großes B f links hakkie x regs hakkie d my 2 links hakkie x regs hakkie
59limnE{|XnX|}=E{limn|XnX|}=0.limit Underscript n right arrow infinity Endscripts upper E left brace StartAbsoluteValue upper X Subscript n Baseline minus upper X EndAbsoluteValue right brace equals upper E left brace limit Underscript n right arrow infinity Endscripts StartAbsoluteValue upper X Subscript n Baseline minus upper X EndAbsoluteValue right brace equals 0 periodlimit Unterschrift n regs pyltjie oneindigheid großes E links krulhakkie StartAbsoluteValue großes X Index n Grundlinie minus großes X EndAbsoluteValue regs krulhakkie is gelyk aan großes E links krulhakkie limit Unterschrift n regs pyltjie oneindigheid StartAbsoluteValue großes X Index n Grundlinie minus großes X EndAbsoluteValue regs krulhakkie is gelyk aan 0 punt
60Pμ,σ{Ylβ(Yn,Sn)}=Pμ,σ{(YYn)/(S·(1+1n)1/2)tβ[n1]}=β,(1)StartLayout 1st Row 1st Column upper P Subscript mu comma sigma Baseline left brace upper Y greater than or equals l Subscript beta Baseline left parenthesis upper Y overbar Subscript n Baseline comma upper S Subscript n Baseline right parenthesis right brace equals upper P Subscript mu comma sigma Baseline left brace left parenthesis upper Y minus upper Y overbar Subscript n Baseline right parenthesis divided by left parenthesis upper S dot left parenthesis 1 plus StartFraction 1 Over n EndFraction right parenthesis Superscript 1 divided by 2 Baseline right parenthesis greater than or equals minus t Subscript beta Baseline left bracket n minus 1 right bracket right brace equals beta comma 2nd Row 1st Column Blank 2nd Column left parenthesis 1 right parenthesis EndLayoutStartLayout 1. Row 1. Column großes P Index my komma sigma Grundlinie links krulhakkie großes Y groter of gelyk aan l Index beta Grundlinie links hakkie großes Y overbar Index n Grundlinie komma großes S Index n Grundlinie regs hakkie regs krulhakkie is gelyk aan großes P Index my komma sigma Grundlinie links krulhakkie links hakkie großes Y minus großes Y overbar Index n Grundlinie regs hakkie divided by links hakkie großes S dot links hakkie 1 plus Anfang Bruch 1 durch n Ende Bruch regs hakkie hoch 1 divided by 2 Grundlinie regs hakkie groter of gelyk aan minus t Index beta Grundlinie links blokhakkie n minus 1 regs blokhakkie regs krulhakkie is gelyk aan beta komma 2. Row 1. Column Blank 2. Column links hakkie 1 regs hakkie EndLayout
61L=(11110011).upper L equals Start 5 By 6 Matrix 1st Row 1st Column 1 2nd Column negative 1 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 2nd Row 1st Column Blank 2nd Column 1 3rd Column negative 1 4th Column Blank 5th Column 0 6th Column Blank 3rd Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 4th Row 1st Column Blank 2nd Column 0 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 5th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column 1 6th Column negative 1 EndMatrix periodgroßes L is gelyk aan Start 5 By 6 Matrix 1. Row 1. Column 1 2. Column negative 1 3. Column Blank 4. Column Blank 5. Column Blank 6. Column Blank 2. Row 1. Column Blank 2. Column 1 3. Column negative 1 4. Column Blank 5. Column 0 6. Column Blank 3. Row 1. Column Blank 2. Column Blank 3. Column Blank 4. Column Blank 5. Column Blank 6. Column Blank 4. Row 1. Column Blank 2. Column 0 3. Column Blank 4. Column Blank 5. Column Blank 6. Column Blank 5. Row 1. Column Blank 2. Column Blank 3. Column Blank 4. Column Blank 5. Column 1 6. Column negative 1 EndMatrix punt
62n[Yn(μ+zβσ)]/Sn~U+nz1β(χ2[n1]/(n1))1/2~t[n1;nz1β],StartRoot n EndRoot left bracket upper Y overbar Subscript n Baseline minus left parenthesis mu plus z Subscript beta Baseline sigma right parenthesis right bracket divided by upper S Subscript n Baseline tilde StartFraction upper U plus StartRoot n EndRoot z Subscript 1 minus beta Baseline Over left parenthesis chi squared left bracket n minus 1 right bracket divided by left parenthesis n minus 1 right parenthesis right parenthesis Superscript 1 divided by 2 Baseline EndFraction tilde t left bracket n minus 1 semicolon StartRoot n EndRoot z Subscript 1 minus beta Baseline right bracket commaAnfang Wurzel n Ende Wurzel links blokhakkie großes Y overbar Index n Grundlinie minus links hakkie my plus z Index beta Grundlinie sigma regs hakkie regs blokhakkie divided by großes S Index n Grundlinie tilde Anfang Bruch großes U plus Anfang Wurzel n Ende Wurzel z Index 1 minus beta Grundlinie durch links hakkie chi squared links blokhakkie n minus 1 regs blokhakkie divided by links hakkie n minus 1 regs hakkie regs hakkie hoch 1 divided by 2 Grundlinie Ende Bruch tilde t links blokhakkie n minus 1 kommapunt Anfang Wurzel n Ende Wurzel z Index 1 minus beta Grundlinie regs blokhakkie komma
63γ=P{Ep,q(X(r),X(s)}=n!(r1)!j=0 sr1(1)jpr+j(nrj)!j!I1q(ns+1,srj).StartLayout 1st Row 1st Column gamma 2nd Column equals upper P left brace upper E Subscript p comma q Baseline subset of left parenthesis upper X Subscript left parenthesis r right parenthesis Baseline comma upper X Subscript left parenthesis s right parenthesis Baseline right brace 2nd Row 1st Column Blank 2nd Column equals StartFraction n factorial Over left parenthesis r minus 1 right parenthesis factorial EndFraction sigma summation Underscript j equals 0 Overscript s minus r minus 1 Endscripts left parenthesis negative 1 right parenthesis Superscript j Baseline StartFraction p Superscript r plus j Baseline Over left parenthesis n minus r minus j right parenthesis factorial j factorial EndFraction upper I Subscript 1 minus q Baseline left parenthesis n minus s plus 1 comma s minus r minus j right parenthesis period EndLayoutStartLayout 1. Row 1. Column gamma 2. Column is gelyk aan großes P links krulhakkie großes E Index p komma q Grundlinie deelversameling van links hakkie großes X Index links hakkie r regs hakkie Grundlinie komma großes X Index links hakkie s regs hakkie Grundlinie regs krulhakkie 2. Row 1. Column Blank 2. Column is gelyk aan Anfang Bruch n factorial durch links hakkie r minus 1 regs hakkie factorial Ende Bruch sigma summation Unterschrift j is gelyk aan 0 Überschrift s minus r minus 1 links hakkie negative 1 regs hakkie hoch j Grundlinie Anfang Bruch p hoch r plus j Grundlinie durch links hakkie n minus r minus j regs hakkie factorial j factorial Ende Bruch großes I Index 1 minus q Grundlinie links hakkie n minus s plus 1 komma s minus r minus j regs hakkie punt EndLayout
64Sitx=1/m0airitx+(i1)/mbi,upper S Subscript i Baseline StartBinomialOrMatrix t Choose x EndBinomialOrMatrix equals Start 2 By 2 Matrix 1st Row 1st Column 1 divided by m 2nd Column 0 2nd Row 1st Column a Subscript i Baseline 2nd Column r Subscript i Baseline EndMatrix StartBinomialOrMatrix t Choose x EndBinomialOrMatrix plus StartBinomialOrMatrix left parenthesis i minus 1 right parenthesis divided by m Choose b Subscript i Baseline EndBinomialOrMatrix commagroßes S Index i Grundlinie StartBinomialOrMatrix t Choose x EndBinomialOrMatrix is gelyk aan Start 2 By 2 Matrix 1. Row 1. Column 1 divided by m 2. Column 0 2. Row 1. Column a Index i Grundlinie 2. Column r Index i Grundlinie EndMatrix StartBinomialOrMatrix t Choose x EndBinomialOrMatrix plus StartBinomialOrMatrix links hakkie i minus 1 regs hakkie divided by m Choose b Index i Grundlinie EndBinomialOrMatrix komma
65c1h42s12TTT(f(t+h)f(t))2dtc2h42sc 1 h Superscript 4 minus 2 s Baseline less than or equals StartFraction 1 Over 2 upper T EndFraction integral Subscript negative upper T Superscript upper T Baseline left parenthesis f left parenthesis t plus h right parenthesis minus f left parenthesis t right parenthesis right parenthesis squared normal d t less than or equals c 2 h Superscript 4 minus 2 sc 1 h hoch 4 minus 2 s Grundlinie kleiner of gelyk aan Anfang Bruch 1 durch 2 großes T Ende Bruch integraal Subscript negative großes T Superscript großes T Baseline links hakkie f links hakkie t plus h regs hakkie minus f links hakkie t regs hakkie regs hakkie squared normales d t kleiner of gelyk aan c 2 h hoch 4 minus 2 s
66C(0)C(h)ch42supper C left parenthesis 0 right parenthesis minus upper C left parenthesis h right parenthesis asymptotically equals c h Superscript 4 minus 2 sgroßes C links hakkie 0 regs hakkie minus großes C links hakkie h regs hakkie assimtoties gelyk aan c h hoch 4 minus 2 s
67S(ω)=limT12TTTf(t)eitωdt2.upper S left parenthesis omega right parenthesis equals limit Underscript upper T right arrow infinity Endscripts StartFraction 1 Over 2 upper T EndFraction StartAbsoluteValue integral Subscript negative upper T Superscript upper T Baseline comma f comma left parenthesis comma t comma right parenthesis comma normal e Superscript italic i t omega Baseline comma normal d comma t EndAbsoluteValue squared periodgroßes S links hakkie omega regs hakkie is gelyk aan limit Unterschrift großes T regs pyltjie oneindigheid Anfang Bruch 1 durch 2 großes T Ende Bruch StartAbsoluteValue integraal Subscript negative großes T Superscript großes T Baseline komma f komma links hakkie komma t komma regs hakkie komma normales e hoch kursives i t omega Grundlinie komma normales d komma t EndAbsoluteValue squared punt
680101[|f(t)f(u)|2+|tu|2]s/2dtdu<integral Subscript 0 Superscript 1 Baseline integral Subscript 0 Superscript 1 Baseline left bracket StartAbsoluteValue f left parenthesis t right parenthesis minus f left parenthesis u right parenthesis EndAbsoluteValue squared plus StartAbsoluteValue t minus u EndAbsoluteValue squared right bracket Superscript negative s divided by 2 Baseline normal d t normal d u less than infinityintegraal Subscript 0 Superscript 1 Baseline integraal Subscript 0 Superscript 1 Baseline links blokhakkie StartAbsoluteValue f links hakkie t regs hakkie minus f links hakkie u regs hakkie EndAbsoluteValue squared plus StartAbsoluteValue t minus u EndAbsoluteValue squared regs blokhakkie hoch negative s divided by 2 Grundlinie normales d t normales d u kleiner as oneindigheid
69EIEk+1|I|s=EIEk|I|sE(R1s+R2s).sans serif upper E left parenthesis sigma summation Underscript upper I element of upper E Subscript k plus 1 Baseline Endscripts StartAbsoluteValue upper I EndAbsoluteValue Superscript s Baseline right parenthesis equals sans serif upper E left parenthesis sigma summation Underscript upper I element of upper E Subscript k Baseline Endscripts StartAbsoluteValue upper I EndAbsoluteValue Superscript s Baseline right parenthesis sans serif upper E left parenthesis upper R 1 Superscript s Baseline plus upper R 2 Superscript s Baseline right parenthesis periodserifenloses großes E links hakkie sigma summation Unterschrift großes I element van großes E Index k plus 1 Grundlinie StartAbsoluteValue großes I EndAbsoluteValue hoch s Grundlinie regs hakkie is gelyk aan serifenloses großes E links hakkie sigma summation Unterschrift großes I element van großes E Index k Grundlinie StartAbsoluteValue großes I EndAbsoluteValue hoch s Grundlinie regs hakkie serifenloses großes E links hakkie großes R 1 hoch s Grundlinie plus großes R 2 hoch s Grundlinie regs hakkie punt
70(x1,y1)left parenthesis x 1 comma y 1 right parenthesislinks hakkie x 1 komma y 1 regs hakkie
71(x2,y2)left parenthesis x 2 comma y 2 right parenthesislinks hakkie x 2 komma y 2 regs hakkie
72d=(x2x1)2+(y2y1)2d equals StartRoot left parenthesis x 2 minus x 1 right parenthesis squared plus left parenthesis y 2 minus y 1 right parenthesis squared EndRootd is gelyk aan Anfang Wurzel links hakkie x 2 minus x 1 regs hakkie squared plus links hakkie y 2 minus y 1 regs hakkie squared Ende Wurzel
73double struck upper Rmit Doppelstrich großes R
74=(,)double struck upper R equals left parenthesis negative infinity comma infinity right parenthesismit Doppelstrich großes R is gelyk aan links hakkie negative oneindigheid komma oneindigheid regs hakkie
75{ 1,2,3 }StartSet 1 comma 2 comma 3 EndSetStartSet 1 komma 2 komma 3 EndSet
761S1 element of upper S1 element van großes S
773S3 element of upper S3 element van großes S
784S4 not an element of upper S4 nie 'n element van großes S
79a=3x1+(1+x)2a equals StartRoot 3 x minus 1 EndRoot plus left parenthesis 1 plus x right parenthesis squareda is gelyk aan Anfang Wurzel 3 x minus 1 Ende Wurzel plus links hakkie 1 plus x regs hakkie squared
80a=(b+c)2d+(e+f)2ga equals StartFraction left parenthesis b plus c right parenthesis squared Over d EndFraction plus StartFraction left parenthesis e plus f right parenthesis squared Over g EndFractiona is gelyk aan Anfang Bruch links hakkie b plus c regs hakkie squared durch d Ende Bruch plus Anfang Bruch links hakkie e plus f regs hakkie squared durch g Ende Bruch
81x=[ (a+b)2(cb)2 ]+[ (d+e)2(fe)2 ]x equals left bracket left parenthesis a plus b right parenthesis squared left parenthesis c minus b right parenthesis squared right bracket plus left bracket left parenthesis d plus e right parenthesis squared left parenthesis f minus e right parenthesis squared right bracketx is gelyk aan links blokhakkie links hakkie a plus b regs hakkie squared links hakkie c minus b regs hakkie squared regs blokhakkie plus links blokhakkie links hakkie d plus e regs hakkie squared links hakkie f minus e regs hakkie squared regs blokhakkie
82x=[ (a+b)2 ]+[ (fe)2 ]x equals left bracket left parenthesis a plus b right parenthesis squared right bracket plus left bracket left parenthesis f minus e right parenthesis squared right bracketx is gelyk aan links blokhakkie links hakkie a plus b regs hakkie squared regs blokhakkie plus links blokhakkie links hakkie f minus e regs hakkie squared regs blokhakkie
83x=[ (a+b)2 ]x equals left bracket left parenthesis a plus b right parenthesis squared right bracketx is gelyk aan links blokhakkie links hakkie a plus b regs hakkie squared regs blokhakkie
84x=(a+b)2x equals left parenthesis a plus b right parenthesis squaredx is gelyk aan links hakkie a plus b regs hakkie squared
85x=a+b2x equals a plus b squaredx is gelyk aan a plus b squared
861234=23StartFraction one half Over three fourths EndFraction equals two thirdsAnfang Bruch een helfte durch drie kwarts Ende Bruch is gelyk aan twee derdes
872((x+1)(x+3)4((x1)(x+2)3))=y2 left parenthesis left parenthesis x plus 1 right parenthesis left parenthesis x plus 3 right parenthesis minus 4 left parenthesis left parenthesis x minus 1 right parenthesis left parenthesis x plus 2 right parenthesis minus 3 right parenthesis right parenthesis equals y2 links hakkie links hakkie x plus 1 regs hakkie links hakkie x plus 3 regs hakkie minus 4 links hakkie links hakkie x minus 1 regs hakkie links hakkie x plus 2 regs hakkie minus 3 regs hakkie regs hakkie is gelyk aan y
88cosx=1x22!+x44!cosine x equals 1 minus StartFraction x squared Over 2 factorial EndFraction plus StartFraction x Superscript 4 Baseline Over 4 factorial EndFraction minus ellipsiskosinus x is gelyk aan 1 minus Anfang Bruch x squared durch 2 factorial Ende Bruch plus Anfang Bruch x hoch 4 Grundlinie durch 4 factorial Ende Bruch minus ellipsis
89x=b±b24ac2ax equals StartFraction negative b plus or minus StartRoot b squared minus 4 a c EndRoot Over 2 a EndFractionx is gelyk aan Anfang Bruch negative b plus of minus Anfang Wurzel b squared minus 4 a c Ende Wurzel durch 2 a Ende Bruch
90x+y2k+1x plus y Superscript StartFraction 2 Over k plus 1 EndFractionx plus y hoch Anfang Bruch 2 durch k plus 1 Ende Bruch
91limx0sinxx=1limit Underscript x right arrow 0 Endscripts StartFraction sine x Over x EndFraction equals 1limit Unterschrift x regs pyltjie 0 Anfang Bruch sinus x durch x Ende Bruch is gelyk aan 1
92d=(x2x1)2+(y2y1)2d equals StartRoot left parenthesis x 2 minus x 1 right parenthesis squared plus left parenthesis y 2 minus y 1 right parenthesis squared EndRootd is gelyk aan Anfang Wurzel links hakkie x 2 minus x 1 regs hakkie squared plus links hakkie y 2 minus y 1 regs hakkie squared Ende Wurzel
93Fn=Fn1+Fn2upper F Subscript n Baseline equals upper F Subscript n minus 1 Baseline plus upper F Subscript n minus 2großes F Index n Grundlinie is gelyk aan großes F Index n minus 1 Grundlinie plus großes F Index n minus 2
94Π=(π11π12π12000π12π11π12000π12π12π11000000π44000000π44000000π44)bold upper Pi equals Start 6 By 6 Matrix 1st Row 1st Column pi 11 2nd Column pi 12 3rd Column pi 12 4th Column 0 5th Column 0 6th Column 0 2nd Row 1st Column pi 12 2nd Column pi 11 3rd Column pi 12 4th Column 0 5th Column 0 6th Column 0 3rd Row 1st Column pi 12 2nd Column pi 12 3rd Column pi 11 4th Column 0 5th Column 0 6th Column 0 4th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column pi 44 5th Column 0 6th Column 0 5th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column pi 44 6th Column 0 6th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 6th Column pi 44 EndMatrixfettes großes Pi is gelyk aan Start 6 By 6 Matrix 1. Row 1. Column pi 11 2. Column pi 12 3. Column pi 12 4. Column 0 5. Column 0 6. Column 0 2. Row 1. Column pi 12 2. Column pi 11 3. Column pi 12 4. Column 0 5. Column 0 6. Column 0 3. Row 1. Column pi 12 2. Column pi 12 3. Column pi 11 4. Column 0 5. Column 0 6. Column 0 4. Row 1. Column 0 2. Column 0 3. Column 0 4. Column pi 44 5. Column 0 6. Column 0 5. Row 1. Column 0 2. Column 0 3. Column 0 4. Column 0 5. Column pi 44 6. Column 0 6. Row 1. Column 0 2. Column 0 3. Column 0 4. Column 0 5. Column 0 6. Column pi 44 EndMatrix
95s11=c11+c12c11c12c11+2c12s 11 equals StartFraction c 11 plus c 12 Over left parenthesis c 11 minus c 12 right parenthesis left parenthesis c 11 plus 2 c 12 right parenthesis EndFractions 11 is gelyk aan Anfang Bruch c 11 plus c 12 durch links hakkie c 11 minus c 12 regs hakkie links hakkie c 11 plus 2 c 12 regs hakkie Ende Bruch
96SiO2+ 6HF H2 SiF6+ 2H2O upper S i normal upper O 2 plus 6 normal upper H normal upper F right arrow normal upper H 2 upper S i normal upper F 6 plus 2 normal upper H 2 normal upper Ogroßes S i normales großes O 2 plus 6 normales großes H normales großes F regs pyltjie normales großes H 2 großes S i normales großes F 6 plus 2 normales großes H 2 normales großes O
97ddx(E(x)A(x)dw(x)dx)+p(x)=0StartFraction d Over d x EndFraction left parenthesis upper E left parenthesis x right parenthesis upper A left parenthesis x right parenthesis StartFraction d w left parenthesis x right parenthesis Over d x EndFraction right parenthesis plus p left parenthesis x right parenthesis equals 0Anfang Bruch d durch d x Ende Bruch links hakkie großes E links hakkie x regs hakkie großes A links hakkie x regs hakkie Anfang Bruch d w links hakkie x regs hakkie durch d x Ende Bruch regs hakkie plus p links hakkie x regs hakkie is gelyk aan 0
98TCSgas=12PsealPmax1TsealTCS Subscript gas Baseline equals minus one half left parenthesis StartFraction upper P Subscript seal Baseline Over upper P Subscript max Baseline EndFraction right parenthesis left parenthesis StartFraction 1 Over upper T Subscript seal Baseline EndFraction right parenthesisTCS Index gas Grundlinie is gelyk aan minus een helfte links hakkie Anfang Bruch großes P Index seal Grundlinie durch großes P Index max Grundlinie Ende Bruch regs hakkie links hakkie Anfang Bruch 1 durch großes T Index seal Grundlinie Ende Bruch regs hakkie
99Bp=7v231+c2a2+c4a4+3v2c21+va21v1c4a41c2a2upper B Subscript p Baseline equals StartStartFraction StartFraction 7 minus v squared Over 3 EndFraction left parenthesis 1 plus StartFraction c squared Over a squared EndFraction plus StartFraction c Superscript 4 Baseline Over a Superscript 4 Baseline EndFraction right parenthesis plus StartFraction left parenthesis 3 minus v right parenthesis squared c squared Over left parenthesis 1 plus v right parenthesis a squared EndFraction OverOver left parenthesis 1 minus v right parenthesis left parenthesis 1 minus StartFraction c Superscript 4 Baseline Over a Superscript 4 Baseline EndFraction right parenthesis left parenthesis 1 minus StartFraction c squared Over a squared EndFraction right parenthesis EndEndFractiongroßes B Index p Grundlinie is gelyk aan Anfang Anfang Bruch Anfang Bruch 7 minus v squared durch 3 Ende Bruch links hakkie 1 plus Anfang Bruch c squared durch a squared Ende Bruch plus Anfang Bruch c hoch 4 Grundlinie durch a hoch 4 Grundlinie Ende Bruch regs hakkie plus Anfang Bruch links hakkie 3 minus v regs hakkie squared c squared durch links hakkie 1 plus v regs hakkie a squared Ende Bruch durch durch links hakkie 1 minus v regs hakkie links hakkie 1 minus Anfang Bruch c hoch 4 Grundlinie durch a hoch 4 Grundlinie Ende Bruch regs hakkie links hakkie 1 minus Anfang Bruch c squared durch a squared Ende Bruch regs hakkie Ende Ende Bruch
100Qtankseries=1RsLsCsupper Q Subscript tank Superscript series Baseline equals StartFraction 1 Over upper R Subscript s Baseline EndFraction StartRoot StartFraction upper L Subscript s Baseline Over upper C Subscript s Baseline EndFraction EndRootgroßes Q Index tank hoch series Grundlinie is gelyk aan Anfang Bruch 1 durch großes R Index s Grundlinie Ende Bruch Anfang Wurzel Anfang Bruch großes L Index s Grundlinie durch großes C Index s Grundlinie Ende Bruch Ende Wurzel
101Δϕpeak=tan1(k2Qtankseries)upper Delta phi Subscript peak Baseline equals tangent Superscript negative 1 Baseline left parenthesis k squared upper Q Subscript tank Superscript series Baseline right parenthesisgroßes Delta phi Index peak Grundlinie is gelyk aan tangens hoch negative 1 Grundlinie links hakkie k squared großes Q Index tank hoch series Grundlinie regs hakkie
102f=1.013WL2Eρ(1+0.293L2EW2σ)f equals 1.013 StartFraction upper W Over upper L squared EndFraction StartRoot StartFraction upper E Over rho EndFraction EndRoot StartRoot left parenthesis 1 plus 0.293 StartFraction upper L squared Over EW squared EndFraction sigma right parenthesis EndRootf is gelyk aan 1.013 Anfang Bruch großes W durch großes L squared Ende Bruch Anfang Wurzel Anfang Bruch großes E durch rho Ende Bruch Ende Wurzel Anfang Wurzel links hakkie 1 plus 0.293 Anfang Bruch großes L squared durch EW squared Ende Bruch sigma regs hakkie Ende Wurzel
103unx=γncoshknxcosknx+sinhknxsinknxu Subscript n Baseline left parenthesis x right parenthesis equals gamma Subscript n Baseline left parenthesis hyperbolic cosine k Subscript n Baseline x minus cosine k Subscript n Baseline x right parenthesis plus left parenthesis hyperbolic sine k Subscript n Baseline x minus sine k Subscript n Baseline x right parenthesisu Index n Grundlinie links hakkie x regs hakkie is gelyk aan gamma Index n Grundlinie links hakkie hiperboliese kosinus k Index n Grundlinie x minus kosinus k Index n Grundlinie x regs hakkie plus links hakkie hiperboliese sinus k Index n Grundlinie x minus sinus k Index n Grundlinie x regs hakkie
104B=F0m(ω02ω2)2+4n2ω2=F0k(1(ω/ω02)2)2+4(n/ω0)2(ω/ω0)2StartLayout 1st Row 1st Column upper B 2nd Column equals StartStartFraction StartFraction upper F 0 Over m EndFraction OverOver StartRoot left parenthesis omega 0 squared minus omega squared right parenthesis squared plus 4 n squared omega squared EndRoot EndEndFraction 2nd Row 1st Column Blank 2nd Column equals StartStartFraction StartFraction upper F 0 Over k EndFraction OverOver StartRoot left parenthesis 1 minus left parenthesis omega divided by omega 0 squared right parenthesis squared right parenthesis squared plus 4 left parenthesis n divided by omega 0 right parenthesis squared left parenthesis omega divided by omega 0 right parenthesis squared EndRoot EndEndFraction EndLayoutStartLayout 1. Row 1. Column großes B 2. Column is gelyk aan Anfang Anfang Bruch Anfang Bruch großes F 0 durch m Ende Bruch durch durch Anfang Wurzel links hakkie omega 0 squared minus omega squared regs hakkie squared plus 4 n squared omega squared Ende Wurzel Ende Ende Bruch 2. Row 1. Column Blank 2. Column is gelyk aan Anfang Anfang Bruch Anfang Bruch großes F 0 durch k Ende Bruch durch durch Anfang Wurzel links hakkie 1 minus links hakkie omega divided by omega 0 squared regs hakkie squared regs hakkie squared plus 4 links hakkie n divided by omega 0 regs hakkie squared links hakkie omega divided by omega 0 regs hakkie squared Ende Wurzel Ende Ende Bruch EndLayout
105p(AandB)=p(A)p(B|A)normal p left parenthesis upper A a n d upper B right parenthesis equals normal p left parenthesis upper A right parenthesis normal p left parenthesis upper B vertical bar upper A right parenthesisnormales p links hakkie großes A a n d großes B regs hakkie is gelyk aan normales p links hakkie großes A regs hakkie normales p links hakkie großes B afstreep großes A regs hakkie
106PMF(x)1xαupper P upper M upper F left parenthesis x right parenthesis proportional to left parenthesis StartFraction 1 Over x EndFraction right parenthesis Superscript alphagroßes P großes M großes F links hakkie x regs hakkie eweredig aan links hakkie Anfang Bruch 1 durch x Ende Bruch regs hakkie hoch alfa
107f(x)=12πexp(-x2/2)f left parenthesis x right parenthesis equals StartFraction 1 Over StartRoot 2 pi EndRoot EndFraction exp left parenthesis minus x squared slash 2 right parenthesisf links hakkie x regs hakkie is gelyk aan Anfang Bruch 1 durch Anfang Wurzel 2 pi Ende Wurzel Ende Bruch exp links hakkie minus x squared skuinsstreep 2 regs hakkie
108dxdθ=βcos2θStartFraction d x Over d theta EndFraction equals StartFraction beta Over cosine squared theta EndFractionAnfang Bruch d x durch d theta Ende Bruch is gelyk aan Anfang Bruch beta durch kosinus squared theta Ende Bruch
109s/2(n-1)s divided by StartRoot 2 left parenthesis n minus 1 right parenthesis EndRoots divided by Anfang Wurzel 2 links hakkie n minus 1 regs hakkie Ende Wurzel

Afrikaans Mathspeak Steve Noble's samples. Locale: af, Style: Brief.

0515623=negative 5 and one fifth minus 6 and two thirds equalsnegative 5 and een vyf de minus 6 and twee derdes is gelyk aan
1734(478)=negative 7 and three fourths minus left p'ren negative 4 and seven eighths right p'ren equalsnegative 7 and drie kwarts minus left p'ren negative 4 and sewe agstes right p'ren is gelyk aan
224.15(13.7)=negative 24.15 minus left p'ren 13.7 right p'ren equalsnegative 24,15 minus left p'ren 13,7 right p'ren is gelyk aan
3(4)×3=12left p'ren negative 4 right p'ren times 3 equals negative 12left p'ren negative 4 right p'ren maal 3 is gelyk aan negative 12
412÷3=4negative 12 divided by 3 equals negative 4negative 12 divided by 3 is gelyk aan negative 4
512÷(4)=3negative 12 divided by left p'ren negative 4 right p'ren equals 3negative 12 divided by left p'ren negative 4 right p'ren is gelyk aan 3
66×56 times 56 maal 5
76×(5)6 times left p'ren negative 5 right p'ren6 maal left p'ren negative 5 right p'ren
86×5negative 6 times 5negative 6 maal 5
96×(5)negative 6 times left p'ren negative 5 right p'rennegative 6 maal left p'ren negative 5 right p'ren
108×7negative 8 times 7negative 8 maal 7
118×(7)negative 8 times left p'ren negative 7 right p'rennegative 8 maal left p'ren negative 7 right p'ren
128×(7)8 times left p'ren negative 7 right p'ren8 maal left p'ren negative 7 right p'ren
138×78 times 78 maal 7
14m1=30°m angle 1 equals 30 degreem hoek 1 is gelyk aan 30 graad
15m2=60° m angle 2 equals 60 degreem hoek 2 is gelyk aan 60 graad
16m1+m2=90° m angle 1 plus m angle 2 equals 90 degreem hoek 1 plus m hoek 2 is gelyk aan 90 graad
17mM+mN=180° m angle upper M plus m angle upper N equals 180 degreem hoek großes M plus m hoek großes N is gelyk aan 180 graad
18A=12bhupper A equals one half b hgroßes A is gelyk aan een helfte b h
19area of trianglearea of square=1 unit216 units2StartFrac area of triangle Over area of square EndFrac equals StartFrac 1 unit squared Over 16 units squared EndFracAnfang Bruch area of triangle durch area of square Ende Bruch is gelyk aan Anfang Bruch 1 unit squared durch 16 units squared Ende Bruch
200.620.6 squared0,6 squared
211.52 1.5 squared1,5 squared
224(2x+3x)4 left p'ren 2 x plus 3 x right p'ren4 left p'ren 2 x plus 3 x right p'ren
2336+4y1y2+5y2236 plus 4 y minus 1 y squared plus 5 y squared minus 236 plus 4 y minus 1 y squared plus 5 y squared minus 2
24(5+9)4+3=left p'ren 5 plus 9 right p'ren minus 4 plus 3 equalsleft p'ren 5 plus 9 right p'ren minus 4 plus 3 is gelyk aan
25BCModAbove upper B upper C With left right arrowModAbove großes B großes C With links regs pyltjie
26PQModAbove upper P upper Q With right arrowModAbove großes P großes Q With regs pyltjie
27GH¯ModAbove upper G upper H With barModAbove großes G großes H With makron
28WX¯YZ¯ModAbove upper W upper X With bar approximately equals ModAbove upper Y upper Z With barModAbove großes W großes X With makron ongeveer gelyk aan ModAbove großes Y großes Z With makron
29BEFangle upper B upper E upper Fhoek großes B großes E großes F
30BEDangle upper B upper E upper Dhoek großes B großes E großes D
31DEFangle upper D upper E upper Fhoek großes D großes E großes F
32x=b±b24ac2ax equals StartFrac negative b plus or minus StartRoot b squared minus 4 a c EndRoot Over 2 a EndFracx is gelyk aan Anfang Bruch negative b plus of minus Anfang Wurzel b squared minus 4 a c Ende Wurzel durch 2 a Ende Bruch
33y=x2+8x+16y equals x squared plus 8 x plus 16y is gelyk aan x squared plus 8 x plus 16
34y=13(3x)y equals one third left p'ren 3 Sup x Base right p'reny is gelyk aan een derde left p'ren 3 hoch x Grund right p'ren
35y=102xy equals 10 minus 2 xy is gelyk aan 10 minus 2 x
36y=2x3+5y equals 2 x cubed plus 5y is gelyk aan 2 x cubed plus 5
37y=(x2+1)(x2+3)y equals left p'ren x squared plus 1 right p'ren left p'ren x squared plus 3 right p'reny is gelyk aan left p'ren x squared plus 1 right p'ren left p'ren x squared plus 3 right p'ren
38y=0.5xy equals 0.5 Sup xy is gelyk aan 0,5 hoch x
39y=222xy equals 22 minus 2 xy is gelyk aan 22 minus 2 x
40y=3xy equals StartFrac 3 Over x EndFracy is gelyk aan Anfang Bruch 3 durch x Ende Bruch
41y=(x+4)(x+4)y equals left p'ren x plus 4 right p'ren left p'ren x plus 4 right p'reny is gelyk aan left p'ren x plus 4 right p'ren left p'ren x plus 4 right p'ren
42y=(4x3)(x+1)y equals left p'ren 4 x minus 3 right p'ren left p'ren x plus 1 right p'reny is gelyk aan left p'ren 4 x minus 3 right p'ren left p'ren x plus 1 right p'ren
43y=20x4x2y equals 20 x minus 4 x squaredy is gelyk aan 20 x minus 4 x squared
44y=x2y equals x squaredy is gelyk aan x squared
45y=3x1y equals 3 Sup x minus 1y is gelyk aan 3 hoch x minus 1
46y=162(x+3)y equals 16 minus 2 left p'ren x plus 3 right p'reny is gelyk aan 16 minus 2 left p'ren x plus 3 right p'ren
47y=4x2x3y equals 4 x squared minus x minus 3y is gelyk aan 4 x squared minus x minus 3
48y=x+1xy equals x plus StartFrac 1 Over x EndFracy is gelyk aan x plus Anfang Bruch 1 durch x Ende Bruch
49y=4x(5x)y equals 4 x left p'ren 5 minus x right p'reny is gelyk aan 4 x left p'ren 5 minus x right p'ren
50y=2(x3)+6(1x)y equals 2 left p'ren x minus 3 right p'ren plus 6 left p'ren 1 minus x right p'reny is gelyk aan 2 left p'ren x minus 3 right p'ren plus 6 left p'ren 1 minus x right p'ren
510.25>516 0.25 greater than five sixteenths0,25 groter as vyf sestien des
5232(57)32 dot left p'ren 5 dot 7 right p'ren32 punt left p'ren 5 punt 7 right p'ren
53(12×12×π×2)+(2×12×π×5)left p'ren one half times one half times pi times 2 right p'ren plus left p'ren 2 times one half times pi times 5 right p'renleft p'ren een helfte maal een helfte maal pi maal 2 right p'ren plus left p'ren 2 maal een helfte maal pi maal 5 right p'ren
54liminfnEn=n1knEk,limsupnEn=n1knEk.liminf Underscript n right arrow infinity Endscripts upper E Sub n Base equals union Underscript n greater than or equals 1 Endscripts intersection Underscript k greater than or equals n Endscripts upper E Sub k Base comma limsup Underscript n right arrow infinity Endscripts upper E Sub n Base equals intersection Underscript n greater than or equals 1 Endscripts union Underscript k greater than or equals n Endscripts upper E Sub k Base periodliminf Unterschrift n regs pyltjie oneindigheid großes E Index n Grund is gelyk aan union Unterschrift n groter of gelyk aan 1 intersection Unterschrift k groter of gelyk aan n großes E Index k Grund komma limsup Unterschrift n regs pyltjie oneindigheid großes E Index n Grund is gelyk aan intersection Unterschrift n groter of gelyk aan 1 union Unterschrift k groter of gelyk aan n großes E Index k Grund punt
55(i)𝒮𝒜;(ii)ifE𝒜thenE𝒜;(iii)ifE1,E2𝒜thenE1E2𝒜.StartLayout 1st Row 1st Column left p'ren i right p'ren 2nd Column script upper S element of script upper A semicolon 2nd Row 1st Column left p'ren ii right p'ren 2nd Column if upper E element of script upper A then upper E overbar element of script upper A semicolon 3rd Row 1st Column left p'ren iii right p'ren 2nd Column if upper E 1 comma upper E 2 element of script upper A then upper E 1 union upper E 2 element of script upper A period EndLayoutStartLayout 1. Row 1. Column left p'ren i right p'ren 2. Column Schreibschrift großes S element van Schreibschrift großes A kommapunt 2. Row 1. Column left p'ren ii right p'ren 2. Column if großes E element van Schreibschrift großes A then großes E overbar element van Schreibschrift großes A kommapunt 3. Row 1. Column left p'ren iii right p'ren 2. Column if großes E 1 komma großes E 2 element van Schreibschrift großes A then großes E 1 eenheid großes E 2 element van Schreibschrift großes A punt EndLayout
56(A.1)IfAthen0P{A}1.(1)(A.2)P{𝒮}=1.(2)(A.3)If{En,n1}is a sequence ofdisjoint(3)StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column left p'ren normal upper A period 1 right p'ren upper I f upper A element of script upper F t h e n 0 less than or equals upper P left brace upper A right brace less than or equals 1 period 4th Column left p'ren 1 right p'ren 2nd Row 1st Column Blank 2nd Column Blank 3rd Column left p'ren normal upper A period 2 right p'ren upper P left brace script upper S right brace equals 1 period 4th Column left p'ren 2 right p'ren 3rd Row 1st Column Blank 2nd Column Blank 3rd Column left p'ren normal upper A period 3 right p'ren upper I f left brace upper E Sub n Base comma n greater than or equals 1 right brace element of script upper F is a sequence of disjoint 4th Column left p'ren 3 right p'ren EndLayoutStartLayout 1. Row 1. Column Blank 2. Column Blank 3. Column left p'ren normales großes A punt 1 right p'ren großes I f großes A element van Schreibschrift großes F t h e n 0 kleiner of gelyk aan großes P links krulhakkie großes A regs krulhakkie kleiner of gelyk aan 1 punt 4. Column left p'ren 1 right p'ren 2. Row 1. Column Blank 2. Column Blank 3. Column left p'ren normales großes A punt 2 right p'ren großes P links krulhakkie Schreibschrift großes S regs krulhakkie is gelyk aan 1 punt 4. Column left p'ren 2 right p'ren 3. Row 1. Column Blank 2. Column Blank 3. Column left p'ren normales großes A punt 3 right p'ren großes I f links krulhakkie großes E Index n Grund komma n groter of gelyk aan 1 regs krulhakkie element van Schreibschrift großes F is a sequence of disjoint 4. Column left p'ren 3 right p'ren EndLayout
57P{Bj|A}=P{Bj}P{A|Bj}jJP{Bj}P{A|Bj}.upper P left brace upper B Sub j Base vertical bar upper A right brace equals StartFrac upper P left brace upper B Sub j Base right brace upper P left brace upper A vertical bar upper B Sub j Base right brace Over sigma summation Underscript j prime element of upper J Endscripts upper P left brace upper B Sub j prime Base right brace upper P left brace upper A vertical bar upper B Sub j prime Base right brace EndFrac periodgroßes P links krulhakkie großes B Index j Grund afstreep großes A regs krulhakkie is gelyk aan Anfang Bruch großes P links krulhakkie großes B Index j Grund regs krulhakkie großes P links krulhakkie großes A afstreep großes B Index j Grund regs krulhakkie durch sigma summation Unterschrift j priem element van großes J großes P links krulhakkie großes B Index j priem Grund regs krulhakkie großes P links krulhakkie großes A afstreep großes B Index j priem Grund regs krulhakkie Ende Bruch punt
58μ1(B)=Bf(x)dμ2(x)mu 1 left p'ren upper B right p'ren equals integral Underscript upper B Endscripts f left p'ren x right p'ren d mu 2 left p'ren x right p'renmy 1 left p'ren großes B right p'ren is gelyk aan integraal Unterschrift großes B f left p'ren x right p'ren d my 2 left p'ren x right p'ren
59limnE{|XnX|}=E{limn|XnX|}=0.limit Underscript n right arrow infinity Endscripts upper E left brace StartAbsoluteValue upper X Sub n Base minus upper X EndAbsoluteValue right brace equals upper E left brace limit Underscript n right arrow infinity Endscripts StartAbsoluteValue upper X Sub n Base minus upper X EndAbsoluteValue right brace equals 0 periodlimit Unterschrift n regs pyltjie oneindigheid großes E links krulhakkie StartAbsoluteValue großes X Index n Grund minus großes X EndAbsoluteValue regs krulhakkie is gelyk aan großes E links krulhakkie limit Unterschrift n regs pyltjie oneindigheid StartAbsoluteValue großes X Index n Grund minus großes X EndAbsoluteValue regs krulhakkie is gelyk aan 0 punt
60Pμ,σ{Ylβ(Yn,Sn)}=Pμ,σ{(YYn)/(S·(1+1n)1/2)tβ[n1]}=β,(1)StartLayout 1st Row 1st Column upper P Sub mu comma sigma Base left brace upper Y greater than or equals l Sub beta Base left p'ren upper Y overbar Sub n Base comma upper S Sub n Base right p'ren right brace equals upper P Sub mu comma sigma Base left brace left p'ren upper Y minus upper Y overbar Sub n Base right p'ren divided by left p'ren upper S dot left p'ren 1 plus StartFrac 1 Over n EndFrac right p'ren Sup 1 divided by 2 Base right p'ren greater than or equals minus t Sub beta Base left brack n minus 1 right brack right brace equals beta comma 2nd Row 1st Column Blank 2nd Column left p'ren 1 right p'ren EndLayoutStartLayout 1. Row 1. Column großes P Index my komma sigma Grund links krulhakkie großes Y groter of gelyk aan l Index beta Grund left p'ren großes Y overbar Index n Grund komma großes S Index n Grund right p'ren regs krulhakkie is gelyk aan großes P Index my komma sigma Grund links krulhakkie left p'ren großes Y minus großes Y overbar Index n Grund right p'ren divided by left p'ren großes S dot left p'ren 1 plus Anfang Bruch 1 durch n Ende Bruch right p'ren hoch 1 divided by 2 Grund right p'ren groter of gelyk aan minus t Index beta Grund left brack n minus 1 right brack regs krulhakkie is gelyk aan beta komma 2. Row 1. Column Blank 2. Column left p'ren 1 right p'ren EndLayout
61L=(11110011).upper L equals Start 5 By 6 Matrix 1st Row 1st Column 1 2nd Column negative 1 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 2nd Row 1st Column Blank 2nd Column 1 3rd Column negative 1 4th Column Blank 5th Column 0 6th Column Blank 3rd Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 4th Row 1st Column Blank 2nd Column 0 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 5th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column 1 6th Column negative 1 EndMatrix periodgroßes L is gelyk aan Start 5 By 6 Matrix 1. Row 1. Column 1 2. Column negative 1 3. Column Blank 4. Column Blank 5. Column Blank 6. Column Blank 2. Row 1. Column Blank 2. Column 1 3. Column negative 1 4. Column Blank 5. Column 0 6. Column Blank 3. Row 1. Column Blank 2. Column Blank 3. Column Blank 4. Column Blank 5. Column Blank 6. Column Blank 4. Row 1. Column Blank 2. Column 0 3. Column Blank 4. Column Blank 5. Column Blank 6. Column Blank 5. Row 1. Column Blank 2. Column Blank 3. Column Blank 4. Column Blank 5. Column 1 6. Column negative 1 EndMatrix punt
62n[Yn(μ+zβσ)]/Sn~U+nz1β(χ2[n1]/(n1))1/2~t[n1;nz1β],StartRoot n EndRoot left brack upper Y overbar Sub n Base minus left p'ren mu plus z Sub beta Base sigma right p'ren right brack divided by upper S Sub n Base tilde StartFrac upper U plus StartRoot n EndRoot z Sub 1 minus beta Base Over left p'ren chi squared left brack n minus 1 right brack divided by left p'ren n minus 1 right p'ren right p'ren Sup 1 divided by 2 Base EndFrac tilde t left brack n minus 1 semicolon StartRoot n EndRoot z Sub 1 minus beta Base right brack commaAnfang Wurzel n Ende Wurzel left brack großes Y overbar Index n Grund minus left p'ren my plus z Index beta Grund sigma right p'ren right brack divided by großes S Index n Grund tilde Anfang Bruch großes U plus Anfang Wurzel n Ende Wurzel z Index 1 minus beta Grund durch left p'ren chi squared left brack n minus 1 right brack divided by left p'ren n minus 1 right p'ren right p'ren hoch 1 divided by 2 Grund Ende Bruch tilde t left brack n minus 1 kommapunt Anfang Wurzel n Ende Wurzel z Index 1 minus beta Grund right brack komma
63γ=P{Ep,q(X(r),X(s)}=n!(r1)!j=0 sr1(1)jpr+j(nrj)!j!I1q(ns+1,srj).StartLayout 1st Row 1st Column gamma 2nd Column equals upper P left brace upper E Sub p comma q Base subset of left p'ren upper X Sub left p'ren r right p'ren Base comma upper X Sub left p'ren s right p'ren Base right brace 2nd Row 1st Column Blank 2nd Column equals StartFrac n factorial Over left p'ren r minus 1 right p'ren factorial EndFrac sigma summation Underscript j equals 0 Overscript s minus r minus 1 Endscripts left p'ren negative 1 right p'ren Sup j Base StartFrac p Sup r plus j Base Over left p'ren n minus r minus j right p'ren factorial j factorial EndFrac upper I Sub 1 minus q Base left p'ren n minus s plus 1 comma s minus r minus j right p'ren period EndLayoutStartLayout 1. Row 1. Column gamma 2. Column is gelyk aan großes P links krulhakkie großes E Index p komma q Grund deelversameling van left p'ren großes X Index left p'ren r right p'ren Grund komma großes X Index left p'ren s right p'ren Grund regs krulhakkie 2. Row 1. Column Blank 2. Column is gelyk aan Anfang Bruch n factorial durch left p'ren r minus 1 right p'ren factorial Ende Bruch sigma summation Unterschrift j is gelyk aan 0 Überschrift s minus r minus 1 left p'ren negative 1 right p'ren hoch j Grund Anfang Bruch p hoch r plus j Grund durch left p'ren n minus r minus j right p'ren factorial j factorial Ende Bruch großes I Index 1 minus q Grund left p'ren n minus s plus 1 komma s minus r minus j right p'ren punt EndLayout
64Sitx=1/m0airitx+(i1)/mbi,upper S Sub i Base StartBinomialOrMatrix t Choose x EndBinomialOrMatrix equals Start 2 By 2 Matrix 1st Row 1st Column 1 divided by m 2nd Column 0 2nd Row 1st Column a Sub i Base 2nd Column r Sub i Base EndMatrix StartBinomialOrMatrix t Choose x EndBinomialOrMatrix plus StartBinomialOrMatrix left p'ren i minus 1 right p'ren divided by m Choose b Sub i Base EndBinomialOrMatrix commagroßes S Index i Grund StartBinomialOrMatrix t Choose x EndBinomialOrMatrix is gelyk aan Start 2 By 2 Matrix 1. Row 1. Column 1 divided by m 2. Column 0 2. Row 1. Column a Index i Grund 2. Column r Index i Grund EndMatrix StartBinomialOrMatrix t Choose x EndBinomialOrMatrix plus StartBinomialOrMatrix left p'ren i minus 1 right p'ren divided by m Choose b Index i Grund EndBinomialOrMatrix komma
65c1h42s12TTT(f(t+h)f(t))2dtc2h42sc 1 h Sup 4 minus 2 s Base less than or equals StartFrac 1 Over 2 upper T EndFrac integral Sub negative upper T Sup upper T Base left p'ren f left p'ren t plus h right p'ren minus f left p'ren t right p'ren right p'ren squared normal d t less than or equals c 2 h Sup 4 minus 2 sc 1 h hoch 4 minus 2 s Grund kleiner of gelyk aan Anfang Bruch 1 durch 2 großes T Ende Bruch integraal Sub negative großes T Sup großes T Base left p'ren f left p'ren t plus h right p'ren minus f left p'ren t right p'ren right p'ren squared normales d t kleiner of gelyk aan c 2 h hoch 4 minus 2 s
66C(0)C(h)ch42supper C left p'ren 0 right p'ren minus upper C left p'ren h right p'ren asymptotically equals c h Sup 4 minus 2 sgroßes C left p'ren 0 right p'ren minus großes C left p'ren h right p'ren assimtoties gelyk aan c h hoch 4 minus 2 s
67S(ω)=limT12TTTf(t)eitωdt2.upper S left p'ren omega right p'ren equals limit Underscript upper T right arrow infinity Endscripts StartFrac 1 Over 2 upper T EndFrac StartAbsoluteValue integral Sub negative upper T Sup upper T Base comma f comma left p'ren comma t comma right p'ren comma normal e Sup italic i t omega Base comma normal d comma t EndAbsoluteValue squared periodgroßes S left p'ren omega right p'ren is gelyk aan limit Unterschrift großes T regs pyltjie oneindigheid Anfang Bruch 1 durch 2 großes T Ende Bruch StartAbsoluteValue integraal Sub negative großes T Sup großes T Base komma f komma left p'ren komma t komma right p'ren komma normales e hoch kursives i t omega Grund komma normales d komma t EndAbsoluteValue squared punt
680101[|f(t)f(u)|2+|tu|2]s/2dtdu<integral Sub 0 Sup 1 Base integral Sub 0 Sup 1 Base left brack StartAbsoluteValue f left p'ren t right p'ren minus f left p'ren u right p'ren EndAbsoluteValue squared plus StartAbsoluteValue t minus u EndAbsoluteValue squared right brack Sup negative s divided by 2 Base normal d t normal d u less than infinityintegraal Sub 0 Sup 1 Base integraal Sub 0 Sup 1 Base left brack StartAbsoluteValue f left p'ren t right p'ren minus f left p'ren u right p'ren EndAbsoluteValue squared plus StartAbsoluteValue t minus u EndAbsoluteValue squared right brack hoch negative s divided by 2 Grund normales d t normales d u kleiner as oneindigheid
69EIEk+1|I|s=EIEk|I|sE(R1s+R2s).sans serif upper E left p'ren sigma summation Underscript upper I element of upper E Sub k plus 1 Base Endscripts StartAbsoluteValue upper I EndAbsoluteValue Sup s Base right p'ren equals sans serif upper E left p'ren sigma summation Underscript upper I element of upper E Sub k Base Endscripts StartAbsoluteValue upper I EndAbsoluteValue Sup s Base right p'ren sans serif upper E left p'ren upper R 1 Sup s Base plus upper R 2 Sup s Base right p'ren periodserifenloses großes E left p'ren sigma summation Unterschrift großes I element van großes E Index k plus 1 Grund StartAbsoluteValue großes I EndAbsoluteValue hoch s Grund right p'ren is gelyk aan serifenloses großes E left p'ren sigma summation Unterschrift großes I element van großes E Index k Grund StartAbsoluteValue großes I EndAbsoluteValue hoch s Grund right p'ren serifenloses großes E left p'ren großes R 1 hoch s Grund plus großes R 2 hoch s Grund right p'ren punt
70(x1,y1)left p'ren x 1 comma y 1 right p'renleft p'ren x 1 komma y 1 right p'ren
71(x2,y2)left p'ren x 2 comma y 2 right p'renleft p'ren x 2 komma y 2 right p'ren
72d=(x2x1)2+(y2y1)2d equals StartRoot left p'ren x 2 minus x 1 right p'ren squared plus left p'ren y 2 minus y 1 right p'ren squared EndRootd is gelyk aan Anfang Wurzel left p'ren x 2 minus x 1 right p'ren squared plus left p'ren y 2 minus y 1 right p'ren squared Ende Wurzel
73double struck upper Rmit Doppelstrich großes R
74=(,)double struck upper R equals left p'ren negative infinity comma infinity right p'renmit Doppelstrich großes R is gelyk aan left p'ren negative oneindigheid komma oneindigheid right p'ren
75{ 1,2,3 }StartSet 1 comma 2 comma 3 EndSetStartSet 1 komma 2 komma 3 EndSet
761S1 element of upper S1 element van großes S
773S3 element of upper S3 element van großes S
784S4 not an element of upper S4 nie 'n element van großes S
79a=3x1+(1+x)2a equals StartRoot 3 x minus 1 EndRoot plus left p'ren 1 plus x right p'ren squareda is gelyk aan Anfang Wurzel 3 x minus 1 Ende Wurzel plus left p'ren 1 plus x right p'ren squared
80a=(b+c)2d+(e+f)2ga equals StartFrac left p'ren b plus c right p'ren squared Over d EndFrac plus StartFrac left p'ren e plus f right p'ren squared Over g EndFraca is gelyk aan Anfang Bruch left p'ren b plus c right p'ren squared durch d Ende Bruch plus Anfang Bruch left p'ren e plus f right p'ren squared durch g Ende Bruch
81x=[ (a+b)2(cb)2 ]+[ (d+e)2(fe)2 ]x equals left brack left p'ren a plus b right p'ren squared left p'ren c minus b right p'ren squared right brack plus left brack left p'ren d plus e right p'ren squared left p'ren f minus e right p'ren squared right brackx is gelyk aan left brack left p'ren a plus b right p'ren squared left p'ren c minus b right p'ren squared right brack plus left brack left p'ren d plus e right p'ren squared left p'ren f minus e right p'ren squared right brack
82x=[ (a+b)2 ]+[ (fe)2 ]x equals left brack left p'ren a plus b right p'ren squared right brack plus left brack left p'ren f minus e right p'ren squared right brackx is gelyk aan left brack left p'ren a plus b right p'ren squared right brack plus left brack left p'ren f minus e right p'ren squared right brack
83x=[ (a+b)2 ]x equals left brack left p'ren a plus b right p'ren squared right brackx is gelyk aan left brack left p'ren a plus b right p'ren squared right brack
84x=(a+b)2x equals left p'ren a plus b right p'ren squaredx is gelyk aan left p'ren a plus b right p'ren squared
85x=a+b2x equals a plus b squaredx is gelyk aan a plus b squared
861234=23StartFrac one half Over three fourths EndFrac equals two thirdsAnfang Bruch een helfte durch drie kwarts Ende Bruch is gelyk aan twee derdes
872((x+1)(x+3)4((x1)(x+2)3))=y2 left p'ren left p'ren x plus 1 right p'ren left p'ren x plus 3 right p'ren minus 4 left p'ren left p'ren x minus 1 right p'ren left p'ren x plus 2 right p'ren minus 3 right p'ren right p'ren equals y2 left p'ren left p'ren x plus 1 right p'ren left p'ren x plus 3 right p'ren minus 4 left p'ren left p'ren x minus 1 right p'ren left p'ren x plus 2 right p'ren minus 3 right p'ren right p'ren is gelyk aan y
88cosx=1x22!+x44!cosine x equals 1 minus StartFrac x squared Over 2 factorial EndFrac plus StartFrac x Sup 4 Base Over 4 factorial EndFrac minus ellipsiskosinus x is gelyk aan 1 minus Anfang Bruch x squared durch 2 factorial Ende Bruch plus Anfang Bruch x hoch 4 Grund durch 4 factorial Ende Bruch minus ellipsis
89x=b±b24ac2ax equals StartFrac negative b plus or minus StartRoot b squared minus 4 a c EndRoot Over 2 a EndFracx is gelyk aan Anfang Bruch negative b plus of minus Anfang Wurzel b squared minus 4 a c Ende Wurzel durch 2 a Ende Bruch
90x+y2k+1x plus y Sup StartFrac 2 Over k plus 1 EndFracx plus y hoch Anfang Bruch 2 durch k plus 1 Ende Bruch
91limx0sinxx=1limit Underscript x right arrow 0 Endscripts StartFrac sine x Over x EndFrac equals 1limit Unterschrift x regs pyltjie 0 Anfang Bruch sinus x durch x Ende Bruch is gelyk aan 1
92d=(x2x1)2+(y2y1)2d equals StartRoot left p'ren x 2 minus x 1 right p'ren squared plus left p'ren y 2 minus y 1 right p'ren squared EndRootd is gelyk aan Anfang Wurzel left p'ren x 2 minus x 1 right p'ren squared plus left p'ren y 2 minus y 1 right p'ren squared Ende Wurzel
93Fn=Fn1+Fn2upper F Sub n Base equals upper F Sub n minus 1 Base plus upper F Sub n minus 2großes F Index n Grund is gelyk aan großes F Index n minus 1 Grund plus großes F Index n minus 2
94Π=(π11π12π12000π12π11π12000π12π12π11000000π44000000π44000000π44)bold upper Pi equals Start 6 By 6 Matrix 1st Row 1st Column pi 11 2nd Column pi 12 3rd Column pi 12 4th Column 0 5th Column 0 6th Column 0 2nd Row 1st Column pi 12 2nd Column pi 11 3rd Column pi 12 4th Column 0 5th Column 0 6th Column 0 3rd Row 1st Column pi 12 2nd Column pi 12 3rd Column pi 11 4th Column 0 5th Column 0 6th Column 0 4th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column pi 44 5th Column 0 6th Column 0 5th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column pi 44 6th Column 0 6th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 6th Column pi 44 EndMatrixfettes großes Pi is gelyk aan Start 6 By 6 Matrix 1. Row 1. Column pi 11 2. Column pi 12 3. Column pi 12 4. Column 0 5. Column 0 6. Column 0 2. Row 1. Column pi 12 2. Column pi 11 3. Column pi 12 4. Column 0 5. Column 0 6. Column 0 3. Row 1. Column pi 12 2. Column pi 12 3. Column pi 11 4. Column 0 5. Column 0 6. Column 0 4. Row 1. Column 0 2. Column 0 3. Column 0 4. Column pi 44 5. Column 0 6. Column 0 5. Row 1. Column 0 2. Column 0 3. Column 0 4. Column 0 5. Column pi 44 6. Column 0 6. Row 1. Column 0 2. Column 0 3. Column 0 4. Column 0 5. Column 0 6. Column pi 44 EndMatrix
95s11=c11+c12c11c12c11+2c12s 11 equals StartFrac c 11 plus c 12 Over left p'ren c 11 minus c 12 right p'ren left p'ren c 11 plus 2 c 12 right p'ren EndFracs 11 is gelyk aan Anfang Bruch c 11 plus c 12 durch left p'ren c 11 minus c 12 right p'ren left p'ren c 11 plus 2 c 12 right p'ren Ende Bruch
96SiO2+ 6HF H2 SiF6+ 2H2O upper S i normal upper O 2 plus 6 normal upper H normal upper F right arrow normal upper H 2 upper S i normal upper F 6 plus 2 normal upper H 2 normal upper Ogroßes S i normales großes O 2 plus 6 normales großes H normales großes F regs pyltjie normales großes H 2 großes S i normales großes F 6 plus 2 normales großes H 2 normales großes O
97ddx(E(x)A(x)dw(x)dx)+p(x)=0StartFrac d Over d x EndFrac left p'ren upper E left p'ren x right p'ren upper A left p'ren x right p'ren StartFrac d w left p'ren x right p'ren Over d x EndFrac right p'ren plus p left p'ren x right p'ren equals 0Anfang Bruch d durch d x Ende Bruch left p'ren großes E left p'ren x right p'ren großes A left p'ren x right p'ren Anfang Bruch d w left p'ren x right p'ren durch d x Ende Bruch right p'ren plus p left p'ren x right p'ren is gelyk aan 0
98TCSgas=12PsealPmax1TsealTCS Sub gas Base equals minus one half left p'ren StartFrac upper P Sub seal Base Over upper P Sub max Base EndFrac right p'ren left p'ren StartFrac 1 Over upper T Sub seal Base EndFrac right p'renTCS Index gas Grund is gelyk aan minus een helfte left p'ren Anfang Bruch großes P Index seal Grund durch großes P Index max Grund Ende Bruch right p'ren left p'ren Anfang Bruch 1 durch großes T Index seal Grund Ende Bruch right p'ren
99Bp=7v231+c2a2+c4a4+3v2c21+va21v1c4a41c2a2upper B Sub p Base equals StartStartFrac StartFrac 7 minus v squared Over 3 EndFrac left p'ren 1 plus StartFrac c squared Over a squared EndFrac plus StartFrac c Sup 4 Base Over a Sup 4 Base EndFrac right p'ren plus StartFrac left p'ren 3 minus v right p'ren squared c squared Over left p'ren 1 plus v right p'ren a squared EndFrac OverOver left p'ren 1 minus v right p'ren left p'ren 1 minus StartFrac c Sup 4 Base Over a Sup 4 Base EndFrac right p'ren left p'ren 1 minus StartFrac c squared Over a squared EndFrac right p'ren EndEndFracgroßes B Index p Grund is gelyk aan Anfang Anfang Bruch Anfang Bruch 7 minus v squared durch 3 Ende Bruch left p'ren 1 plus Anfang Bruch c squared durch a squared Ende Bruch plus Anfang Bruch c hoch 4 Grund durch a hoch 4 Grund Ende Bruch right p'ren plus Anfang Bruch left p'ren 3 minus v right p'ren squared c squared durch left p'ren 1 plus v right p'ren a squared Ende Bruch durch durch left p'ren 1 minus v right p'ren left p'ren 1 minus Anfang Bruch c hoch 4 Grund durch a hoch 4 Grund Ende Bruch right p'ren left p'ren 1 minus Anfang Bruch c squared durch a squared Ende Bruch right p'ren Ende Ende Bruch
100Qtankseries=1RsLsCsupper Q Sub tank Sup series Base equals StartFrac 1 Over upper R Sub s Base EndFrac StartRoot StartFrac upper L Sub s Base Over upper C Sub s Base EndFrac EndRootgroßes Q Index tank hoch series Grund is gelyk aan Anfang Bruch 1 durch großes R Index s Grund Ende Bruch Anfang Wurzel Anfang Bruch großes L Index s Grund durch großes C Index s Grund Ende Bruch Ende Wurzel
101Δϕpeak=tan1(k2Qtankseries)upper Delta phi Sub peak Base equals tangent Sup negative 1 Base left p'ren k squared upper Q Sub tank Sup series Base right p'rengroßes Delta phi Index peak Grund is gelyk aan tangens hoch negative 1 Grund left p'ren k squared großes Q Index tank hoch series Grund right p'ren
102f=1.013WL2Eρ(1+0.293L2EW2σ)f equals 1.013 StartFrac upper W Over upper L squared EndFrac StartRoot StartFrac upper E Over rho EndFrac EndRoot StartRoot left p'ren 1 plus 0.293 StartFrac upper L squared Over EW squared EndFrac sigma right p'ren EndRootf is gelyk aan 1.013 Anfang Bruch großes W durch großes L squared Ende Bruch Anfang Wurzel Anfang Bruch großes E durch rho Ende Bruch Ende Wurzel Anfang Wurzel left p'ren 1 plus 0.293 Anfang Bruch großes L squared durch EW squared Ende Bruch sigma right p'ren Ende Wurzel
103unx=γncoshknxcosknx+sinhknxsinknxu Sub n Base left p'ren x right p'ren equals gamma Sub n Base left p'ren hyperbolic cosine k Sub n Base x minus cosine k Sub n Base x right p'ren plus left p'ren hyperbolic sine k Sub n Base x minus sine k Sub n Base x right p'renu Index n Grund left p'ren x right p'ren is gelyk aan gamma Index n Grund left p'ren hiperboliese kosinus k Index n Grund x minus kosinus k Index n Grund x right p'ren plus left p'ren hiperboliese sinus k Index n Grund x minus sinus k Index n Grund x right p'ren
104B=F0m(ω02ω2)2+4n2ω2=F0k(1(ω/ω02)2)2+4(n/ω0)2(ω/ω0)2StartLayout 1st Row 1st Column upper B 2nd Column equals StartStartFrac StartFrac upper F 0 Over m EndFrac OverOver StartRoot left p'ren omega 0 squared minus omega squared right p'ren squared plus 4 n squared omega squared EndRoot EndEndFrac 2nd Row 1st Column Blank 2nd Column equals StartStartFrac StartFrac upper F 0 Over k EndFrac OverOver StartRoot left p'ren 1 minus left p'ren omega divided by omega 0 squared right p'ren squared right p'ren squared plus 4 left p'ren n divided by omega 0 right p'ren squared left p'ren omega divided by omega 0 right p'ren squared EndRoot EndEndFrac EndLayoutStartLayout 1. Row 1. Column großes B 2. Column is gelyk aan Anfang Anfang Bruch Anfang Bruch großes F 0 durch m Ende Bruch durch durch Anfang Wurzel left p'ren omega 0 squared minus omega squared right p'ren squared plus 4 n squared omega squared Ende Wurzel Ende Ende Bruch 2. Row 1. Column Blank 2. Column is gelyk aan Anfang Anfang Bruch Anfang Bruch großes F 0 durch k Ende Bruch durch durch Anfang Wurzel left p'ren 1 minus left p'ren omega divided by omega 0 squared right p'ren squared right p'ren squared plus 4 left p'ren n divided by omega 0 right p'ren squared left p'ren omega divided by omega 0 right p'ren squared Ende Wurzel Ende Ende Bruch EndLayout
105p(AandB)=p(A)p(B|A)normal p left p'ren upper A a n d upper B right p'ren equals normal p left p'ren upper A right p'ren normal p left p'ren upper B vertical bar upper A right p'rennormales p left p'ren großes A a n d großes B right p'ren is gelyk aan normales p left p'ren großes A right p'ren normales p left p'ren großes B afstreep großes A right p'ren
106PMF(x)1xαupper P upper M upper F left p'ren x right p'ren proportional to left p'ren StartFrac 1 Over x EndFrac right p'ren Sup alphagroßes P großes M großes F left p'ren x right p'ren eweredig aan left p'ren Anfang Bruch 1 durch x Ende Bruch right p'ren hoch alfa
107f(x)=12πexp(-x2/2)f left p'ren x right p'ren equals StartFrac 1 Over StartRoot 2 pi EndRoot EndFrac exp left p'ren minus x squared slash 2 right p'renf left p'ren x right p'ren is gelyk aan Anfang Bruch 1 durch Anfang Wurzel 2 pi Ende Wurzel Ende Bruch exp left p'ren minus x squared skuinsstreep 2 right p'ren
108dxdθ=βcos2θStartFrac d x Over d theta EndFrac equals StartFrac beta Over cosine squared theta EndFracAnfang Bruch d x durch d theta Ende Bruch is gelyk aan Anfang Bruch beta durch kosinus squared theta Ende Bruch
109s/2(n-1)s divided by StartRoot 2 left p'ren n minus 1 right p'ren EndRoots divided by Anfang Wurzel 2 left p'ren n minus 1 right p'ren Ende Wurzel

Afrikaans Mathspeak Steve Noble's samples. Locale: af, Style: Superbrief.

0515623=negative 5 and one fifth minus 6 and two thirds equalsnegative 5 and een vyf de minus 6 and twee derdes is gelyk aan
1734(478)=negative 7 and three fourths minus L p'ren negative 4 and seven eighths R p'ren equalsnegative 7 and drie kwarts minus L p'ren negative 4 and sewe agstes R p'ren is gelyk aan
224.15(13.7)=negative 24.15 minus L p'ren 13.7 R p'ren equalsnegative 24,15 minus L p'ren 13,7 R p'ren is gelyk aan
3(4)×3=12L p'ren negative 4 R p'ren times 3 equals negative 12L p'ren negative 4 R p'ren maal 3 is gelyk aan negative 12
412÷3=4negative 12 divided by 3 equals negative 4negative 12 divided by 3 is gelyk aan negative 4
512÷(4)=3negative 12 divided by L p'ren negative 4 R p'ren equals 3negative 12 divided by L p'ren negative 4 R p'ren is gelyk aan 3
66×56 times 56 maal 5
76×(5)6 times L p'ren negative 5 R p'ren6 maal L p'ren negative 5 R p'ren
86×5negative 6 times 5negative 6 maal 5
96×(5)negative 6 times L p'ren negative 5 R p'rennegative 6 maal L p'ren negative 5 R p'ren
108×7negative 8 times 7negative 8 maal 7
118×(7)negative 8 times L p'ren negative 7 R p'rennegative 8 maal L p'ren negative 7 R p'ren
128×(7)8 times L p'ren negative 7 R p'ren8 maal L p'ren negative 7 R p'ren
138×78 times 78 maal 7
14m1=30°m angle 1 equals 30 degreem hoek 1 is gelyk aan 30 graad
15m2=60° m angle 2 equals 60 degreem hoek 2 is gelyk aan 60 graad
16m1+m2=90° m angle 1 plus m angle 2 equals 90 degreem hoek 1 plus m hoek 2 is gelyk aan 90 graad
17mM+mN=180° m angle upper M plus m angle upper N equals 180 degreem hoek großes M plus m hoek großes N is gelyk aan 180 graad
18A=12bhupper A equals one half b hgroßes A is gelyk aan een helfte b h
19area of trianglearea of square=1 unit216 units2Frac area of triangle Over area of square EndFrac equals Frac 1 unit squared Over 16 units squared EndFracBruch area of triangle durch area of square Ende Bruch is gelyk aan Bruch 1 unit squared durch 16 units squared Ende Bruch
200.620.6 squared0,6 squared
211.52 1.5 squared1,5 squared
224(2x+3x)4 L p'ren 2 x plus 3 x R p'ren4 L p'ren 2 x plus 3 x R p'ren
2336+4y1y2+5y2236 plus 4 y minus 1 y squared plus 5 y squared minus 236 plus 4 y minus 1 y squared plus 5 y squared minus 2
24(5+9)4+3=L p'ren 5 plus 9 R p'ren minus 4 plus 3 equalsL p'ren 5 plus 9 R p'ren minus 4 plus 3 is gelyk aan
25BCModAbove upper B upper C With L R arrowModAbove großes B großes C With L R arrow
26PQModAbove upper P upper Q With R arrowModAbove großes P großes Q With R arrow
27GH¯ModAbove upper G upper H With barModAbove großes G großes H With makron
28WX¯YZ¯ModAbove upper W upper X With bar approximately equals ModAbove upper Y upper Z With barModAbove großes W großes X With makron ongeveer gelyk aan ModAbove großes Y großes Z With makron
29BEFangle upper B upper E upper Fhoek großes B großes E großes F
30BEDangle upper B upper E upper Dhoek großes B großes E großes D
31DEFangle upper D upper E upper Fhoek großes D großes E großes F
32x=b±b24ac2ax equals Frac negative b plus or minus Root b squared minus 4 a c EndRoot Over 2 a EndFracx is gelyk aan Bruch negative b plus of minus Wurzel b squared minus 4 a c Ende Wurzel durch 2 a Ende Bruch
33y=x2+8x+16y equals x squared plus 8 x plus 16y is gelyk aan x squared plus 8 x plus 16
34y=13(3x)y equals one third L p'ren 3 Sup x Base R p'reny is gelyk aan een derde L p'ren 3 hoch x Grund R p'ren
35y=102xy equals 10 minus 2 xy is gelyk aan 10 minus 2 x
36y=2x3+5y equals 2 x cubed plus 5y is gelyk aan 2 x cubed plus 5
37y=(x2+1)(x2+3)y equals L p'ren x squared plus 1 R p'ren L p'ren x squared plus 3 R p'reny is gelyk aan L p'ren x squared plus 1 R p'ren L p'ren x squared plus 3 R p'ren
38y=0.5xy equals 0.5 Sup xy is gelyk aan 0,5 hoch x
39y=222xy equals 22 minus 2 xy is gelyk aan 22 minus 2 x
40y=3xy equals Frac 3 Over x EndFracy is gelyk aan Bruch 3 durch x Ende Bruch
41y=(x+4)(x+4)y equals L p'ren x plus 4 R p'ren L p'ren x plus 4 R p'reny is gelyk aan L p'ren x plus 4 R p'ren L p'ren x plus 4 R p'ren
42y=(4x3)(x+1)y equals L p'ren 4 x minus 3 R p'ren L p'ren x plus 1 R p'reny is gelyk aan L p'ren 4 x minus 3 R p'ren L p'ren x plus 1 R p'ren
43y=20x4x2y equals 20 x minus 4 x squaredy is gelyk aan 20 x minus 4 x squared
44y=x2y equals x squaredy is gelyk aan x squared
45y=3x1y equals 3 Sup x minus 1y is gelyk aan 3 hoch x minus 1
46y=162(x+3)y equals 16 minus 2 L p'ren x plus 3 R p'reny is gelyk aan 16 minus 2 L p'ren x plus 3 R p'ren
47y=4x2x3y equals 4 x squared minus x minus 3y is gelyk aan 4 x squared minus x minus 3
48y=x+1xy equals x plus Frac 1 Over x EndFracy is gelyk aan x plus Bruch 1 durch x Ende Bruch
49y=4x(5x)y equals 4 x L p'ren 5 minus x R p'reny is gelyk aan 4 x L p'ren 5 minus x R p'ren
50y=2(x3)+6(1x)y equals 2 L p'ren x minus 3 R p'ren plus 6 L p'ren 1 minus x R p'reny is gelyk aan 2 L p'ren x minus 3 R p'ren plus 6 L p'ren 1 minus x R p'ren
510.25>516 0.25 greater than five sixteenths0,25 groter as vyf sestien des
5232(57)32 dot L p'ren 5 dot 7 R p'ren32 punt L p'ren 5 punt 7 R p'ren
53(12×12×π×2)+(2×12×π×5)L p'ren one half times one half times pi times 2 R p'ren plus L p'ren 2 times one half times pi times 5 R p'renL p'ren een helfte maal een helfte maal pi maal 2 R p'ren plus L p'ren 2 maal een helfte maal pi maal 5 R p'ren
54liminfnEn=n1knEk,limsupnEn=n1knEk.liminf Underscript n R arrow infinity Endscripts upper E Sub n Base equals union Underscript n greater than or equals 1 Endscripts intersection Underscript k greater than or equals n Endscripts upper E Sub k Base comma limsup Underscript n R arrow infinity Endscripts upper E Sub n Base equals intersection Underscript n greater than or equals 1 Endscripts union Underscript k greater than or equals n Endscripts upper E Sub k Base periodliminf Unterschrift n R arrow oneindigheid großes E Index n Grund is gelyk aan union Unterschrift n groter of gelyk aan 1 intersection Unterschrift k groter of gelyk aan n großes E Index k Grund komma limsup Unterschrift n R arrow oneindigheid großes E Index n Grund is gelyk aan intersection Unterschrift n groter of gelyk aan 1 union Unterschrift k groter of gelyk aan n großes E Index k Grund punt
55(i)𝒮𝒜;(ii)ifE𝒜thenE𝒜;(iii)ifE1,E2𝒜thenE1E2𝒜.Layout 1st Row 1st Column L p'ren i R p'ren 2nd Column script upper S element of script upper A semicolon 2nd Row 1st Column L p'ren ii R p'ren 2nd Column if upper E element of script upper A then upper E overbar element of script upper A semicolon 3rd Row 1st Column L p'ren iii R p'ren 2nd Column if upper E 1 comma upper E 2 element of script upper A then upper E 1 union upper E 2 element of script upper A period EndLayoutLayout 1. Row 1. Column L p'ren i R p'ren 2. Column Schreibschrift großes S element van Schreibschrift großes A kommapunt 2. Row 1. Column L p'ren ii R p'ren 2. Column if großes E element van Schreibschrift großes A then großes E overbar element van Schreibschrift großes A kommapunt 3. Row 1. Column L p'ren iii R p'ren 2. Column if großes E 1 komma großes E 2 element van Schreibschrift großes A then großes E 1 eenheid großes E 2 element van Schreibschrift großes A punt EndLayout
56(A.1)IfAthen0P{A}1.(1)(A.2)P{𝒮}=1.(2)(A.3)If{En,n1}is a sequence ofdisjoint(3)Layout 1st Row 1st Column Blank 2nd Column Blank 3rd Column L p'ren normal upper A period 1 R p'ren upper I f upper A element of script upper F t h e n 0 less than or equals upper P L brace upper A R brace less than or equals 1 period 4th Column L p'ren 1 R p'ren 2nd Row 1st Column Blank 2nd Column Blank 3rd Column L p'ren normal upper A period 2 R p'ren upper P L brace script upper S R brace equals 1 period 4th Column L p'ren 2 R p'ren 3rd Row 1st Column Blank 2nd Column Blank 3rd Column L p'ren normal upper A period 3 R p'ren upper I f L brace upper E Sub n Base comma n greater than or equals 1 R brace element of script upper F is a sequence of disjoint 4th Column L p'ren 3 R p'ren EndLayoutLayout 1. Row 1. Column Blank 2. Column Blank 3. Column L p'ren normales großes A punt 1 R p'ren großes I f großes A element van Schreibschrift großes F t h e n 0 kleiner of gelyk aan großes P L brace großes A R brace kleiner of gelyk aan 1 punt 4. Column L p'ren 1 R p'ren 2. Row 1. Column Blank 2. Column Blank 3. Column L p'ren normales großes A punt 2 R p'ren großes P L brace Schreibschrift großes S R brace is gelyk aan 1 punt 4. Column L p'ren 2 R p'ren 3. Row 1. Column Blank 2. Column Blank 3. Column L p'ren normales großes A punt 3 R p'ren großes I f L brace großes E Index n Grund komma n groter of gelyk aan 1 R brace element van Schreibschrift großes F is a sequence of disjoint 4. Column L p'ren 3 R p'ren EndLayout
57P{Bj|A}=P{Bj}P{A|Bj}jJP{Bj}P{A|Bj}.upper P L brace upper B Sub j Base vertical bar upper A R brace equals Frac upper P L brace upper B Sub j Base R brace upper P L brace upper A vertical bar upper B Sub j Base R brace Over sigma summation Underscript j prime element of upper J Endscripts upper P L brace upper B Sub j prime Base R brace upper P L brace upper A vertical bar upper B Sub j prime Base R brace EndFrac periodgroßes P L brace großes B Index j Grund afstreep großes A R brace is gelyk aan Bruch großes P L brace großes B Index j Grund R brace großes P L brace großes A afstreep großes B Index j Grund R brace durch sigma summation Unterschrift j priem element van großes J großes P L brace großes B Index j priem Grund R brace großes P L brace großes A afstreep großes B Index j priem Grund R brace Ende Bruch punt
58μ1(B)=Bf(x)dμ2(x)mu 1 L p'ren upper B R p'ren equals integral Underscript upper B Endscripts f L p'ren x R p'ren d mu 2 L p'ren x R p'renmy 1 L p'ren großes B R p'ren is gelyk aan integraal Unterschrift großes B f L p'ren x R p'ren d my 2 L p'ren x R p'ren
59limnE{|XnX|}=E{limn|XnX|}=0.limit Underscript n R arrow infinity Endscripts upper E L brace AbsoluteValue upper X Sub n Base minus upper X EndAbsoluteValue R brace equals upper E L brace limit Underscript n R arrow infinity Endscripts AbsoluteValue upper X Sub n Base minus upper X EndAbsoluteValue R brace equals 0 periodlimit Unterschrift n R arrow oneindigheid großes E L brace AbsoluteValue großes X Index n Grund minus großes X EndAbsoluteValue R brace is gelyk aan großes E L brace limit Unterschrift n R arrow oneindigheid AbsoluteValue großes X Index n Grund minus großes X EndAbsoluteValue R brace is gelyk aan 0 punt
60Pμ,σ{Ylβ(Yn,Sn)}=Pμ,σ{(YYn)/(S·(1+1n)1/2)tβ[n1]}=β,(1)Layout 1st Row 1st Column upper P Sub mu comma sigma Base L brace upper Y greater than or equals l Sub beta Base L p'ren upper Y overbar Sub n Base comma upper S Sub n Base R p'ren R brace equals upper P Sub mu comma sigma Base L brace L p'ren upper Y minus upper Y overbar Sub n Base R p'ren divided by L p'ren upper S dot L p'ren 1 plus Frac 1 Over n EndFrac R p'ren Sup 1 divided by 2 Base R p'ren greater than or equals minus t Sub beta Base L brack n minus 1 R brack R brace equals beta comma 2nd Row 1st Column Blank 2nd Column L p'ren 1 R p'ren EndLayoutLayout 1. Row 1. Column großes P Index my komma sigma Grund L brace großes Y groter of gelyk aan l Index beta Grund L p'ren großes Y overbar Index n Grund komma großes S Index n Grund R p'ren R brace is gelyk aan großes P Index my komma sigma Grund L brace L p'ren großes Y minus großes Y overbar Index n Grund R p'ren divided by L p'ren großes S dot L p'ren 1 plus Bruch 1 durch n Ende Bruch R p'ren hoch 1 divided by 2 Grund R p'ren groter of gelyk aan minus t Index beta Grund L brack n minus 1 R brack R brace is gelyk aan beta komma 2. Row 1. Column Blank 2. Column L p'ren 1 R p'ren EndLayout
61L=(11110011).upper L equals 5 By 6 Matrix 1st Row 1st Column 1 2nd Column negative 1 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 2nd Row 1st Column Blank 2nd Column 1 3rd Column negative 1 4th Column Blank 5th Column 0 6th Column Blank 3rd Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 4th Row 1st Column Blank 2nd Column 0 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 5th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column 1 6th Column negative 1 EndMatrix periodgroßes L is gelyk aan 5 By 6 Matrix 1. Row 1. Column 1 2. Column negative 1 3. Column Blank 4. Column Blank 5. Column Blank 6. Column Blank 2. Row 1. Column Blank 2. Column 1 3. Column negative 1 4. Column Blank 5. Column 0 6. Column Blank 3. Row 1. Column Blank 2. Column Blank 3. Column Blank 4. Column Blank 5. Column Blank 6. Column Blank 4. Row 1. Column Blank 2. Column 0 3. Column Blank 4. Column Blank 5. Column Blank 6. Column Blank 5. Row 1. Column Blank 2. Column Blank 3. Column Blank 4. Column Blank 5. Column 1 6. Column negative 1 EndMatrix punt
62n[Yn(μ+zβσ)]/Sn~U+nz1β(χ2[n1]/(n1))1/2~t[n1;nz1β],Root n EndRoot L brack upper Y overbar Sub n Base minus L p'ren mu plus z Sub beta Base sigma R p'ren R brack divided by upper S Sub n Base tilde Frac upper U plus Root n EndRoot z Sub 1 minus beta Base Over L p'ren chi squared L brack n minus 1 R brack divided by L p'ren n minus 1 R p'ren R p'ren Sup 1 divided by 2 Base EndFrac tilde t L brack n minus 1 semicolon Root n EndRoot z Sub 1 minus beta Base R brack commaWurzel n Ende Wurzel L brack großes Y overbar Index n Grund minus L p'ren my plus z Index beta Grund sigma R p'ren R brack divided by großes S Index n Grund tilde Bruch großes U plus Wurzel n Ende Wurzel z Index 1 minus beta Grund durch L p'ren chi squared L brack n minus 1 R brack divided by L p'ren n minus 1 R p'ren R p'ren hoch 1 divided by 2 Grund Ende Bruch tilde t L brack n minus 1 kommapunt Wurzel n Ende Wurzel z Index 1 minus beta Grund R brack komma
63γ=P{Ep,q(X(r),X(s)}=n!(r1)!j=0 sr1(1)jpr+j(nrj)!j!I1q(ns+1,srj).Layout 1st Row 1st Column gamma 2nd Column equals upper P L brace upper E Sub p comma q Base subset of L p'ren upper X Sub L p'ren r R p'ren Base comma upper X Sub L p'ren s R p'ren Base R brace 2nd Row 1st Column Blank 2nd Column equals Frac n factorial Over L p'ren r minus 1 R p'ren factorial EndFrac sigma summation Underscript j equals 0 Overscript s minus r minus 1 Endscripts L p'ren negative 1 R p'ren Sup j Base Frac p Sup r plus j Base Over L p'ren n minus r minus j R p'ren factorial j factorial EndFrac upper I Sub 1 minus q Base L p'ren n minus s plus 1 comma s minus r minus j R p'ren period EndLayoutLayout 1. Row 1. Column gamma 2. Column is gelyk aan großes P L brace großes E Index p komma q Grund deelversameling van L p'ren großes X Index L p'ren r R p'ren Grund komma großes X Index L p'ren s R p'ren Grund R brace 2. Row 1. Column Blank 2. Column is gelyk aan Bruch n factorial durch L p'ren r minus 1 R p'ren factorial Ende Bruch sigma summation Unterschrift j is gelyk aan 0 Überschrift s minus r minus 1 L p'ren negative 1 R p'ren hoch j Grund Bruch p hoch r plus j Grund durch L p'ren n minus r minus j R p'ren factorial j factorial Ende Bruch großes I Index 1 minus q Grund L p'ren n minus s plus 1 komma s minus r minus j R p'ren punt EndLayout
64Sitx=1/m0airitx+(i1)/mbi,upper S Sub i Base BinomialOrMatrix t Choose x EndBinomialOrMatrix equals 2 By 2 Matrix 1st Row 1st Column 1 divided by m 2nd Column 0 2nd Row 1st Column a Sub i Base 2nd Column r Sub i Base EndMatrix BinomialOrMatrix t Choose x EndBinomialOrMatrix plus BinomialOrMatrix L p'ren i minus 1 R p'ren divided by m Choose b Sub i Base EndBinomialOrMatrix commagroßes S Index i Grund BinomialOrMatrix t Choose x EndBinomialOrMatrix is gelyk aan 2 By 2 Matrix 1. Row 1. Column 1 divided by m 2. Column 0 2. Row 1. Column a Index i Grund 2. Column r Index i Grund EndMatrix BinomialOrMatrix t Choose x EndBinomialOrMatrix plus BinomialOrMatrix L p'ren i minus 1 R p'ren divided by m Choose b Index i Grund EndBinomialOrMatrix komma
65c1h42s12TTT(f(t+h)f(t))2dtc2h42sc 1 h Sup 4 minus 2 s Base less than or equals Frac 1 Over 2 upper T EndFrac integral Sub negative upper T Sup upper T Base L p'ren f L p'ren t plus h R p'ren minus f L p'ren t R p'ren R p'ren squared normal d t less than or equals c 2 h Sup 4 minus 2 sc 1 h hoch 4 minus 2 s Grund kleiner of gelyk aan Bruch 1 durch 2 großes T Ende Bruch integraal Sub negative großes T Sup großes T Base L p'ren f L p'ren t plus h R p'ren minus f L p'ren t R p'ren R p'ren squared normales d t kleiner of gelyk aan c 2 h hoch 4 minus 2 s
66C(0)C(h)ch42supper C L p'ren 0 R p'ren minus upper C L p'ren h R p'ren asymptotically equals c h Sup 4 minus 2 sgroßes C L p'ren 0 R p'ren minus großes C L p'ren h R p'ren assimtoties gelyk aan c h hoch 4 minus 2 s
67S(ω)=limT12TTTf(t)eitωdt2.upper S L p'ren omega R p'ren equals limit Underscript upper T R arrow infinity Endscripts Frac 1 Over 2 upper T EndFrac AbsoluteValue integral Sub negative upper T Sup upper T Base comma f comma L p'ren comma t comma R p'ren comma normal e Sup italic i t omega Base comma normal d comma t EndAbsoluteValue squared periodgroßes S L p'ren omega R p'ren is gelyk aan limit Unterschrift großes T R arrow oneindigheid Bruch 1 durch 2 großes T Ende Bruch AbsoluteValue integraal Sub negative großes T Sup großes T Base komma f komma L p'ren komma t komma R p'ren komma normales e hoch kursives i t omega Grund komma normales d komma t EndAbsoluteValue squared punt
680101[|f(t)f(u)|2+|tu|2]s/2dtdu<integral Sub 0 Sup 1 Base integral Sub 0 Sup 1 Base L brack AbsoluteValue f L p'ren t R p'ren minus f L p'ren u R p'ren EndAbsoluteValue squared plus AbsoluteValue t minus u EndAbsoluteValue squared R brack Sup negative s divided by 2 Base normal d t normal d u less than infinityintegraal Sub 0 Sup 1 Base integraal Sub 0 Sup 1 Base L brack AbsoluteValue f L p'ren t R p'ren minus f L p'ren u R p'ren EndAbsoluteValue squared plus AbsoluteValue t minus u EndAbsoluteValue squared R brack hoch negative s divided by 2 Grund normales d t normales d u kleiner as oneindigheid
69EIEk+1|I|s=EIEk|I|sE(R1s+R2s).sans serif upper E L p'ren sigma summation Underscript upper I element of upper E Sub k plus 1 Base Endscripts AbsoluteValue upper I EndAbsoluteValue Sup s Base R p'ren equals sans serif upper E L p'ren sigma summation Underscript upper I element of upper E Sub k Base Endscripts AbsoluteValue upper I EndAbsoluteValue Sup s Base R p'ren sans serif upper E L p'ren upper R 1 Sup s Base plus upper R 2 Sup s Base R p'ren periodserifenloses großes E L p'ren sigma summation Unterschrift großes I element van großes E Index k plus 1 Grund AbsoluteValue großes I EndAbsoluteValue hoch s Grund R p'ren is gelyk aan serifenloses großes E L p'ren sigma summation Unterschrift großes I element van großes E Index k Grund AbsoluteValue großes I EndAbsoluteValue hoch s Grund R p'ren serifenloses großes E L p'ren großes R 1 hoch s Grund plus großes R 2 hoch s Grund R p'ren punt
70(x1,y1)L p'ren x 1 comma y 1 R p'renL p'ren x 1 komma y 1 R p'ren
71(x2,y2)L p'ren x 2 comma y 2 R p'renL p'ren x 2 komma y 2 R p'ren
72d=(x2x1)2+(y2y1)2d equals Root L p'ren x 2 minus x 1 R p'ren squared plus L p'ren y 2 minus y 1 R p'ren squared EndRootd is gelyk aan Wurzel L p'ren x 2 minus x 1 R p'ren squared plus L p'ren y 2 minus y 1 R p'ren squared Ende Wurzel
73double struck upper Rmit Doppelstrich großes R
74=(,)double struck upper R equals L p'ren negative infinity comma infinity R p'renmit Doppelstrich großes R is gelyk aan L p'ren negative oneindigheid komma oneindigheid R p'ren
75{ 1,2,3 }Set 1 comma 2 comma 3 EndSetSet 1 komma 2 komma 3 EndSet
761S1 element of upper S1 element van großes S
773S3 element of upper S3 element van großes S
784S4 not an element of upper S4 nie 'n element van großes S
79a=3x1+(1+x)2a equals Root 3 x minus 1 EndRoot plus L p'ren 1 plus x R p'ren squareda is gelyk aan Wurzel 3 x minus 1 Ende Wurzel plus L p'ren 1 plus x R p'ren squared
80a=(b+c)2d+(e+f)2ga equals Frac L p'ren b plus c R p'ren squared Over d EndFrac plus Frac L p'ren e plus f R p'ren squared Over g EndFraca is gelyk aan Bruch L p'ren b plus c R p'ren squared durch d Ende Bruch plus Bruch L p'ren e plus f R p'ren squared durch g Ende Bruch
81x=[ (a+b)2(cb)2 ]+[ (d+e)2(fe)2 ]x equals L brack L p'ren a plus b R p'ren squared L p'ren c minus b R p'ren squared R brack plus L brack L p'ren d plus e R p'ren squared L p'ren f minus e R p'ren squared R brackx is gelyk aan L brack L p'ren a plus b R p'ren squared L p'ren c minus b R p'ren squared R brack plus L brack L p'ren d plus e R p'ren squared L p'ren f minus e R p'ren squared R brack
82x=[ (a+b)2 ]+[ (fe)2 ]x equals L brack L p'ren a plus b R p'ren squared R brack plus L brack L p'ren f minus e R p'ren squared R brackx is gelyk aan L brack L p'ren a plus b R p'ren squared R brack plus L brack L p'ren f minus e R p'ren squared R brack
83x=[ (a+b)2 ]x equals L brack L p'ren a plus b R p'ren squared R brackx is gelyk aan L brack L p'ren a plus b R p'ren squared R brack
84x=(a+b)2x equals L p'ren a plus b R p'ren squaredx is gelyk aan L p'ren a plus b R p'ren squared
85x=a+b2x equals a plus b squaredx is gelyk aan a plus b squared
861234=23Frac one half Over three fourths EndFrac equals two thirdsBruch een helfte durch drie kwarts Ende Bruch is gelyk aan twee derdes
872((x+1)(x+3)4((x1)(x+2)3))=y2 L p'ren L p'ren x plus 1 R p'ren L p'ren x plus 3 R p'ren minus 4 L p'ren L p'ren x minus 1 R p'ren L p'ren x plus 2 R p'ren minus 3 R p'ren R p'ren equals y2 L p'ren L p'ren x plus 1 R p'ren L p'ren x plus 3 R p'ren minus 4 L p'ren L p'ren x minus 1 R p'ren L p'ren x plus 2 R p'ren minus 3 R p'ren R p'ren is gelyk aan y
88cosx=1x22!+x44!cosine x equals 1 minus Frac x squared Over 2 factorial EndFrac plus Frac x Sup 4 Base Over 4 factorial EndFrac minus ellipsiskosinus x is gelyk aan 1 minus Bruch x squared durch 2 factorial Ende Bruch plus Bruch x hoch 4 Grund durch 4 factorial Ende Bruch minus ellipsis
89x=b±b24ac2ax equals Frac negative b plus or minus Root b squared minus 4 a c EndRoot Over 2 a EndFracx is gelyk aan Bruch negative b plus of minus Wurzel b squared minus 4 a c Ende Wurzel durch 2 a Ende Bruch
90x+y2k+1x plus y Sup Frac 2 Over k plus 1 EndFracx plus y hoch Bruch 2 durch k plus 1 Ende Bruch
91limx0sinxx=1limit Underscript x R arrow 0 Endscripts Frac sine x Over x EndFrac equals 1limit Unterschrift x R arrow 0 Bruch sinus x durch x Ende Bruch is gelyk aan 1
92d=(x2x1)2+(y2y1)2d equals Root L p'ren x 2 minus x 1 R p'ren squared plus L p'ren y 2 minus y 1 R p'ren squared EndRootd is gelyk aan Wurzel L p'ren x 2 minus x 1 R p'ren squared plus L p'ren y 2 minus y 1 R p'ren squared Ende Wurzel
93Fn=Fn1+Fn2upper F Sub n Base equals upper F Sub n minus 1 Base plus upper F Sub n minus 2großes F Index n Grund is gelyk aan großes F Index n minus 1 Grund plus großes F Index n minus 2
94Π=(π11π12π12000π12π11π12000π12π12π11000000π44000000π44000000π44)bold upper Pi equals 6 By 6 Matrix 1st Row 1st Column pi 11 2nd Column pi 12 3rd Column pi 12 4th Column 0 5th Column 0 6th Column 0 2nd Row 1st Column pi 12 2nd Column pi 11 3rd Column pi 12 4th Column 0 5th Column 0 6th Column 0 3rd Row 1st Column pi 12 2nd Column pi 12 3rd Column pi 11 4th Column 0 5th Column 0 6th Column 0 4th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column pi 44 5th Column 0 6th Column 0 5th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column pi 44 6th Column 0 6th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 6th Column pi 44 EndMatrixfettes großes Pi is gelyk aan 6 By 6 Matrix 1. Row 1. Column pi 11 2. Column pi 12 3. Column pi 12 4. Column 0 5. Column 0 6. Column 0 2. Row 1. Column pi 12 2. Column pi 11 3. Column pi 12 4. Column 0 5. Column 0 6. Column 0 3. Row 1. Column pi 12 2. Column pi 12 3. Column pi 11 4. Column 0 5. Column 0 6. Column 0 4. Row 1. Column 0 2. Column 0 3. Column 0 4. Column pi 44 5. Column 0 6. Column 0 5. Row 1. Column 0 2. Column 0 3. Column 0 4. Column 0 5. Column pi 44 6. Column 0 6. Row 1. Column 0 2. Column 0 3. Column 0 4. Column 0 5. Column 0 6. Column pi 44 EndMatrix
95s11=c11+c12c11c12c11+2c12s 11 equals Frac c 11 plus c 12 Over L p'ren c 11 minus c 12 R p'ren L p'ren c 11 plus 2 c 12 R p'ren EndFracs 11 is gelyk aan Bruch c 11 plus c 12 durch L p'ren c 11 minus c 12 R p'ren L p'ren c 11 plus 2 c 12 R p'ren Ende Bruch
96SiO2+ 6HF H2 SiF6+ 2H2O upper S i normal upper O 2 plus 6 normal upper H normal upper F R arrow normal upper H 2 upper S i normal upper F 6 plus 2 normal upper H 2 normal upper Ogroßes S i normales großes O 2 plus 6 normales großes H normales großes F R arrow normales großes H 2 großes S i normales großes F 6 plus 2 normales großes H 2 normales großes O
97ddx(E(x)A(x)dw(x)dx)+p(x)=0Frac d Over d x EndFrac L p'ren upper E L p'ren x R p'ren upper A L p'ren x R p'ren Frac d w L p'ren x R p'ren Over d x EndFrac R p'ren plus p L p'ren x R p'ren equals 0Bruch d durch d x Ende Bruch L p'ren großes E L p'ren x R p'ren großes A L p'ren x R p'ren Bruch d w L p'ren x R p'ren durch d x Ende Bruch R p'ren plus p L p'ren x R p'ren is gelyk aan 0
98TCSgas=12PsealPmax1TsealTCS Sub gas Base equals minus one half L p'ren Frac upper P Sub seal Base Over upper P Sub max Base EndFrac R p'ren L p'ren Frac 1 Over upper T Sub seal Base EndFrac R p'renTCS Index gas Grund is gelyk aan minus een helfte L p'ren Bruch großes P Index seal Grund durch großes P Index max Grund Ende Bruch R p'ren L p'ren Bruch 1 durch großes T Index seal Grund Ende Bruch R p'ren
99Bp=7v231+c2a2+c4a4+3v2c21+va21v1c4a41c2a2upper B Sub p Base equals NestFrac Frac 7 minus v squared Over 3 EndFrac L p'ren 1 plus Frac c squared Over a squared EndFrac plus Frac c Sup 4 Base Over a Sup 4 Base EndFrac R p'ren plus Frac L p'ren 3 minus v R p'ren squared c squared Over L p'ren 1 plus v R p'ren a squared EndFrac NestOver L p'ren 1 minus v R p'ren L p'ren 1 minus Frac c Sup 4 Base Over a Sup 4 Base EndFrac R p'ren L p'ren 1 minus Frac c squared Over a squared EndFrac R p'ren NestEndFracgroßes B Index p Grund is gelyk aan geschachteltBruch Bruch 7 minus v squared durch 3 Ende Bruch L p'ren 1 plus Bruch c squared durch a squared Ende Bruch plus Bruch c hoch 4 Grund durch a hoch 4 Grund Ende Bruch R p'ren plus Bruch L p'ren 3 minus v R p'ren squared c squared durch L p'ren 1 plus v R p'ren a squared Ende Bruch geschachteltdurch L p'ren 1 minus v R p'ren L p'ren 1 minus Bruch c hoch 4 Grund durch a hoch 4 Grund Ende Bruch R p'ren L p'ren 1 minus Bruch c squared durch a squared Ende Bruch R p'ren geschachteltEnde Bruch
100Qtankseries=1RsLsCsupper Q Sub tank Sup series Base equals Frac 1 Over upper R Sub s Base EndFrac Root Frac upper L Sub s Base Over upper C Sub s Base EndFrac EndRootgroßes Q Index tank hoch series Grund is gelyk aan Bruch 1 durch großes R Index s Grund Ende Bruch Wurzel Bruch großes L Index s Grund durch großes C Index s Grund Ende Bruch Ende Wurzel
101Δϕpeak=tan1(k2Qtankseries)upper Delta phi Sub peak Base equals tangent Sup negative 1 Base L p'ren k squared upper Q Sub tank Sup series Base R p'rengroßes Delta phi Index peak Grund is gelyk aan tangens hoch negative 1 Grund L p'ren k squared großes Q Index tank hoch series Grund R p'ren
102f=1.013WL2Eρ(1+0.293L2EW2σ)f equals 1.013 Frac upper W Over upper L squared EndFrac Root Frac upper E Over rho EndFrac EndRoot Root L p'ren 1 plus 0.293 Frac upper L squared Over EW squared EndFrac sigma R p'ren EndRootf is gelyk aan 1.013 Bruch großes W durch großes L squared Ende Bruch Wurzel Bruch großes E durch rho Ende Bruch Ende Wurzel Wurzel L p'ren 1 plus 0.293 Bruch großes L squared durch EW squared Ende Bruch sigma R p'ren Ende Wurzel
103unx=γncoshknxcosknx+sinhknxsinknxu Sub n Base L p'ren x R p'ren equals gamma Sub n Base L p'ren hyperbolic cosine k Sub n Base x minus cosine k Sub n Base x R p'ren plus L p'ren hyperbolic sine k Sub n Base x minus sine k Sub n Base x R p'renu Index n Grund L p'ren x R p'ren is gelyk aan gamma Index n Grund L p'ren hiperboliese kosinus k Index n Grund x minus kosinus k Index n Grund x R p'ren plus L p'ren hiperboliese sinus k Index n Grund x minus sinus k Index n Grund x R p'ren
104B=F0m(ω02ω2)2+4n2ω2=F0k(1(ω/ω02)2)2+4(n/ω0)2(ω/ω0)2Layout 1st Row 1st Column upper B 2nd Column equals NestFrac Frac upper F 0 Over m EndFrac NestOver Root L p'ren omega 0 squared minus omega squared R p'ren squared plus 4 n squared omega squared EndRoot NestEndFrac 2nd Row 1st Column Blank 2nd Column equals NestFrac Frac upper F 0 Over k EndFrac NestOver Root L p'ren 1 minus L p'ren omega divided by omega 0 squared R p'ren squared R p'ren squared plus 4 L p'ren n divided by omega 0 R p'ren squared L p'ren omega divided by omega 0 R p'ren squared EndRoot NestEndFrac EndLayoutLayout 1. Row 1. Column großes B 2. Column is gelyk aan geschachteltBruch Bruch großes F 0 durch m Ende Bruch geschachteltdurch Wurzel L p'ren omega 0 squared minus omega squared R p'ren squared plus 4 n squared omega squared Ende Wurzel geschachteltEnde Bruch 2. Row 1. Column Blank 2. Column is gelyk aan geschachteltBruch Bruch großes F 0 durch k Ende Bruch geschachteltdurch Wurzel L p'ren 1 minus L p'ren omega divided by omega 0 squared R p'ren squared R p'ren squared plus 4 L p'ren n divided by omega 0 R p'ren squared L p'ren omega divided by omega 0 R p'ren squared Ende Wurzel geschachteltEnde Bruch EndLayout
105p(AandB)=p(A)p(B|A)normal p L p'ren upper A a n d upper B R p'ren equals normal p L p'ren upper A R p'ren normal p L p'ren upper B vertical bar upper A R p'rennormales p L p'ren großes A a n d großes B R p'ren is gelyk aan normales p L p'ren großes A R p'ren normales p L p'ren großes B afstreep großes A R p'ren
106PMF(x)1xαupper P upper M upper F L p'ren x R p'ren proportional to L p'ren Frac 1 Over x EndFrac R p'ren Sup alphagroßes P großes M großes F L p'ren x R p'ren eweredig aan L p'ren Bruch 1 durch x Ende Bruch R p'ren hoch alfa
107f(x)=12πexp(-x2/2)f L p'ren x R p'ren equals Frac 1 Over Root 2 pi EndRoot EndFrac exp L p'ren minus x squared slash 2 R p'renf L p'ren x R p'ren is gelyk aan Bruch 1 durch Wurzel 2 pi Ende Wurzel Ende Bruch exp L p'ren minus x squared skuinsstreep 2 R p'ren
108dxdθ=βcos2θFrac d x Over d theta EndFrac equals Frac beta Over cosine squared theta EndFracBruch d x durch d theta Ende Bruch is gelyk aan Bruch beta durch kosinus squared theta Ende Bruch
109s/2(n-1)s divided by Root 2 L p'ren n minus 1 R p'ren EndRoots divided by Wurzel 2 L p'ren n minus 1 R p'ren Ende Wurzel