math cheat sheet.docx

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MATH CHEAT SHEET

Basic Math and Pre-Algebra Cheat SheetGroups of Numbers

Natural or counting numbers1, 2, 3, 4,

!hole numbers0, 1, 2, 3,

"ntegers-3, -2, -1, 0, 1, 2, 3,

#ational numbersIntegers and fractions.

"rrational numbersCannot be written as fractions: 3 or .

Prime numbersDivisible onl b 1 and itself: 2, 3, !, ", 11, 13, #0 and 1 are not$ri%e or co%$osite.&

Composite numbersDivisible b %ore t'an ()st 1: 4, *, +, , 10, 12,

Properties of Addition and Multiplication

Closurell answers fall into original set.

Commutati$erder does not %a/e an difference: a b b a, ab ba.

Associati$ero)$ing does not %a/e an difference: #a b& c a #b c&,#ab&c a#bc&.

"dentit%0 for addition, 1 for %)lti$lication.

"n$erseegative for addition, reci$rocal for %)lti$lication.

&rder of &perations

1. !or' (ithin parentheses ) *+ brac'ets , + and braces . / from innermost and(or' out(ard0

2. Simplif% e1ponents and roots (or'ing from left to right0

3. 2o multiplication and di$ision+ (hiche$er comes first left to right0

4. 2o addition and subtraction+ (hiche$er comes first left to right0

#ounding &ff

1. 3nderline the place $alue to (hich %ou4re rounding off0

2. 5oo' to the immediate right )one place* of %our underlined place $alue0


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3. "dentif% the number )the one to the right*0

If it is ! or 'ig'er, ro)nd o)r )nderlined $lace val)e )$ 1 and c'ange all t'e ot'ern)%bers to its rig't to eros. If less t'an !, leave o)r )nderlined $lace val)e as it isand c'ange all t'e ot'er n)%bers to t'e rig't to eros.

2ecimals

5o add or s)btract deci%als, si%$l line )$ t'e deci%al $oints and t'en add ors)btract as )s)al.

5o %)lti$l deci%als, ()st %)lti$l as )s)al and t'en co)nt t'e total n)%ber of digitsabove t'e line t'at are to t'e rig't of all deci%al $oints. 6lace t'e deci%al $oint ino)r answer so t'at t'ere are t'e sa%e n)%ber of digits to t'e rig't of t'e deci%al$oint as t'ere are above t'e line.

5o divide deci%als, if t'e n)%ber o)7re dividing b 'as a deci%al, %ove t'e deci%alto t'e rig't as %an $laces as $ossible and t'en %ove it )nder t'e division sign ()stas %an $laces #add eros if necessar&. 8ove t'e deci%al )$ to o)r answer.

6ractions

5o add or s)btract fractions, o) %)st 'ave a co%%on deno%inator.

If two fractions 'ave a co%%on deno%inator #t'ese are called like fractions&, o)si%$l add or s)btract t'e n)%erator and /ee$ t'e sa%e deno%inator. #9or ea%$le,1;! 2;! 3;!.&

If two fractions do not 'ave a co%%on deno%inator (unlike fractions),find a lowestco%%on deno%inator #'en s)btracting %ied n)%bers, o) %a 'ave to ?borrow? fro% t'e w'ole n)%ber.>'en o) borrow 1 fro% t'e w'ole n)%ber, t'e 1 %)st be c'anged to a fraction.

5o %)lti$l fractions, si%$l %)lti$l t'e n)%erators and t'en %)lti$l t'edeno%inators. #9or ea%$le, 2;3 @ 1;! 2;1!.& Aed)ce to lowest ter%s if necessar.

5o divide fractions, invert t'e second fraction and t'en %)lti$l. #9or ea%$le, 1;! B1;4 1;! @ 4;1 4;!.&

A

additi$e in$erse5'e o$$osite #negative& of a n)%ber. n n)%ber $l)s its additive inverse

e=)als 0.

associati$e propert%ro)$ing of ele%ents %a/es no difference in t'e o)tco%e. 5'is is onltr)e for %)lti$lication and addition.

B


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bracesro)$ing s%bols )sed after t'e )se of brac/ets signs E )sed to re$resent a set.

brac'etsro)$ing s%bols F G, )sed after t'e )se of $arent'eses.

C

cancelingIn %)lti$lication of fractions, dividing t'e sa%e n)%ber into bot' a n)%erator anda deno%inator.

circumference5'e distance aro)nd a circle e=)als 2 @ H @ t'e radi)s or H @ t'e dia%eter#C 2Hr or Hd&.

closure propert%>'en all answers fall into t'e original set.

combinations5'e total n)%ber of inde$endent $ossible c'oices.

common denominator n)%ber t'at can be divided evenl b all deno%inators in t'e$roble%.

common factors9actors t'at are t'e sa%e for two or %ore n)%bers.

common multiples8)lti$les t'at are t'e sa%e for two or %ore n)%bers.

commutati$e propert%5'e order of ele%ents does not %a/e an difference in t'e o)tco%e.5'is is onl tr)e for %)lti$lication and addition.

comple1 fraction fraction 'aving a fraction or fractions in t'e n)%erator and;ordeno%inator.

composite number n)%ber divisible b %ore t'an ()st 1 and itself.

cube5'e res)lt w'en a n)%ber is %)lti$lied b itself twice.

cube root n)%ber t'at w'en %)lti$lied b itself twice gives o) t'e original n)%ber its

s%bol is3

.

2

decimal fraction9raction wit' a deno%inator 10, 100, 1,000, and so on, written )sing adeci%al $oint for ea%$le, .3 and .2"!.

decimal point $oint )sed to disting)is' deci%al fractions fro% w'ole n)%bers.

denominator5'e botto% s%bol or n)%ber of a fraction.

dependent e$ents>'en t'e o)tco%e of one event 'as a bearing or effect on t'e o)tco%eof anot'er event.

difference5'e res)lt of s)btraction.

distributi$e propert%5'e $rocess of distrib)ting a n)%ber on t'e o)tside of t'e$arent'eses to eac' n)%ber on t'e inside a#b c& ab ac.

E

e$en numbern integer #$ositive w'ole n)%bers, ero, and negative w'ole n)%bers&divisible b 2 #wit' no re%ainder&.


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e1panded notation6ointing o)t t'e $lace val)e of a digit b writing a n)%ber as t'e digit @its $lace val)e. 9or ea%$le, 342 #3 @ 102& #4 @ 101& #2 @ 10&.

e1ponent s%all n)%ber $laced above and to t'e rig't of a n)%ber e$resses t'e $owerto w'ic' t'e =)antit is to be raised or lowered.

6

factor )noun* n)%ber or s%bol t'at divides evenl into a larger n)%ber. 9or ea%$le, *is a factor of 24.

factor )$erb*5o find two or %ore =)antities w'ose $rod)ct e=)als t'e original =)antit.

fractions%bol t'at e$resses $art of a w'ole and consists of a n)%erator and adeno%inator for ea%$le, 3;!.

G

greatest common factor5'e largest factor co%%on to two or %ore n)%bers.

"

identit% element for addition 7n n)%ber added to 0 gives t'e original n)%ber.

identit% element for multiplication 8n n)%ber %)lti$lied b 1 gives t'e originaln)%ber.

improper fraction fraction in w'ic' t'e n)%erator is greater t'an t'e deno%inator forea%$le, 3;2.

independent e$ents>'en t'e o)tco%e of one event 'as no bearing or effect on t'eo)tco%e of anot'er event.

integer w'ole n)%ber, eit'er $ositive, negative, or ero.

in$ert5)rn )$side down, as in ?invert 2;3 3;2.?

irrational number n)%ber t'at is not rational #cannot be written as a fraction ;, wit' a nat)ral n)%ber and an integer& for ea%$le, 3 or H.

5

least common multiple5'e s%allest %)lti$le t'at is co%%on to two or %ore n)%bers.

lo(est common denominator5'e s%allest n)%ber t'at can be divided evenl b all

deno%inators in t'e $roble%.

M

mean )arithmetic*5'e average of a n)%ber of ite%s in a gro)$ #t'e total ite%s divided bt'e n)%ber of ite%s&.

median5'e %iddle ite% in an ordered gro)$. If t'e gro)$ 'as an even n)%ber of ite%s, t'e%edian is t'e average of t'e two %iddle ter%s.


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mi1ed number n)%ber containing bot' a w'ole n)%ber and a fraction for ea%$le, !.

mode5'e n)%ber a$$earing %ost fre=)entl in a gro)$.

multiples)%bers fo)nd b %)lti$ling a n)%ber b 2, b 3, b 4, and so on.

multiplicati$e in$erse5'e reci$rocal of a n)%ber. n n)%ber %)lti$lied b its

%)lti$licative inverse e=)als 1.

N

natural number co)nting n)%ber 1, 2, 3, 4, and so on.

negati$e number n)%ber less t'an 0.

number series se=)ence of n)%bers wit' so%e $attern. ne n)%ber follows anot'er inso%e defined %anner.

numerator5'e to$ s%bol or n)%ber of a fraction.

&

odd numbern integer not divisible b 2.

operation8)lti$lication, addition, s)btraction, or division.

order of operations5'e $riorit given to an o$eration relative to ot'er o$erations. 9orea%$le, %)lti$lication is $erfor%ed before addition.

P

parenthesesro)$ing s%bols # &.

percent or percentage co%%on fraction wit' 100 as its deno%inator. 9or ea%$le, 3"Jis 3";100.

permutations5'e total n)%ber of de$endent c'oices.

place $alue5'e val)e given a digit b t'e $osition of a digit in t'e n)%ber.

positi$e number n)%ber greater t'an ero.

po(er $rod)ct of e=)al factors. 4 @ 4 @ 4 43, reads ?fo)r to t'e t'ird $ower? or ?t'et'ird $ower of fo)r.? 6ower and e$onent are so%e@ )sed interc'angeabl.

prime number n)%ber t'at can be divided b onl itself and one. probabilit%5'en)%erical %eas)re of t'e c'ance of an o)tco%e or event occ)rring.

product5'e res)lt of %)lti$lication.

proper fraction fraction in w'ic' t'e n)%erator is less t'an t'e deno%inator for ea%$le,2;3.

proportion>ritten as two e=)al ratios. 9or ea%$le, ! is to 4 as 10 is to +, or !;4 10;+.

9


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:uotient5'e res)lt of division.

#

range5'e difference between t'e largest and t'e s%allest n)%ber in a set of n)%bers.

ratio co%$arison between two n)%bers or s%bols %a be written :, ;, or is to .

rational numbern integer or fraction s)c' as ";+ or ;4 or !;1. n n)%ber t'at can bewritten as a fraction ; wit' a nat)ral n)%ber and an integer.

real numbern rational or irrational n)%ber.

reciprocal5'e %)lti$licative inverse of a n)%ber. 9or ea%$le, 2;3 is t'e reci$rocal of 3;2.

reducingC'anging a fraction into its lowest ter%s. 9or ea%$le, 2;4 is red)ced to 1;2.

rounding offC'anging a n)%ber to t'e nearest $lace val)e as s$ecified a %et'od ofa$$roi%ating.

S

scientific notation n)%ber between 1 and 10 and %)lti$lied b a $ower of 10, )sed forwriting ver large or ver s%all n)%bers for ea%$le, 2.! @ 104.

s:uare5'e res)lt w'en a n)%ber is %)lti$lied b itself.

s:uare root n)%ber t'at w'en %)lti$lied b itself gives o) t'e original n)%ber itss%bol is . 9or ea%$le, ! is t'e s=)are root of 2! 2! !.

sum5'e res)lt of addition.

T

tenth5'e first deci%al $lace to t'e rig't of t'e deci%al $oint. 9or ea%$le, ." is seven-tent's.

!

(eighted mean5'e %ean of a set of n)%bers t'at 'ave been weig'ted #%)lti$lied b t'eirrelative i%$ortance or @ of occ)rrence&.

(hole number0, 1, 2, 3, and so on.

Algebra " Cheat SheetA1ioms of E:ualit%

#efle1i$e a1ioma aS%mmetric a1iomIf a b, t'en b aTransiti$e a1iomIf a b and b c, t'en a cAdditi$e a1iomIf a b and c d, t'en a c b dMultiplicati$e a1iomIf a b and c d, t'en ac bdSol$ing E:uations

1. Ki%$lif if necessar.


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2. et t'e variable on one side of t'e e=)al sign and n)%bers on t'e ot'er.

3. Divide b t'e n)%ber in front of t'e variable.

Sol$ing S%stems of E:uations

Addition;Subtraction MethodCo%bine e=)ations to eli%inate one variable. 5'e e=)ations%a need to be %)lti$lied b a co%%on %)lti$le first.Substitution MethodKolve one e=)ation for one variable and s)bstit)te t'at variable intoot'er e=)ations.Graphing Methodra$' eac' e=)ation on t'e sa%e gra$'. 5'e coordinates of t'eintersection are t'e sol)tion.Monomials

monomialis an algebraic e$ression t'at consists of onl one ter%.

dd or s)btract %ono%ials wit' li/e ter%s onl: 3xy 2xy !xy.

5o %)lti$l %ono%ials, add t'e e$onents of t'e sa%e bases:x4#x3& x".

5o divide %ono%ials, s)btract t'e e$onent of t'e divisor fro% t'e e$onent of t'edividend of t'e sa%e base:x+;x3x!.

Pol%nomials

polynomialis an algebraic e$ression of two or %ore ter%s, s)c' asx y. Binomialsconsistof eactl two ter%s. Trinomialsconsist of eactl t'ree ter%s.

5o add or s)btract $olno%ials, add or s)btract li/e ter%s onl.

5o %)lti$l two $olno%ials, %)lti$l eac' ter% in one $olno%ial b eac' ter% in t'eot'er $olno%ial.

5'e 9..I.


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Difference of two s=)ares

1. 9ind t'e s=)are root of t'e first ter% and t'e second ter%.

2. L$ress o)r answer as t'e $rod)ct of t'e s)% and difference of t'ose =)antities.La%$le: 2- # 3 - 3&

5rino%ials

1. C'ec/ to see if o) can %ono%ial factor.

2. Mse do)ble $arent'eses and factor t'e first ter% and $lace t'e factors in t'e left sideof t'e $arent'esis.

3. 9actor t'e last ter% and $lace t'e factors in t'e rig't sides of t'e $arent'eses.

4. Deciding t'e signs of t'e n)%bers, and t'e n)%bers t'e%selves, %a ta/e trial anderror. 8)lti$l t'e %eans and etre%es t'eir s)% %)st e=)al t'e %iddle ter%.La%$le: 2 3 2 # 2 1&

A1ioms of "ne:ualit%

Trichotom% a1ioma N b, a b, or a O b.Transiti$e a1iomIf a N b and b N c, t'en a N c.Additi$e a1iomIf a N b, t'en a c N b c.Positi$e multiplication a1iomIf c N 0, t'en a N b if, and onl if, ac N bc.Negati$e multiplication a1iomIf c O 0, t'en a N b if, and onl if, ac O bc.Sol$ing 9uadratic E:uations

B% factoring6)t all ter%s on one side of t'e e=)al sign and factor. Ket eac' factor to eroand solve.

B% using the :uadratic formula

6l)g into t'e for%)laB% completing the s:uare6)t t'e e=)ation in t'e for% of a2 b -c #%a/e a -1 bdividing if necessar&. dd #b;2&2to bot' sides of t'e e=)ation to for% a $erfect s=)are on t'eleft side of t'e e=)ation. 9ind t'e s=)are root of bot' sides of t'e e=)ation. Kolve t'e res)ltinge=)ation.

Algebra "" Cheat Sheet

Geometr% 6ormulas

6eri%eter P s)% of sides

rea of a triangle A bh

rea of a rectangle A lw

rea of a $arallelogra% A bh

rea of a tra$eoid A P(b1+ b2)Ph

rea of a circle A r2


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!ord Problem 6ormulas

Distance istance rate Q ti%e

Interest !nterest princi$al Q rate Q ti%e

Co%$o)nd Interest

>or/ acco%$lis'ed wor/ acco%$lis'ed #rate of wor/& Q #ti%e wor/ed&

Graphing 6ormulas

Distance #for gra$'s&

8id$oint

Klo$e

Klo$e-Interce$t 9or% y mx b

9uadratic E:uations

Ktandard 9or% ax 2 bx c 0

Difference of s=)ares x 2- b 2 #x b- b&

R)adratic for%)la

Conic Sections

Circle "entere# at (',/) with a ra#ius of r$

#x- h&2 #y- k&2 r2

Llli$se "entere# at (',/) with its ma%or axis parallel to the&axis$

"entere# at (', /) with its ma%or axis parallel to the&axis$

6arabola !f parabola opens 'ertically$

y a#x- h&2 k

!f parabola opens horiontally$

x a#y- k&2 h

S$erbola !f trans'erse axis is horiontal$


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!f trans'erse axis is 'ertical$

Se:uences and Series

rit'%etic Ke=)ence an a1 #n- 1

eo%etric Ke=)ence an a1r n&1

rit'%etic Keries

eo%etric Keries

Geometr% Cheat Sheet

, s ngle, angles

% T 8eas)re of angle T

rc T

8eas)re of arc T

Circle, circles

Congr)ent

ot congr)ent

Degree

L=)al

U ot e=)al

N reater t'anO


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K ngle-side-angle, for $roving triangles congr)ent

cos Cosine

cot Cotangent

C6C5C Congr)ent $arts of congr)ent triangles are congr)ent

csc Cosecant

CKK56 Corres$onding sides of si%ilar triangles are $ro$ortionalKK Kide-angle-side, for $roving triangles congr)ent

sin Kine

tan 5angent

Commonl% 3sed


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TCDLSI


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diametera line seg%ent t'at contains t'e center and 'as its end$oints on t'e circle. lso,t'e lengt' of t'is seg%ent. # c'ord t'ro)g' t'e center of t'e circle.&

E

e:uilateral trianglea triangle in w'ic' all t'ree angles are e=)al in %eas)re and all t'ree

sides 'ave t'e sa%e lengt'.

e1terior anglean angle for%ed o)tside t'e $olgon b etending one side. In a triangle, t'e%eas)re of an eterior angle e=)als t'e s)% of t'e %eas)res of t'e two re%ote interiorangles.

H

heightaltit)de. 9ro% t'e 'ig'est $oint, a $er$endic)lar drawn to t'e base.

heptagona $lane closed fig)re wit' seven sides and seven angles.

he1agona $lane closed fig)re wit' si sides and si angles.

h%potenusein a rig't triangle, t'e side o$$osite t'e 0^ angle.

"

inscribed anglein a circle, an angle for%ed b two c'ords. Its verte is on t'e circle. 5'e%eas)re of an inscribed angle e=)als one-'alf t'e %eas)re of its arc.

interior anglesangles for%ed inside t'e s'a$e or wit'in two $arallel lines.

intersecting lineslines t'at %eet at a $oint.

isosceles right trianglea triangle 'aving two e=)al sides, two e=)al angles, and one 0^angle. Its sides are alwas in t'e ratio 1, 1, 2.

isosceles trianglea triangle 'aving two e=)al sides #and t')s two e=)al angles across fro%t'ose sides&.

5

legsin a rig't triangle, t'e two sides for%ing t'e 0^ angle. In a tra$eoid, t'e non$arallelsides.

line segmenta $art of a line 'as two end$oints

M

medianin a triangle, a line seg%ent drawn fro% a verte to t'e %id$oint of t'e o$$ositeside.

medianin a tra$eoid, a line seg%ent $arallel to t'e bases and bisecting t'e legs.

midpointt'e 'alfwa $oint of a line seg%ent, e=)idistant fro% eac' end$oint.

minutea s)bdivision of an angle, one-sitiet' of a degree.


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N

nonagona $lane closed fig)re wit' nine sides and nine angles

&

obtuse anglean angle greater t'an 0^ b)t less t'an 1+0^

obtuse trianglea triangle containing an obt)se angle

octagona $lane closed fig)re wit' eig't sides and eig't angles

P

parallel linestwo or %ore lines, alwas t'e sa%e distance a$art. 6arallel lines never %eet.

parallelograma fo)r-sided $lane closed fig)re 'aving o$$osite sides e=)al and $arallel.#$$osite angles are e=)al, and consec)tive angles are s)$$le%entar.&

pentagona five-sided $lane closed fig)re. 5'e s)% of its five angles is !40^.

perimetert'e total distance aro)nd t'e o)tside of an $olgon. 5'e total lengt' of all t'esides.

perpendicular linestwo lines t'at intersect at rig't angles.

pi )*a constant )sed in deter%ining a circle7s area or circ)%ference. L=)als a$$roi%atel3.14 or 22;"

planeoften described as a flat s)rface.

plane figures'a$e 'aving onl lengt' and widt' #two di%ensional&.

plane geometr%t'e st)d of s'a$es and fig)res in two di%ensions.

pointa basic ele%ent of geo%etr, a location. If two lines intersect, t'e do so at a $oint.

pol%gon%an-sided $lane closed fig)re. 5riangle, =)adrilateral, $entagon, and so on.

prisma t'ree-di%ensional s'a$e bo)nded b congr)ent $arallel faces and a set of$arallelogra%s for%ed b (oining t'e corres$onding vertices of t'e bases.

P%thagorean theorema t'eore% t'at a$$lies to rig't triangles. 5'e s)% of t'e s=)ares of arig't triangle7s two legs e=)als t'e s=)are of t'e '$oten)se #a2 b2 c2&.

9

:uadrilaterala fo)r-sided $lane closed fig)re. 5'e s)% of its fo)r angles e=)als 3*0^.

#

radii$l)ral of radi)s.

radiusa line seg%ent w'ose end$oints lie one at t'e center of a circle and one on t'e circle.lso, t'e lengt' of t'is seg%ent.


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ra%a 'alf-line. Contin)es forever in one direction. Sas one end$oint.

rectanglea fo)r-sided $lane closed fig)re 'aving o$$osite sides e=)al and $arallel and fo)rrig't angles.

regular pol%gona $olgon in w'ic' sides and angles are all e=)al. 9or ea%$le, a reg)lar$entagon 'as five e=)al angles and five e=)al sides.

rhombusa $arallelogra% wit' fo)r e=)al sides.

right anglean angle w'ose %eas)re is e=)al to 0^.

right circular c%lindera solid s'a$ed li/e a can. Tase %eets side at a rig't angle.

right trianglea triangle containing a 0^ angle.

S

scalene trianglea triangle 'aving none of its sides e=)al #or angles e=)al&.

similar'aving t'e sa%e s'a$e b)t not t'e sa%e sie, in $ro$ortion.

solid geometr%t'e st)d of s'a$es and fig)res in t'ree di%ensions: lengt', widt', andt'ic/ness.

s:uarea fo)r-sided $lane closed fig)re 'aving e=)al sides and fo)r rig't angles. Its o$$ositesides are $arallel.

straight anglean angle e=)al to 1+0^. ften called a line.

straight lineoften described as t'e s'ortest distance between two $oints. Contin)es foreverin bot' directions. #oida fo)r-sided $lane closed fig)re wit' onl one $air of $arallel sides, called bases.

trianglea t'ree-sided $lane closed fig)re. Contains t'ree angles t'e s)% of w'ose %eas)resis 1+0^.

-scorea )nit of %eas)re%ent obtained b s)btracting t'e %ean and dividing b t'estandard deviation.

Courtesy: Cliff Notes.

math cheat sheet.docx

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