# List of math formulas

Here you can find a summary of the main formulas you need to know. This list was not organized by years of schooling but thematically. Just choose one of the topics and you will be able to view the formulas related to this subject. This is not an exhaustive list, ie it's not here all math formulas that are used in mathematics class, only those that were considered most important.

#### Areas

 Square A=l^2 l : length of side Rectangle A=wxxh w : widthh : height Triangle A=(bxxh)/2 b : baseh : height Rhombus A=(Dxxd)/2 D : large diagonald : small diagonal Trapezoid A=(B+b)/2xxh B : large sideb : small sideh: height Regular polygon A=P/2xxa P : perimetera : apothem Circle A=pir^2P=2pir r : radiusP : perimeter Cone(lateral surface) A=pirxxs r : radiuss : slant height Sphere(surface area) A=4pir^2 r: radius

#### Volumes

 Cube V=s^3 s: side Parallelepiped V=lxxwxxh l: lengthw: widthh: height Regular prism V=bxxh b: baseh: height Cylinder V=pir^2xxh r: radiush: height Cone (or pyramid) V=1/3bxxh b: baseh: height Sphere V=4/3pir^3 r: radius

#### Functions and Equations

 Directly Proportional y = kx                k = y/x k: Constant of Proportionality Inversely Proportional y = k/x                k = yx ax^2+bx+c=0 Quadratic formula x=(-b +- sqrt(b^2 - 4ac))/(2a) Concavity Concave up: a > 0 Concave down: a < 0 Discriminant Delta = b^2 - 4ac Vertex of the parabola V((-b)/(2a),(-Delta)/(4a)) y=a(x-h)^2+k Concavity Concave up: a > 0 Concave down: a < 0 Vertex of the parabola V(h, k) Zero-product property AxxB=0 hArr A=0 vv B=0 ex : (x+2)xx(x-1)=0 hArr x+2=0 vv x-1=0 hArr x=-2 vv x=1 Difference of two squares (a-b)(a+b)=a^2 - b^2 ex : (x-2)(x+2)=x^2 - 2^2=x^2 - 4 Perfect square trinomial (a+b)^2=a^2 + 2ab + b^2 ex : (2x+3)^2=(2x)^2 + 2*2x*3 +3^2=4x^2 + 12x + 9 Binomial theorem (x + y)^n = sum_(k=0)^n text( )^nC_k text( ) x^(n-k) text( ) y^k

#### Exponents

 Product a^mxxa^n=a^(m+n) ex : 3^5xx3^2=3^(5+2)=3^7 a^mxxb^m=(axxb)^m ex : 3^5xx2^5=(3xx2)^5=6^5 Quotient a^m-:a^n=a^(m-n) ex : 3^7-:3^2=3^(7-2)=3^5 a^m-:b^m=(a-:b)^m ex : 6^5-:2^5=(6-:2)^5=3^5ex : 5^3-:2^3=(5/2)^3 Power of Power (a^m)^p=a^(mxxp) ex : (5^2)^3=5^(2xx3)=5^6 Zero Exponents a^0=1 ex : 8^0=1 Negative Exponents a^-n=(1/a)^n ex : 3^-2=(1/3)^2ex : (2/3)^-4=(3/2)^4 Fractional Exponents a^(p/q)=root(q)(a^p) ex : 2^(4/3) = root(3)(2^4)

 Multiplication root(n)(x)xxroot(n)(y)=root(n)(x xx y) ex : root(3)(2)xxroot(3)(5)=root(3)(2xx5) hArr root(3)(10) Division root(n)(x)-:root(n)(y)=root(n)(x/y) ex : root(4)(8)-:root(4)(3)=root(4)(8/3) Addition a root(n)(x)+-b root(n)(x)=(a+-b)root(n)(x) ex : 4root(3)(5)-2root(3)(5)=(4-2)root(3)(5) hArr 2root(3)(5) Exponents (root(n)(x))^p=root(n)(x^p) ex : (sqrt 2)^3=sqrt (2^3) hArr sqrt 8 Radicals root(n)(root(p)(x))=root(n*p)(x) ex : root(3)(sqrt 5)=root (3xx2)(5) hArr root(6)(5) Exponentiation root(n)(a^m)=a^(m/n) ex : root(3)(4^5)=4^(5/3) Simplifying Radicals (root(n)(a))^n=a ex : (sqrt(3))^2=3 (root(n)(a))^m=root(n)(a^m) ex : (sqrt(4))^5=sqrt(4^5)

#### Trigonometry

 Trigonometry Ratios sin alpha=(opp.)/ (hip.) opp.: oppositehip.: hypotenuse cos alpha=(adj.)/(hip.) adj.: adjacenthip.: hypotenuse tan alpha=(opp.)/(adj.) opp.: oppositeadj.: adjacent Fundamental Identities sin^2 alpha + cos^2 alpha=1 tan alpha=(sin alpha)/(cos alpha) tan^2 alpha + 1 = 1/(cos^2 alpha) Law of Sines(aka sine rule) (sin A)/a = (sin B)/b = (sin C)/c Law of Cosines(aka cosine rule) a^2=b^2+c^2-2bc cos A Heron's formula A=sqrt(s(s-a)(s-b)(s-c))s=(a+b+c)/2 Exact Values sin (pi/6)=1/2 cos (pi/6)=sqrt(3)/2 tan (pi/6)=sqrt(3)/3 sin (pi/4)=sqrt(2)/2 cos (pi/4)=sqrt(2)/2 tan (pi/4)=1 sin (pi/3)=sqrt(3)/2 cos (pi/3)=1/2 tan (pi/3)=sqrt(3) Angle Relationships sin (-alpha)=-sin alpha cos (- alpha)=cos alpha tan (-alpha)=-tan alpha sin (pi - alpha)=sin alpha cos (pi - alpha)=-cos alpha tan (pi - alpha)=-tan alpha sin (pi + alpha)=-sin alpha cos (pi + alpha)=-cos alpha tan (pi + alpha)=tan alpha sin (pi/2 - alpha)=cos alpha cos (pi/2 - alpha)=sin alpha tan (pi/2 - alpha)=1/(tan alpha) sin (pi/2 + alpha)=cos alpha cos (pi/2 + alpha)=-sin alpha tan (pi/2 + alpha)=-1/(tan alpha) sin ((3pi)/2 - alpha)=-cos alpha cos ((3pi)/2 - alpha)=-sin alpha tan ((3pi)/2 - alpha)=1/(tan alpha) sin ((3pi)/2 + alpha)=-cos alpha cos ((3pi)/2 + alpha)=sin alpha tan ((3pi)/2 + alpha)=-1/(tan alpha) Trigonometric Equations sin x=sin alpha hArr x = alpha + 2kpi vv x = pi - alpha + 2kpi, k in ZZ  cos x=cos alpha hArr x = alpha + 2kpi vv x = - alpha + 2kpi, k in ZZ  tan x=tan alpha hArr x = alpha + kpi, k in ZZ  Sum Formulas sin (a+b)=sin a xx cos b + sin b xx cos a cos (a+b)=cos a xx cos b - sin a xx sin b tan (a+b)=(tan a + tan b) / (1 - tan a xx tan b) Difference Formulas sin (a-b)=sin a xx cos b - sin b xx cos a cos (a-b)=cos a xx cos b + sin a xx sin b tan (a-b)=(tan a - tan b) / (1 + tan a xx tan b) Double Angle Formulas sin (2a)=2xxsin a xx cos a cos (2a)=cos ^2 a - sin^2 a tan (2a)=(2 xx tan a) / (1 - tan^2 a)

#### Geometry

 Euler's Polyhedral Formula F + V = E + 2 F: FaceV: VertexE: Edge Sum of interior angles of a polygon S_i=(n-2)xx180º n: Number of sides Pythagorean theorem H^2=C_1^2+C_2^2 Hypotenuse: HLeg: C_1 e C_2 Distance between two points bar (AB)=sqrt((x_1-x_2)^2+(y_1-y_2)^2) ex: A(8,2) e B(4,-1)bar (AB)=sqrt((8-4)^2+(2+1)^2) hArrbar(AB)=sqrt(16+9) hArr bar(AB)=5 Midpoints M((x_1+x_2)/2,(y_1+y_2)/2) ex: A(2,6) e B(4,-2)M((2+4)/2,(6-2)/2) hArr M(3,2) Equation of a straight line Slope–intercept formSlope: m, Y intercept: b y=mx+b Vector FormDirection vector: vec u(u_1,u_2,u_3)Point(x_0,y_0,z_0) (x,y,z)=(x_0,y_0,z_0)+k(u_1,u_2,u_3), k in RR Cartesian FormDirection vector: vec u(u_1,u_2,u_3)Point(x_0,y_0,z_0) (x - x_0)/u_1=(y - y_0)/u_2=(z - z_0)/u_3 Parametric FormDirection vector: vec u(u_1,u_2,u_3)Point(x_0,y_0,z_0) {(x = x_0 + Ku_1),(y = y_0 + Ku_2),(z = z_0 + Ku_3):}, k in RR Equation of a plane Cartesian FormNormal vector: vec u(n_1,n_2,n_3)Point(x_0,y_0,z_0) n_1(x-x_0)+n_2(y-y_0)+n_3(z-z_0)=0 Scalar FormNormal vector: vec u(n_1,n_2,n_3) n_1x + n_2y + n_3z +d = 0 Equation of a circle Center (x_0,y_0) and radius r (x-x_0)^2+(y-y_0)^2=r^2 Equation of a Sphere Center (x_0,y_0,z_0) and radius r (x-x_0)^2+(y-y_0)^2+(z-z_0)^2=r^2 Equation of an Ellipse Center (h, k) Axis a and b ((x-h)/a)^2+((y-k)/b)^2=1

#### Logic

Conjunction      Disjunction      Implication
pqp ^^ q   pqp vv q   pqp rArr q
VVV   VVV   VVV
VFF   VFV   VFF
FVF   FVV   FVV
FFF   FFF   FFV
 Law of noncontradiction p ^^ ~p hArr F Law of the excluded middle p vv ~p hArr V Double Negation ~(~p) hArr p Commutativity Conjunction p ^^ q hArr q ^^ p Disjunction p vv q hArr q vv p Associativity Conjunction (p ^^ q) ^^ r hArr p ^^ (q ^^ r) Disjunction (p vv q) vv r hArr p vv (q vv r) Neutral Element Conjunction p ^^ V hArr p Disjunction p vv F hArr p Absorbing Element Conjunction p ^^ F hArr F Disjunction p vv V hArr V Idempotence Conjunction p ^^ p hArr p Disjunction p vv p hArr p Distributive Property Conjunction over Disjunction p ^^ (q vv r) hArr (p ^^ q) vv (p ^^ r) Disjunction over Conjunction p vv (q ^^ r) hArr (p vv q) ^^ (p vv r) Properties of Implication Transitive (p rArr q) ^^ (q rArr r) rArr (p rArr r) Implication and Disjunction (p rArr q) hArr ~p vv q Negation ~(p rArr q) hArr p ^^ ~q Contrapositive of an Implication (p rArr q) hArr (~q rArr ~p) Properties of Equivalence Double implication (p hArr q) hArr [(p rArr q) ^^ (q rArr p)] Transitive [(p hArr q) ^^ (q hArr r)] rArr (p hArr r) Negation ~(p hArr q) hArr [(p ^^ ~q) vv (q ^^ ~p)] De Morgan's laws Negation of a Conjunction ~(p ^^ q) hArr ~p vv ~q Negation of a Disjunction ~(p vv q) hArr ~p ^^ ~q De Morgan's laws Negation of Universal Quantifier ~(AAx, p(x)) hArr EEx: ~p(x) Negation of Existential Quantifier ~(EEx: p(x)) hArr AAx, ~p(x)

#### Vectors

 Notation vec(AB)=B - A = (b_1-a_1,b_2-a_2) ex : A(3,2) and B(4,5)vec(AB)=(4,5)-(3,2)=(4-3,5-2)=(1,3) Magnitude ||vec u||=sqrt((u_1)^2 + (u_2)^2) ex : vec u(3,2)||vec u||=sqrt(3^2+2^2) hArr ||vec u||=sqrt 13 Square of magnitude of a vector (vec u)^2 = ||vec u||^2 ex : vec u(4,3) and ||vec u||=5 then (vec u)^2 = 5^2 Calculations A+vec u=(a_1+u_1, a_2+u_2) ex : A(4,5) and vec u(3,2)A+vec u=(4+3, 5+2) hArr A+vec u=(7, 7) vec u+vec v=(u_1+v_1, u_2+v_2) ex : vec u(6,3) and vec v(2,1)vec u+vec v=(6+2, 3+1) hArr vec u+vec v=(8, 4) kxxvec u=(kxxu_1, kxxu_2) ex : k=2 and vec u(3,4)kxxvec u=(2xx3, 2xx4) hArr kxxvec u=(6, 8) The Scalar or Dot Product vec u.vec v=u_1xxv_1+u_2xxv_2 ex : vec u(2,1) and vec v(0,3)vec u.vec v=2xx0+1xx3vec u.vec v=3 vec u.vec v=||vec u||xx||vec v||xxcos(vec u \^ vec v) Angle between two lines Direction vector of lines: vec u and vec vangle: alpha cos alpha=|vec u.vec v|/(||vec u||xx||vec v||) To use the above concepts in space, just add a third coordinate.

#### Statistic

 Summation Rules and Properties sum_(i=p)^n lambda = (n-p+1)lambda sum_(i=1)^n lambda x_i = lambda sum_(i=1)^n x_i sum_(i=1)^n (x_i + y_i) = sum_(i=1)^n x_i + sum_(i=1)^n y_i sum_(i=1)^n x_i = sum_(i=1)^p x_i + sum_(i=p+1)^n x_i Used Symbols Statistical sample x = (x_1, x_2, x_3, ..., x_n) Sample size N Absolute Frequency n_i Relative Frequency f_i = n_i / N Cumulative (Absolute) Frequency N_i Cumulative Relative Frequency F_i Sample Mean Ungrouped Data bar(x) = (sum_(i=1)^k x_i)/N Grouped Data bar(x) = (sum_(i=1)^k n_i x_i)/N bar(x) = sum_(i=1)^k f_i x_i Median If N is odd Me = x_k, k = (N+1)/2 If N is even Me = (x_k + x_(k+1))/2, k = N/2 Sum of Deviationsfrom the Mean sum_(i=1)^k d_i = sum_(i=1)^k (x_i - bar(x)) = 0 Sum of Squared Deviationsfrom the Mean Ungrouped Data SS_x = sum_(i=1)^k (x_i - bar(x))^2 SS_x = sum_(i=1)^k x_i^2 - k bar(x)^2 Grouped Data SS_x = sum_(i=1)^k (x_i - bar(x))^2 n_i Sample Variance S_x^2 = (SS_x)/(N-1) Sample Standard Deviation S_x = sqrt((SS_x)/(N-1))

#### Sequences

 Arithmetic sequences Common difference r = u_(n+1) - u_n Expression for the nth term u_n=u_1+(n-1)r Monotonicity Increasing if r>0Decreasing if r < 0 Sum of the first n terms S_n=(u_1+u_n)/2xxn Geometric sequences Common ratio r = u_(n+1) / u_n Expression for the nth term u_n=u_1xxr^(n-1) Monotonicity Increasing if u_1>0 ^^ r>1Decreasing if u_1 < 0 ^^ r>1Not Monotonic if r < 0 Sum of the first n terms S_n=u_1xx(1-r^n)/(1-r) Simple Interest FV = P xx (1 + r xx t) FV : Future ValueP : Principalt : timer : interest rate Compound Interest FV = P xx (1 + r)^t

#### Derivatives

 Average rate of change between two points Slope of the Secant Line [a,b] SSL=(f(b)-f(a))/(b-a) Rate of change at a point f'(x_0)=lim_(x->x_0)(f(x)-f(x_0))/(x-x_0) f'(x_0)=lim_(h->0)(f(x_0+h)-f(x_0))/h Constant a'=0 ex : 4'=0 Multiplication by constant (mx)'=m ex : (3x)'=3 Power Rule (u^n)'=nxxu^(n-1)xxu' ex : ((6x)^5)'=5(6x)^4xx(6x)'=5(6x)^4xx6 Root (root(n)(u))'=(u')/(n xx root(n)(u^(n-1))) ex : (sqrt(2x))'=((2x)')/(2 xx sqrt(2x))=1/(sqrt(2x)) Exponential (a^u)'=u'xxa^uxxln a ex : (7^(3x))'=3xx7^(3x)xxln7 Exponential base e (e^u)'=u'xxe^u ex : (e^(2x))'=2xxe^(2x) Sum Rule (u+v)'=u'+v' ex : (2x+5)'=(2x)'+5'=2 Product Rule (uxxv)'=u'v + uv' ex : (x^2xxe^x)=(x^2)'e^x+x^2(e^x)'=2xe^x+x^2e^x Quotient Rule (u/v)'=(u'v - uv')/v^2 ex : ((x+1)/(2x))' = ((x+1)'xx(2x) - (x+1)xx(2x)')/(2x)^2 Chain Rule (g o f)'=g'(f) xx f' ex : g(x)=2x^2;g'(x)=4x;f(x)=2x;f'(x)=2      (gof)'=4(2x)xx2 Sine (sin u)'=u'xxcosu ex : (sin(6x))'=6xxcos(6x) Cosine (cos u)'=-u'xxsinu ex : (cos(3x))'=-3xxsin(3x) Tangent (tan u)'=(u')/(cos^2u) ex : (tan(x))'=1/(cos^2x) Logarithms (log_a u)'=(u')/(uxxln a) ex : (log_4 (6x))'=((6x)´)/(6xln 4)=6/(6xln 4)=1/(xln 4) Natural logarithm (ln u)'=(u')/(u) ex : (ln (5x))'=((5x)´)/(5x)=5/(5x)=1/x

#### Probability and Sets

 Commutative A uu B = B uu A A nn B = B nn A Associative A uu (B uu C) = A uu (B uu C) A nn (B nn C) = A nn (B nn C) Neutral element A uu O/ = A A nn E = A Absorbing element A uu E = E A nn O/ = O/ Distributive A uu (B nn C) = (A uu B) nn (A uu C) A nn (B uu C) = (A nn B) uu (A nn C) De Morgan's laws bar(A nn B) = bar(A) uu bar(B) bar(A uu B) = bar(A) nn bar(B) Laplace laws P(A) = text(Number of ways it can happen)/text(Total number of outcomes) Complement of an Event P(bar(A)) = 1 - P(A) Union of Events P(A uu B) = P(A) + P(B) - P(A nn B) Conditional Probability P(A | B) = (P(A nn B)) / (P(B)) Independent Events P(A | B) = P(A) P(A nn B) = P(A) xx P(B) Permutation P_n = n! = n xx (n - 1) xx ... xx 2 xx 1 ex : P_4 = 4! = 4 xx 3 xx 2 xx 1 = 24 Permutations without repetition text()^nA_p = (n!)/((n-p)!) ex : text()^6A_2 = (6!)/((6-2)!)=30 Permutations with repetition text()^nA_p^' = n^p ex : text()^5A_3^' = 5^3=125 Combination text()^nC_p = (text()^nA_p)/(p!)=(n!)/((n-p)! xx p!) ex : text()^5C_4 = (text()^5A_4)/(4!)=5 ProbabilityDistribution Average value mu = x_1p_1 + x_2p_2 + ... + x_kp_k Standard deviation sigma=sqrt(sum_(i=1)^k p_i(x_i-mu)^2 Binomial distribution P(X=k) = text()^nC_k.p^k.(1-p)^(n-k) ex : B(10;0,6)P(X=3) = text()^10C_3xx0,6^3xx0,4^7

#### logarithms

 Definition log_a b = x hArr b=a^x ex : 3^x=15 hArr x=log_3 15 log_a 1 = 0 ex : log_3 1 = 0 log_a a = 1 ex : log 10 = 1 log_a a^b = b ex : ln e^2 = 2 Product log_a (uxxv) = log_a u + log_a v ex : log_6 10 + log_6 2 = log_6 (10xx2) = log_6 20 Quotient log_a (u/v) = log_a u - log_a v ex : log_4 9 - log_4 3 = log_4 (9/3) = log_4 3 Exponential log_a u^v = vxxlog_a u ex : log_4 36 = log_4 6^2= 2xxlog_4 6 Change of Base log_a u = (log_b u)/(log_b a) ex : log_4 5 xx log_5 6 = log_4 5 xx (log_4 6)/(log_4 5) = log_4 6

#### Special Limits

 lim_(x->+oo) a^x/x^p = +oo      (a, p in RR) lim_(x->+oo) (log_a x) / x = 0      (a > 1, a in RR) lim_(x->0) (e^x - 1)/x = 1 lim_(x->0) (ln (x+1)) / x = 1 lim_(x->0) sin x/x = 1 lim_(x->+oo) sin x/x = 0 lim_(u_n->+oo)(1 + k/(u_n))^(u_n) = e^k lim (1 + 1/n)^n = e      (n in NN)

#### Integrals and primitives

 Common primitives int 1 dx = x + c, c in RR int (u(x))^alpha.u'(x) dx = ((u(x))^(alpha + 1))/(alpha + 1) + c, alpha in RR\\{0,-1}, c in RR int (u'(x))/(u(x)) dx = ln(abs(u(x))) + c, c in RR int e^u(x).u'(x) dx = e^u(x) + c, c in RR int sin(u(x)).u'(x) dx = - cos (u(x)) + c, c in RR int cos(u(x)).u'(x) dx = sin (u(x)) + c, c in RR Linearity rulesof integration int (f(x) + g(x)) dx = int f(x) dx + int g(x) dx int k.f(x) dx = k int f(x) dx Integration by parts(or partial integration) int u dv = uv - int v du Properties ofDefinite Integrals int_b^a f(x) dx = - int_a^b f(x) dx  int_a^a f(x) dx = 0 int_a^b f(x) dx = int_a^c f(x) dx + int_c^b f(x) dx int_a^b (f(x) + g(x)) dx = int_a^b f(x) dx + int_a^b g(x) dx int_a^b k.f(x) dx = k int_a^b f(x) dx Barrow's rule int_a^b f(x) dx = F(b) - F(a), where F is primitive from f in the interval [a,b]

#### Complex Numbers

 Algebraic Form Complex number z = a + bi Conjugate bar z = a -bi Symmetry -z = -a -bi Equality a + bi = c + di hArr a = c ^^ b = d Addition (a+bi)+(c+di)=(a+c)+(b+d)i Subtraction (a+bi)−(c+di)=(a−c)+(b−d)i Multiplication (a+bi)xx(c+di)=(ac−bd)+(ad+bc)i Division (a+bi)/(c+di)=(a+bi)/(c+di)xx(c−di)/(c−di)=(ac+bd)/(c^2+d^2)+(bc−ad)/(c^2+d^2)i Inverse z^-1 = 1/z z^-1 = 1/(|z|^2). bar z Properties bar bar z = z |z| = |bar z| |z|^2 = z.bar z Re(z) = (z + bar z)/2 Im(z) = (z - bar z)/(2i) Exponential to Algebraicform conversion Angle arg(z) = theta theta = tan^(-1)(b/a) Distance |z| |z| = sqrt(a^2 + b^2) Exponential form Complex number z = |z| . e^(i theta) z = |z| . (cos theta + i sin theta) Conjugate bar z = |z| . e^(i(-theta)) Symmetry -z = |z| . e^(i(theta + pi)) Multiplication z_1 = |z_1| . e^(i theta_1)   z_2 = |z_2| . e^(i theta_2) z_1 xx z_2 = |z_1| |z_2| . e^(i (theta_1 + theta_2)) Division z_1 / z_2 = |z_1| / |z_2| . e^(i (theta_1 - theta_2)) Exponentiation z^n = |z|^n . e^(i n theta) Radicals root(n)(|z| . e^(i theta)) = root(n)(|z|) . e^(i ((theta + 2 k pi)/n)), k in {0,...,n-1), n in NN

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