List of math formulas

notebook

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 the formulas used in mathematics, only those that were considered most important.

SquareSquare`A=l^2``l` : length of side
RectangleRectangle`A=wxxh``w` : width
`h` : height
TriangleTriangle`A=(bxxh)/2``b` : base
`h` : height
RhombusRhombus`A=(Dxxd)/2``D` : large diagonal
`d` : small diagonal
TrapezoidTrapezoid`A=(B+b)/2xxh``B` : large side
`b` : small side
`h`: height
Regular polygonRegular polygon`A=P/2xxa``P` : perimeter
`a` : apothem
CircleCircle`A=pir^2`
`P=2pir`
`r` : radius
`P` : perimeter
Cone
(lateral surface)
cone`A=pirxxs``r` : radius
`s` : slant height
Sphere
(surface area)
sphere`A=4pir^2``r`: radius
CubeCube`V=s^3``s`: side
ParallelepipedParallelepiped`V=lxxwxxh``l`: length
`w`: width
`h`: height
Regular prismPrism`V=bxxh``b`: base
`h`: height
CylinderCylinder`V=pir^2xxh``r`: radius
`h`: height
Cone (or pyramid)Cone`V=1/3bxxh``b`: base
`h`: height
SphereSphere`V=4/3pir^3``r`: radius
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)`
ConcavityConcave 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`ConcavityConcave 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`
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=3^5`
ex : `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)^2`
ex : `(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 Ratiosright triangle`sin alpha=(opp.)/ (hip.)``opp.`: opposite
`hip.`: hypotenuse
`cos alpha=(adj.)/(hip.)``adj.`: adjacent
`hip.`: hypotenuse
`tan alpha=(opp.)/(adj.)``opp.`: opposite
`adj.`: adjacent
Fundamental Identities`sin^2 alpha + cos^2 alpha=1``tan alpha=(sin alpha)/(cos alpha)``tan^2 alpha + 1 = 1/(cos^2 alpha)`
acutangle triangleLaw 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)`
Euler's Polyhedral Formula`F + V = E + 2``F`: Face
`V`: Vertex
`E`: Edge
Sum of interior angles of a regular polygon`S_i=(n-2)xx180º``n`: Number of sides
Pythagorean theorem`H^2=C_1^2+C_2^2`Hypotenuse: `H`
Leg: `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) hArr`
`bar(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 lineSlope–intercept form
Slope: `m`, Y intercept: `b`
`y=mx+b`
Vector Form
Direction 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 Form
Direction 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 Form
Direction 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 planeCartesian Form
Normal 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 Form
Normal vector: `vec u(n_1,n_2,n_3)`
`n_1x + n_2y + n_3z +d = 0`
Equation of a circleCenter `(x_0,y_0)` and radius `r``(x-x_0)^2+(y-y_0)^2=r^2`
Equation of a SphereCenter `(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 EllipseCenter `(h, k)` Axis `a` and `b``((x-h)/a)^2+((y-k)/b)^2=1`
      Conjunction      Disjunction      Implication
   `p``q``p ^^ q`   `p``q``p vv q`   `p``q``p 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`
CommutativityConjunction`p ^^ q hArr q ^^ p`
Disjunction`p vv q hArr q vv p`
AssociativityConjunction`(p ^^ q) ^^ r hArr p ^^ (q ^^ r)`
Disjunction`(p vv q) vv r hArr p vv (q vv r)`
Neutral ElementConjunction`p ^^ V hArr p`
Disjunction`p vv F hArr p`
Absorbing ElementConjunction`p ^^ F hArr F`
Disjunction`p vv V hArr V`
IdempotenceConjunction`p ^^ p hArr p`
Disjunction`p vv p hArr p`
Distributive PropertyConjunction 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 ImplicationTransitive`(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 EquivalenceDouble 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 lawsNegation of a Conjunction`~(p ^^ q) hArr ~p vv ~q`
Negation of a Disjunction`~(p vv q) hArr ~p ^^ ~q`
De Morgan's lawsNegation of Universal Quantifier`~(AAx, p(x)) hArr EEx: ~p(x)`
Negation of Existential Quantifier`~(EEx: p(x)) hArr AAx, ~p(x)`
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+1xx3`
`vec u.vec v=3`
`vec u.vec v=||vec u||xx||vec v||xxcos(vec u \^ vec v)`
Angle between two linesDirection vector of lines: `vec u` and `vec v`
angle: `alpha`
`cos alpha=|vec u.vec v|/(||vec u||xx||vec v||)`
To use the above concepts in space, just add a third coordinate.
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 SymbolsStatistical 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 MeanUngrouped 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`
MedianIf 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 Deviations
from the Mean
`sum_(i=1)^k d_i = sum_(i=1)^k (x_i - bar(x)) = 0`
Sum of Squared Deviations
from 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))`
Arithmetic sequencesCommon difference`r = u_(n+1) - u_n`
Expression for the nth term`u_n=u_1+(n-1)r`
MonotonicityIncreasing if `r>0`
Decreasing if `r < 0`
Sum of the first n terms`S_n=(u_1+u_n)/2xxn`
Geometric sequencesCommon ratio`r = u_(n+1) / u_n`
Expression for the nth term`u_n=u_1xxr^(n-1)`
MonotonicityIncreasing if `u_1>0 ^^ r>1`
Decreasing if `u_1 < 0 ^^ r>1`
Not 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 Value
`P` : Principal
`t` : time
`r` : interest rate
Compound Interest`FV = P xx (1 + r)^t`
Average rate of change between two pointsSlope 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`
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`
Probability
Distribution
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`
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`
`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)`
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 rules
of 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`
Properties of
Definite 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]`
Algebraic FormComplex 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 Algebraic
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Angle`arg(z) = theta``theta = tan^(-1)(b/a)`
Distance`|z|``|z| = sqrt(a^2 + b^2)`
Exponential formComplex 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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