List of Math Formulas

2023-05-29 22:53| 来源: 网络整理| 查看: 265

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` : width`h` : height Triangle`A=(bxxh)/2``b` : base`h` : height Rhombus`A=(Dxxd)/2``D` : large diagonal`d` : small diagonal Trapezoid`A=(B+b)/2xxh``B` : large side`b` : small side`h`: height Regular polygon`A=P/2xxa``P` : perimeter`a` : apothem Circle`A=pir^2``P=2pir``r` : radius`P` : perimeter Cone(lateral surface)`A=pirxxs``r` : radius`s` : slant height Sphere(surface area)`A=4pir^2``r`: radius Save Formulas Like it? Share it! Volumes Cube`V=s^3``s`: side Parallelepiped`V=lxxwxxh``l`: length`w`: width`h`: height Regular prism`V=bxxh``b`: base`h`: height Cylinder`V=pir^2xxh``r`: radius`h`: height Cone (or pyramid)`V=1/3bxxh``b`: base`h`: height Sphere`V=4/3pir^3``r`: radius Save Formulas Like it? Share it! 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)` 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` Save Formulas Like it? Share it! 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=3^5`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)` Save Formulas Like it? Share it! Radicals 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)` Save Formulas Like it? Share it! Trigonometry Trigonometry Ratios`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)` 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)` Save Formulas Like it? Share it! Geometry Euler's Polyhedral Formula`F + V = E + 2``F`: Face`V`: Vertex`E`: 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: `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 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 planeCartesian 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 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` Save Formulas Like it? Share it! Logic 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)` Save Formulas Like it? Share it! 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+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. Save Formulas Like it? Share it! 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 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 Deviationsfrom the Mean`sum_(i=1)^k d_i = sum_(i=1)^k (x_i - bar(x)) = 0` Sum of Squared Deviationsfrom the MeanUngrouped 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))` Save Formulas Like it? Share it! Sequences 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` Save Formulas Like it? Share it! Derivatives 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` Save Formulas Like it? Share it! 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` ProbabilityDistributionAverage 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` Save Formulas Like it? Share it! 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` Save Formulas Like it? Share it! 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)` Save Formulas Like it? Share it! 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]` Save Formulas Like it? Share it! Complex Numbers 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 Algebraicform conversionAngle`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` Save Formulas Like it? Share it!

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