Complex numbers are beneficial to finding the square roots of negative numbers.

A complex number is a sum of a real number and a imaginary number.

Imaginary numbers:

sqrt(-1)=i

i^2=-1

i^3=-i

i^4=1

Modulus of a complex number:

In the imaginary plane, a complex number point is (a,ib) and the distance between that point to (0,0) is r = |√a^2+b^2|

The point that is plotted and the origin point forms a right triangle, and the angle (θ) that is nearest to the origin will be trigged. Arctan(b/a)=argz(θ), arg is the argument (θ) angle of the thing.

Use that, and the length of the line on the origin is rsinθ and the height is rcosθ.

The complex number z=a+ib can be represented as z=r(cosθ+isinθ) inn polar form.

Conjugate of a complex number:

z=a+ib

=a-ib

z+=(a+ib)+(a-ib)

z=a^2+b^2

Reciprocal:

z^-1=i(-b)/a^2+b^2

Two complex numbers will be equal if there constant components are equal.

Euler's formula:

Addition of complex numbers:

z1=a+ib, z2=c+id

z1+z2=(a+c)+i(b+d)

Closure law:

if z1 and z2 are complex numbers, their sum is also a complex number.

Commutative law:

z1+z2=z2+z1

associative law:

Additive identity:

z + 0 = 0 + z = 0

Additive inverses:

z=a+ib

-z=-a-ib

z + (-z) = (-z) + z = 0

Subtraction of complex numbers:

Multiplication of complex numbers:

z1=a+ib. z2=c+id

z1z2= (ca - bd) + i(ad + bc)

Division of complex numbers:

Algebraic complex formulas: