Complex Numbers

What does i equal?

√-1

What does i² equal?

-1

How to get the modulus of a complex number?

Square the individual Numbers. Get rid of the i. Square root

How to add and subtract real/imaginary numbers?

Add real with real and imaginary with imaginary

If a complex number is multiplied by i. How does it affect it when plotted on an argand diagram?

Rotation through 90 degrees anti-clockwise about the origin

If a complex number is multiplied by -i. How does it affect it when plotted on an argand diagram?

Rotation through 90 degrees clockwise about the origin

What is Re()?

The real part of a complex number

How to get the conjugate of a complex number?

Change the sign of the imaginary part

Symbol of a complex number

z

Symbol of the conjugate of a complex number

ˉz

Relationship between a complex number and its conjugate when plotted

Conjugate is the image of the complex number by axial symmetry in the real axis

Polar form of a complex number

z = r(Cosθ + iSinθ)

What is r in the polar form of a complex number?

The modolus of the complex number

How to get theta in the polar form of a complex number?

Argument of z

What is the argument of z?

Anticlockwise angle from the positive x axis to the line from the origin to z

How to get θ if z is in first quadrant?

Find angle in triangle made by line segment of z

How to get θ if z is in second quadrant?

θ = π - angle made by z

How to get θ if z is in third quadrant?

θ = π + angle made by z

How to get θ if z is in fourth quadrant?

θ = 2π - angle made by z

Formula for multiplying numbers in polar form

z1z2 = r1r2 (Cos(Q1+Q2) + iSin(Q1+Q2))

How to multiply numbers in polar form?

Multiply the modulus and add arguments

Dividing numbers in polar form formula

z1/z2 = r1/r2 (Cos(Q1 - Q2) + iSin(Q1 - Q2))

How to divide numbers in polar form?

Divide the modulus and minus arguments

How to divide a complex number by a real number?

Write each number in the complex number as a fraction with the real number and simplify

How to divide a complex number by another complex number?

Multiply top and bottom by the conjugate of the bottom

De Moivre's Theorem

z^n = r^n (Cosnθ + iSinnθ)

General polar form of a complex number

r(Cos(θ + 2nπ) + i Sin(θ + 2nπ))

How do you now when to stop subbing numbers in when finding the solutions of a complex number?

Same amount of solutions as the number the z is brought up to start by subbing in 0 and keep going until solution amount is reached

De Moivre's Theorem Proof (Base Case)

(r(Cosθ + iSinθ))^1 = r^1(Coseθ + iSinθ)^1 = r(Coseθ+iSinθ)

De Moivre's Theorem Proof (Assume for n=k k>=1)

(r(Cosθ+ iSinθ))^k = r^k(Coskθ + iSinkθ)

De Moivre's Theorem Proof (Prove n = k+ 1)

(r(Cosθ + iSinθ))^k+1 = (r(Cosθ + iSinθ))^k . (r(Cosθ + iSinθ)) = r^k(Coskθ+iSinkθ) . r(Cosθ+ iSinθ) = r^k+1 (CoskθCosθ + i CoskθSinθ + i SinkθCosθ - 1sinkθsinθ) = r^k+1 (CoskθCosθ - SinkθSinθ + i (CoskθSinθ + SinkθCosθ) = r^k+1 (Cos(kθ+θ) + iSin(kθ+θ) = r^k+1 (Cos(k+1)θ + i SIn(k+1)θ))