Complex numbers

Imaginary number

# Assume that
i² = −1
# therefore the imaginary unit is
i = √(-1)

Powers of Imaginary Unit:

The pattern is from i to i⁴, then it repeats.
To evaluate iⁿ, just to get the remainder n % 4 and see which term it is in the pattern. e.g., i¹⁰ is equal to -1 because 10 % 4 = 2 which is 2nd term in the pattern.

Complex Plane

Conjugates

Its mainly usage is for the complex number’s division.

Forms of complex number

refer to khan academy article.

• Rectangle form

This form is good for plotting in the complex plane.

• Polar form

This form only refers to the Absolute value and Angle,
which are also called the Modulus and the Argument.

• Exponential form

Its reasoning is too complicated, so just to remember it.
This form is very good for complex number's multiplication and division or further operations.

Operations of complex numbers

It’s easier just see the i as an unknown variable, then to do normal operations.
Refer to maths is fun.

• Addition & Subtraction:

• Multiplication:

• Division:

Need to use the notion conjugates

• Powers:

Better to convert it to Exponential form first, and do powers.

Convert between Rectangle form & Polar form

FORMS AND TECHNIQUES HERE, ARE RATHER APPLICABLE TO MANY TOPICS THAN MERELY FOR COMPLEX NUMBER. IT’S ACTUALLY FOR ALL TRIGONOMETRY RELATED TOPICS!!

Absolute value of complex number

The absolute value of it literarily means the DISTANCE from the point of complex number to the origin.

How to understand this term?

IT SPEND ME A LOT OF TIME TO UNDERSTAND AT THE FIRST TIME.
In the example 5+6i,
to calculate the absolute value of it |5+6i|, I know it's using the Pythagorean theorem.

But why it ISN’T doing √(5²+6²i²) instead of √(5²+6²)?
It is the most confusing part, and hard to consult the Internet with a clear answer that why do we Take out the i from calculating the distance.

The simple answer is that:
No matter it’s i or i², it is either √(-1) or -1, in another word it actually is just a NEGATIVE ONE,
but when we're counting a DISTANCE, it's always a POSITIVE, so we have to CANCEL OUT the -1 from the number. That's the point we could ignore it when doing absolute value.

Angle of complex number

Note:
It only regards to the tangent knowledge of trig functions.
Better to review tangent identities to find all possible solutions of tan⁻¹(θ).

tan(θ) = tan(-π + θ)

Steps:

• Find all solutions for angle θ.
• Plot rectangle form of complex number and figure out the Quadrant.
• Select the right solution at the right quadrant.

Example: Find angle for z = −3 − 6i
Solve:

• Principle solution: θ = arctan(-6/-3) = 1.107 which is at 1st Quadrant.
• Another pair solution would be -π + θ = -2.03 which is at 3rd Quadrant.
• -3-6i should be at the 3rd Quadrant in complex plane, so answer is -2.03.

For some exact values, we also need the tangent values of unit circle:

Convert between Exponential form & Polar form

Example: Find the solution of z⁴ = -625 in Rectangle form and Polar form, which argument in [270°, 360°]
Solve:
It's actually a process of conversion:

• -625 could be represented as -625 +0i in Rectangle form
• Modulus r is √(-625²+0²) which is equal to 625.
• tanθ = 0/-625 = 0, check the tangent unit circle table to know the angle could be 0° or 180°, according to its rectangle form, it's certain to be on the negative X-axis, so it's 180°.
• Add all possible solutions and get the angle is 180° + n*360°.
• Polar form is: 625[cos(180° + n*360°) + i*sin(180° + n*360°)]
• since the complex number is z⁴, so:
• Modulus = ⁴√625 = 5
• Argument = (180° + n*360°) / 4 = (45°+n*90)
• Range of argument is [270°, 360°], so the argument is 315°
• So the Polar form is: 5(cos315 + i*sin315).
• Rectangle form is: 5√2/2 - i*5√2/2

Powers of complex numbers

Example

Solve: