Complex numbers

Question 1

What is the imaginary unit i defined as?

Answer 1

i = √(-1)

Question 2

What is the pattern for powers of i?

Answer 2

i¹ = i
i² = -1
i³ = -i
i⁴ = 1
Then the pattern repeats every 4 powers

Question 3

How do you find i²⁵

Answer 3

Divide 25 by 4, remainder is 1
So i²⁵ = i¹ = i

Question 4

Simplify √(-16)

Answer 4

√(-16)
= √(16 × -1)
= √16 × √(-1)
= 4i

Question 5

What is the standard form of a complex number?

Answer 5

a + bi, where a is the real part and b is the imaginary part

Question 6

What is a purely real complex number?

Answer 6

A complex number where Im(z) = 0, like z = 7

Question 7

What is a purely imaginary complex number?

Answer 7

A complex number where Re(z) = 0, like z = -3i

Question 8

How do you add complex numbers?

Answer 8

Add real parts together and imaginary parts together: (a + bi) + (c + di) = (a + c) + (b + d)i

Question 9

How do you multiply complex numbers?

Answer 9

Use FOIL and remember

Question 10

How do you divide complex numbers?

Answer 10

Multiply numerator and denominator by the conjugate of the denominator

Question 11

What is the conjugate of z = a + bi?

Answer 11

z̄ = a - bi (change the sign of the imaginary part)

Question 12

What is z + z̄ equal to?

Answer 12

2a (a real number - the imaginary parts cancel)

Question 13

What is z × z̄ equal to in rectangular form?

Answer 13

a² + b² (always real and positive)

Question 14

What is z - z̄ equal to?

Answer 14

2bi (imaginary number - the real parts cancel)

Question 15

What does the conjugate of the conjugate return?

Answer 15

z

Question 16

What are the axes on an Argand diagram?

Answer 16

Horizontal axis = Real axis
Vertical axis = Imaginary axis

Question 17

How do you plot z = 3 + 2i on an Argand diagram?

Answer 17

Plot the point (3, 2) - 3 units right, 2 units up

Question 18

Where is z = -2 + 4i on the Argand diagram?

Answer 18

In Quadrant 2 at point (-2, 4)

Question 19

What is the formula for modulus of z = a + bi?

Answer 19

|z| = √(a² + b²)

Question 20

What does the modulus represent geometrically?

Answer 20

The distance from the origin to the point on the Argand diagram

Question 21

What is |z̄| compared to |z|?

Answer 21

|z̄| = |z|
(conjugates have the same modulus)

Question 22

What is |z₁ × z₂| equal to?

Answer 22

|z₁| × |z₂|

Question 23

What is |z₁ / z₂| equal to?

Answer 23

|z₁| / |z₂|

Question 24

When is |z| = 0

Answer 24

When z = 0

Question 25

Why is |z| >= 0

Answer 25

It's a square root of squared terms (always positive or zero) It represents distance (which can't be negative) The only exception is z = 0, where |z| = 0

Question 26

What is the argument of a complex number?

Answer 26

The angle θ between the positive real axis and the line from origin to z

Question 27

What is the formula for finding the argument?

Answer 27

an(θ) = b/a, then θ = arctan(b/a) Adjusted for the correct quadrant

Question 28

How do you determine which quadrant adjustment to use?

Answer 28

Look at the signs of a and b: Q1 (a > 0, b > 0): use θ directly Q2 (a < 0, b > 0): θ = 180° + arctan(b/a) Q3 (a < 0, b < 0): θ = 180° + arctan(b/a) Q4 (a > 0, b < 0): use θ directly (negative)

Question 29

When do you use quadrant adjustments?

Answer 29

Only when a point is inside a quadrant, not on an axis

Question 30

What is the principal argument range?

Answer 30

-180° < θ ≤ 180°

Question 31

What is the polar form of a complex number?

Answer 31

z = r(cos θ + i sin θ) z = r cis θ

Question 32

In polar form z = r cis θ, what are r and θ?

Answer 32

r = |z| and θ = argument

Question 33

How do you convert from rectangular to polar form?

Answer 33

1. Find r = √(a² + b²), 2. Find θ = arctan(b/a) adjusted for quadrant 3. Write z = r cis θ

Question 34

How do you convert from polar to rectangular form?

Answer 34

1. Calculate a = r cos θ and b = r sin θ 2. Write z = a + bi

Question 35

What happens to the argument of z = 0?

Answer 35

The argument is undefined (no direction from origin to itself)

Question 36

What is method 1 of graphing polar form?

Answer 36

1. Start at the origin 2. Rotate ccw by θ from the positive real axis 3. Move outward distance of r units in that direction

Question 37

What is method 2 of graphing polar form?

Answer 37

Convert to rectangular form

Question 38

What is the conjugate of z = r cis θ in polar form?

Answer 38

z̄ = r cis(-θ) (same modulus, negative angle)

Question 39

Why is the conjugate r cis(-θ)?

Answer 39

Because conjugate reflects across the real axis, which flips the angle from θ to -θ while keeping distance r the same

Question 40

What is z × z̄ equal to in polar form?

Answer 40

r²

Question 41

How do you multiply two complex numbers in polar form?

Answer 41

(r₁ cis θ₁)(r₂ cis θ₂) = r₁r₂ cis(θ₁ + θ₂)

Question 42

How do you divide two complex numbers in polar form?

Answer 42

(r₁ cis θ₁)/(r₂ cis θ₂) = (r₁/r₂) cis(θ₁ - θ₂)

Question 43

What is the modulus of z when written as z = r cis θ?

Answer 43

The modulus is r (it's already shown directly in polar form)

Question 44

What is De Moivre's Theorem?

Answer 44

(r cis θ)ⁿ = rⁿ cis(nθ)

Question 45

What is Euler's Formula?

Answer 45

re^(iθ)

Question 46

Convert rectangular to Euler's

Answer 46

1. Calculate r 2. Calculate argument

Question 47

Convert Euler's to rectangular

Answer 47

1. Find a & b 2. z = r(cos θ + i sin θ) where a = r cos θ, b = r sin θ

Question 48

What is the general formula for nth roots in polar form?

Answer 48

ⁿ√r x cis[ (θ+2πk) / n ]

Question 49

How many nth roots does a non-zero complex number have?

Answer 49

Exactly n distinct roots

Question 50

What values of k do you use to find all nth roots?

Answer 50

k = 0, 1, 2, ..., n - 1 There are n cases

Question 51

How are the nth roots geometrically arranged?

Answer 51

Evenly spaced around a circle of radius ⁿ√r, separated by angles of 2π/n radians

Question 52

When does a quadratic equation have complex roots?

Answer 52

When D < 0

Question 53

What does cis(2π) = cis(0) mean?

Answer 53

Complex numbers are periodic with period 2π. Adding 2π to the angle gives the same complex number

Question 54

What do exponents in complex numbers represent geometrically?

Answer 54

Rotation and scaling. e^(iθ) represents a rotation by angle θ. r·e^(iθ) scales by r and rotates by θ.

Question 55

How do you multiply complex numbers using exponents?

Answer 55

Multiply the moduli and add the angles.

Question 56

How do you solve a quadratic equation with complex roots?

Answer 56

Use the quadratic formula. When discriminant < 0, simplify √(negative) = i√(positive).

Question 57

What is Euler's Identity?

Answer 57

e^(iπ) + 1 = 0

Question 58

What does arg(z) represent?

Answer 58

arg(z) is the argument (or angle θ) of z

Question 59

Method: If a + bi = 1 - 8i, how do you solve for a and b?

Answer 59

Equate real and imaginary parts separately

Question 60

arg(z₁z₂) is equal to what?

Answer 60

arg(z₁) + arg(z₂)

Question 61

arg(z₁/z₂) is equal to what?

Answer 61

arg(z₁) - arg(z₂)