Complex numbers

Question 1

x² = -1 has no real solution so we introduce

√-1 = i

i² =

Answer 1

i² = -1

Question 2

√-9 =

Answer 2

√-9 = √9√-1 = 3i

Question 3

What is the standard form for a complex number?

Answer 3

z = a + bi

a is real part

bi is imaginary part

Question 4

What are the rules with adding/subtractin and multiplying complex numbers?

Answer 4

For addition and subtraction you group like terms (real and imaginary).

e.g. (a + bi) + (c + di) = (a+c) + (b+d)i

For multiplicaiton, use FOIL.

Question 5

What is the conjugate (z) of z = a + bi?

zz = ?

How can this result be used?

Answer 5

Conjugate z = a - bi

zz = a² + b²

The conjugate is most useful when there is a complex number in the denominator of a fraction. Multiplying by the conjugate removes the complex number,

Question 6

zz = ?

Answer 6

zz = a² + b²

Question 7

How can you obtain the product of 2 complex numbers?

Answer 7

1. Multiply moduli
2. Add arguments

Question 8

How can you obtain the quotient of two complex numbers?

Answer 8

1. Divide the moduli.
2. Subtract the arguments

Question 9

What is the formula for the power of a complex number?

Answer 9

zⁿ = rⁿ(cos nØ +isin nØ)

Question 10

What is the modulus of z = a + bi

Answer 10

|z| = √(a² + b²)

Question 11

Binomial theorem is…

Answer 11

(a+b)ⁿ = ⁿC_{<strong>k</strong>}aⁿ + ⁿC_ka^n-1b + ⁿC_ka^n-2b² + ⁿC_kbⁿ

^e.g.

(a+b)⁴ = 1a⁴ + 4a³b + 6a²b² + 4ab³ + 1b⁴

Remember Pascals triangle and you’ll be right.

Question 12

Pascals triangle

Answer 12

Question 13

How do you find the roots of a complex number?

Answer 13

1. Rearrange to remove √. eg √-4 => z² = -4
2. Find modulus (√(a² +b²) and angle (graph).

θ = angle + 2kπ

3. Write in exponential form. re^iθ
   e. g. z² = 4e^{i(π + 2kπ)}
4. Sove for z
   eg. z = 2e^{i(π/2 + kπ)}
5. Sub in k-values starting from 0 to find roots.