Pure 1: Complex Numbers

Question 1

What is an imaginary number?

Answer 1

Any number ‘bi’ where ‘b’ is a real number

Question 2

What is i?

Answer 2

✔️-1

Question 3

What is a complex number?

Answer 3

Any number ‘a + bi’ where ‘a’ and ‘b’ are real numbers

Question 4

How are complex numbers added?

Answer 4

Add real and imaginary parts separately before combining

Question 5

How are complex numbers multiplied?

Answer 5

Multiply each part by every part of the other number (like expanding brackets)

Question 6

What are the powers of i? (5)

Answer 6

i^0 = 1
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1

Question 7

What is the discriminant? (4)

Answer 7

b^2 - 4ac
>0 = 2 real distinct solutions
=0 = 1 repeated solution
<0 = No real solutions

Question 8

What is a complex conjugate

Answer 8

For any complex number ‘z=a+bi’ the complex conjugate pair is ‘z*=a-bi’

Question 9

What always happens when a complex conjugate pair is multiplied together?

Answer 9

The answer is real

Question 10

How are complex numbers divided?

Answer 10

‘Rationalise’ the denominator (multiply by complex conjugate) and simplify as far as possible

Question 11

What are the possible variations of solutions of a cubic equation with real coefficients? (2)

Answer 11

Three real roots

One real root, one complex conjugate pair

Question 12

What are the possible variations of solutions of a quartic equation with real coefficients? (3)

Answer 12

4 real solutions
2 real solutions, 1 complex conjugate pair
2 complex conjugate pairs