Let’s do some Further Maths

What sorts of numbers do we already know about?

Natural numbers: 1,2,3,4...

Integers: 0, -1, -2, -3

Rational numbers: ½, ¼,... Irrational numbers: π, √2

Let’s do some Further Maths

Natural numbers: 1,2,3,4...

Integers: 0, -1, -2, -3

Rational numbers: ½, ¼,... Irrational numbers: π, √2

Complex numbers

Why would we ever need complex numbers? Solving quadratic equations: ax2+bx+c=0

Solving cubic equations square roots of negative numbers appear in

intermediate steps even when the roots are real Fundamental Theorem of Algebra- every

polynomial of degree n has n roots

We only need one new number

We define the number i = √-1 so that i2=-1 All square roots of negative numbers can be

written using i √-4 = √4 x √-1 = 2i

Exercise

Write the following using i:1. √-36 2. √-121 3. √-10 4. √-18 5. -√-75

We treat i a little like xExerciseSimplify:1. 3i + 2i 2. 16i – 5i 3. (2i)(3i) 4. i(4i)(6i)

5. Copy and complete: i0 =i1 =i2 =i3 =i4 =i5 =i6 =

6. i12 = 7. i25 = 8. i1026 =

Complex Numbers Multiples of i are imaginary numbers Real numbers can be added to imaginary numbers

to form complex numbers like 3+2i or -1/2 -√2i We can add, subtract and multiply complex

numbers (dividing is a little more complicated!) Exercise1. (3+2i) + (-1-4i) = 2. (2-i) – (1+5i) =

3. (2-2i)(1+3i) =

Where are complex numbers on the number line?

Is i positive or negative? Suppose i > 0. Then i2 > 0. In other words, -1 > 0.

So i < 0. Then i + (-i) < 0 +(-i). So –i > 0. And (-i)2 > 0. In other words, -1 > 0.

We need a 2-D number line!

A

B

CD

EF

G

H

I

Exercise

Name the complex numbers represented by each point on the Argand diagram.

What will squaring do to this man?

Oops!