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 1

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 2

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 3

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 4

We treat i a little like x Exercise Simplify: 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 = 5

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!) Exercise 1. (3+2i) + (-1-4i) = 2. (2-i) – (1+5i) = 3. (2-2i)(1+3i) = 6

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.

We need a 2-D number line! C D F B E Exercise Name the complex numbers represented by each point on the Argand diagram. A G I H

What will squaring do to this man?

Oops!