1. 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

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 2

3. 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

4. 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. √ √ √ √ √-75 4

5. 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

6. 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

7. Where are complex numbers on the number line?

8. 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.

9. 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

10. What will squaring do to this man?

11. Oops!