### Number Types

Before you get started, you will need the following skills:
• Foil (Algebra - factoring)
• Exponent rules
• How to plot a graph
• How to divide and find the remainder
• Surds
• The different number types and be able to tell the difference.
• What a complex number is.
• How to add/subtract and multiply/divide complex numbers

This ——> ● is a multiplication sign; i have used it in some places instead of x to avoid confusion

## Both rational and irrational are real numbers

### Rational numbers

Can be expressed as a decimal or a repeating decimal like 3.333333…. or a fraction like 3/1 or 2/3.

Rational numbers come in different types

Natural
1, 2, 3, 4, 5

Whole
0, 1, 2, 3, 4

Integer
…-3, -2, -1, 0, 1, 2, 3…

The difference between natural numbers and whole numbers is whole numbers includes 0 and natural numbers do not.  Integers have negatives, whereas whole and natural numbers do not.  Furthermore, all natural numbers are whole numbers and all whole numbers are integers.

### Irrational number

Can be expressed as a surd; it is a decimal with no repeating pattern.   π (Pi) and e are both examples of an irrational number. In addition, surds that are squares are not irrational for example; the square root of 25 is just 5.  However, the square root of 3 is irrational.

## Imaginary numbers

I will use an example to describe imaginary numbers.  What is the square root of -4? -2 squared would give 4, while 2 squared still gives 4.  The reason is that you cannot have a minus radicand.  √ is the radical, while the value under it is the radicand.

√-1 is an imaginary numbers called ‘i’.
√-1 = i
√-1 ● √-1 = -1  (because of the surd rule √a ● √a = a)

When ‘i’ is raised to the power of:

i0 = 1
i1 = i
i2 = -1
i3 = -i

This pattern repeats, which means you can divide big exponents by 4.

For example, i16 = 1.  This is because:
i4 =1,

i16 i4 ● i● i4 ● i
= 1 ● 1 ● 1 ● 1
= 1

Next example, i234

Divide the exponent by 4
234/4= 58 r 2  (the remainder forms the exponent)
I232 ● i2  is the same as i234  (exponent rule an ● am = an+m)
1 ● i2  = 1 ● -1
= -1

## Complex numbers

•         Multiplying and dividing

Complex numbers have real and imaginary terms.
(4 +2i)

Example 1
(4 +3i) + (2 -2i) = ?
Combine the like terms to get:
6 +i
You can now plot the above answer on a graph.

Example 2
(2 +√-16) + (3 +√-4)
You can tidy this up by multiplying √16 and √4 by √-1 give:
(2 (+√16)(√-1)) + (3 (+√4)(√-1)) = (2 + 4i) + (3 +2i)
= 5 +6i
Again, you can plot the answer on a graph.

### Subtracting

Example

(4 -5i) – (2 -2i)

Note the minus sign will change the signs in the second set of brackets.

4 -+2 = 2
5i - -2i = 7i
2 +7i is the final answer, which can be plotted on a graph.

### Multiplying

Use foil
(2 +3i)(3-2i)
F  = 2 x 3 = 6
O = 2 x -2i = -4i
I   = 3i x 3 = 9i
L  = 3i x -2i = -6i

6 -4i +9i -6i2 now combine the like terms
6 +5i -6i2 Remember that i2 =-1; therefore, -6 x -1 = 6
6 +5i +6 combine the like terms
12 +5i is the final answer and can be plotted on a graph.

### Dividing

(3 -2i)/(2 -4i)
Multiply both denominator and numerator by the denominator’s conjugate

Top (numerator)
F)  3 x 2 = 6
O) 3 x 4i = 12i
I)  -2i x 2 = -4i
L) -2i x 4i = -8i2

6 +12i -4i -8i2  The i squared is = -1
6 +12i -4i -8(-1)
6 +12i -4i +8 combine the like terms to give
14 +16i

Bottom (denominator)
F)  2 x 2 = 4
O) 2 x 4i = 8i
I) -4i x 2 = -8i
L) -4i x 4i =16i2

4 +8i -8i +16i2  the i squared is = -1
4 +8i -8i +16(-1)
4 +8i -8i -16  combine the like terms
-11

You cannot simplify this any further.

Until next time peace.