### 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

You will learn about:

- 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

##
__Both rational and irrational are real numbers__

__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__

__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:

i

^{0}= 1
i

^{1}= i
i

^{2}= -1
i

^{3 }= -i
This pattern repeats, which means you can divide big
exponents by 4.

**For example**, i

^{16 }= 1. This is because:

i

i

= 1 ● 1 ● 1 ● 1

^{4}=1,i

^{16 }= i^{4}● i^{4 }● i^{4}● i^{4 }= 1 ● 1 ● 1 ● 1

= 1

**Next example**, i

^{234 }

Divide the exponent by 4

234/4= 58 r 2 (the
remainder forms the exponent)

I

^{232}● i^{2 }is the same as i^{234 }(exponent rule a^{n }● a^{m}= a^{n+m})
1 ● i

^{2}= 1 ● -1
= -1

##
__Complex numbers__

__Complex numbers__

- Adding and subtracting
- Multiplying and dividing

Complex numbers have real and
imaginary terms.

(4 +2i)

###
__Adding__

__Adding__

*Example 1*
(4 +3i) + (2 -2i) = ?

Combine the like terms to get:

6 +i

*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 __

__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 __

__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 -6i

^{2}now combine the like terms
6 +5i -6i

^{2}Remember that i^{2}=-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__

__Dividing__

(3 -2i)/(2 -4i)

**Top (numerator)**

F) 3 x 2 = 6

O) 3 x 4i = 12i

I) -2i x 2 = -4i

L) -2i x 4i = -8i

^{2}
6 +12i -4i -8i

^{2}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 =16i

^{2}
4 +8i -8i +16i

^{2}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.

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