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

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 ● i⁴ ● i⁴ ● i⁴ 
     = 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 ● a^m = a^n+m)

1 ● i²  = 1 ● -1

          = -1

Complex numbers

•         Adding and subtracting
•         Multiplying and dividing

Complex numbers have real and imaginary terms.

(4 +2i)

Adding

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.