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
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
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 ● am =
an+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
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)
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.
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