1.1 Numbers: Directed numbers, Types of numbers and Factors and Multiples

September 09, 2020

In this lesson, we are going to talk about:

1. Directed numbers and their practical uses
2. Types of numbers
3. Multiples and factors

Directed numbers and their practical use

Directed numbers are numbers we use to explain the world around us! And that includes the following:

- The height of this box is 40m
- The temperature outside is 18°
- My phone's battery is 50.8%

Types of numbers

1. Natural number - 1 and any number obtained by adding one to it repeatedly

1, 2, 3, 4, ...

2. Integer (positive, negative and zero) - A number that is not a fraction

- 2, -1, 0, 1, 2, ...

3. Prime number - An integer than one that cannot be divided by any integer (other than 1 and itself) to give another integer

2, 3, 5, 7, ...

4. Composite number - A quantity expressed in two different units

5. Rational number - An integer or a fraction

- 2, -⅓, 0, ¾, 1

N.B. 0.5 = ½, so 0.5 qualify to be a rational number but π doesn't qualify because it cannot be expressed as a fraction; π = ²²⁄₇

6. Irrational number - A real number that cannot be expressed as a rational number

π, e, √2

7. Real number - any rational or irrational number

All numbers that are part of rational and irrational numbers, e.g.

- 2, 0, 1, π

Common factors and common multiples

Now that we know of a few types of numbers, let's put our knowledge to use!

We are going to talk about multiples and factors; let's start with multiples!

Multiples
Given a number 2;
2×1 = 2
2×2 = 4
2×3 = 6

Then 2, 4, 6 are multiples of 2. We multiply a number 2 by natural numbers to find it's multiples.

Given another number 4;
4×1 = 4
4×2 = 8
4×3 = 12

Then 4, 8, 12 are multiples of 4. We multiply a number 4 by natural numbers to find it's multiples.

Generally:
Given any number n
n × 1 = n
n × 2 = 2_n

n× 3 = 3_n

Then n, 2_n, 3_n, ... are multiples of n!

Factors:
Given a number 6;
1×6 = 6
2×3 = 6

So 1, 2, 3, 6 are factors of 6. All integers that gives 6 when multiplied are factors of 6.

Given another number 8;
1×8 = 8
2×4 = 8
4×4 = 8

So 1, 2, 4, 8 are factors of 8. All integers that gives 8 when multiplied are factors of 8.

Generally:
Given a number n
If a×b = n, then a and b are factors.

Our knowledge of multiples and factors should help us to find the Lowest Common Multiples and Highest Common Factors

Highest Common Factor (HCF)

Examples on HCF:
1. Find the HCF of 6 and 16

Factors of 6 = {1, 2, 3, 6}
Factors of 16 = {1, 2, 4, 8, 16}
Common factors = {1, 2}
The greatest is 2 so HCF of 6 and 16 is 2

2. Find the HCF of 24 and 54

Factors of 24 = {1, 2, 3, 4, 6, 8, 12, 24}
Factors of 54 = {1, 2, 3, 6, 9, 18,27,54}
Common factors = {1, 2, 3, 6}
The greatest is 6 so HCF of 24 and 54 is 6

Exercises on HCF

1. Find the HCF between the following;
a. 2 and 6
b. 12 and 36
c. 54 and 62
Bonus: 72 and 108

Lowest Common Multiple (LCM)

Examples on LCM

1. Find the LCM of 4 and 6

Multiples of 4 = {4, 8, 12, 16, 20, 24, ...}
Multiples of 6 = {6, 12, 18, 24, 30, 36, ...}

We might have too many multiples, but we need the lowest. A few of them are {12, 24, ...}
The lowest multiple is 12.

2. Find the LCM of 5 and 7

Multiples of 5 = {5, 10, 15, 20, 25, 30, 35, ...}
Multiples of 7 = {7, 14, 21, 28, 35, ...}

The multiples are still numerous but we need the lowest. Check if 70 is a multiple too.
The lowest multiple is 35.

Exercises on LCM:
[Try this ones, if you encounter any problem, refer back to the examples. Good Luck!!!]
1. Find the LCM of the following;
a. 2 and 8
b. 3 and 5
c. 13 and 15
d. 26 and 17