Objectives: By the end of this subtopic learners should be able to:
Identify the different types of numbers.
Use the number line to manipulate number problems.
Manipulate word problems involving different types of numbers.
Demonstrate familiarity with common fractions, decimal fractions and percentages and convert from one to the other using three forms in appropriate contexts.
Types of numbers
Numbers can be classified according to the properties they have, this brings us to the different types of numbers and these types are discussed below;
Rational numbers
Any number that can be expressed in the form $\frac{x}{y}$ where x and y are both integers are called rational numbers and this include 0. This means that rational numbers include integers, fractions.
Examples
- $\frac{5}{7}$; -2; 0; $\frac{3}{5}$; 0,6; 7
Irrational numbers
Those numbers which are not exact and cannot be expressed in the form x/yare called irrational numbers. The values of irrational numbers can only be approximated and they neither terminate nor recur.
Examples
√2; √3; π; 0,22375809125…
Integer
An integer is a negative or positive whole number including zero.
Examples
-2; -1; 0; 1; 2
Natural numbers
These are numbers that are used in everyday lives for counting, they can also be called counting numbers. For instance counting people in a room one starts from 1, so natural numbers start from 1.
Examples
1; 2; 3; 4; 5…
Odd numbers
An odd number is a number that is not exactly divisible by 2, that is if divided by 2 it leaves a remainder of 1.
Examples
1; 3; 5; 7; 9…
Even numbers
An even number is a number that is exactly divisible by 2, i.e. if divided by 2 it does not leave a remainder.
Examples
2; 4; 6; 8; 10…
Prime numbers
A prime number is a number greater than 0 and is divisible by 1 and itself only, meaning that it has only two factors.2 is the only even number that is prime and not every even number.
Examples
2; 3; 5; 7; 11…
Surds
Surds are those roots (cube roots and higher roots) which are not exact. They are classified under irrational numbers.
Examples
√3; √5; ∛35; ∜21…
Example 1
Given the list -9;-8 $\frac{1}{2}$;-7;-2 $\frac{1}{7}$; -1; 0; √2; 2;√3; 3; 3$\frac{3}{7}$; 4 List all the
a) Natural numbers b) Whole numbers c) Irrational numbers d) Integers
Solution
Natural numbers 2; 3; 4 whole numbers 0; 2; 3; 4 Irrational numbers √2; √3 Integers -9; -7; -1; 0; 2; 3; 4 Given the set of natural numbers between 10 and 30 List a) Prime numbers b)Even numbers c) Odd numbers
Solution
List all the numbers between 10 and 30 11; 12; 13; 14; 15;16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29 Prime numbers, those numbers with factors 1 and itself 11; 13;17; 19;23; 29 Even numbers, those numbers exactly divisible by 2 12; 14; 16; 18; 20; 22; 24; 26;28 Odd numbers, those numbers which can not exactly be divide by 2 11; 13; 15; 17; 19; 21; 23; 25; 27; 29
The number line and directed numbers
Directed numbers Any number that has a sign before it is a directed number. They give direction in which the number is suppose to move a plus showing right movement and a negative showing left movement e.g. -2, -5, +1, +5, +100. Numbers with negative signs e.g. -5 are read as minus 5 and those with positive signs e.g. +5 are read as plus 5. However those with positive signs can be written without the sign e.g. +4 can be written as 4. Addition and subtraction of directed numbers Addition and subtraction of directed numbers can be done using any one of the following methods,
A number line consists of integers and it can be used to add, subtract integers.
Examples 2
(a) +1+ 2 (b) -4+2 (c) +3-2 (d) -3-4
Solution
Start from 1 on the number line, count 2 steps to the right and theanswer is 3.
Start from -4 on the number line, count 2 steps to the right and the answer is -2.
Start from+3 on the number line, count 2 steps to the left and the answer is 1 Start from -3 on the number line, count 4 steps to the left and the answer is -7. Some questions may include large numbers and it will be difficult to use the number line and the following methods may be considered.
1. Simplifying directed numbers with same signs
There are 3 steps to follow;
Ignore the signs.
Find the sum of the two numbers.
Write down the sign on both numbers.
Example 3
+55 +121
= +176
2.-101 -221
=-322
2. Simplifying directed numbers with different signs
There are three steps to follow when simplifying directed numbers with different signs, for example -10+6.
1.Ignore the signs.
2.Find the difference between the two numbers.
3.Take the sign on the bigger number and write it down in your answer.
Examples
-81+23
= -58
2.90 – 35
= 55
Application of directed numbers
Directed numbers can be applied in our everyday lives for instance in financial loss or gain and temperature.
Example
The temperature at 0800 is 12 at 1600 is 28Find the difference between the 2 temperatures.
2.The temperature below the sea level was14and at the top it was 28. What is the difference between the two temperatures?
Solution
28℃- 12℃
=16℃
2.$14\xb0C-(-28\xb0C)$$\mathrm{=14}\xb0C+28\xb0C$
=$42\xb0C$
FRACTIONS
Fractions can be expressed as common fractions, percentages or decimal fractions. Common fractions are expressed as $\frac{x}{y}$ where x and y are integers,where {y: y>0 } $\frac{\mathrm{Numerator}}{\mathrm{Denominator}}$ Division line. Examples of fractions $\frac{3}{5}$;$\frac{9}{10}$;$\frac{108}{115}$ These forms can be changed from one form to the other, for instance converting common fractions to decimals and vice versa.
Examples
1.Change the following to a decimal fraction (a) $\frac{2}{5}\mathrm{}$ (b)$\frac{3}{7}\mathrm{}$ Solution To change a fraction to decimal fraction, divide the numerator by the denominator. (a) $\frac{2}{5}=0,4$ (2÷5=0 r 2 ,then add 0 to make the remainder 20 then divide by 5) (b)$\frac{3}{7}$ = 0, 4285…
2.Change the following to percentages. (a) $\frac{3}{5}$ (b) 0, 65
Solution To change a common fraction or a decimal fraction to an equivalent percentage, multiply the fraction by 100. a)$\frac{3}{5}$× 100 = 60% b)0.65 × 100 = 65%
3.Express as a fraction, a)0,8 b)0,07 c)2,5
Solution Express 0.8 as a common fraction a)0, 8=$\frac{8}{10}$ =$\frac{4}{5}$ b)0,07=$\frac{7}{100}$ c)2,5=$\frac{25}{100}$ =$\frac{1}{4}$
4.Find 30% of $2100. Solution $\frac{30}{100}$×$2100 = $630
5.A packet of sweets cost $12.40 and there are 20 sweets in the packets, what is the cost of 1 sweet. Solution $\frac{\mathrm{\$12,40}}{20}\mathrm{}$ = $ 0,62