Whole numbers

1. Basic Properties of
Numbers and the
Number System
WHOLE NUMBERS

2. 1.1 WHOLE NUMBERS
 Simplest numbers are represented by the symbols 0, 1, 2, 3 etc.
 These symbols are numerals that represent numbers. In general practice,
‘numbers’ is often used to mean ‘numerals’.
 These numerals represent the natural numbers, used to count whole objects
rather than fraction of them.
 Natural numbers are also called whole number or the integers.
 An integer greater than zero (i.e. 1,2,3,4,5……) is called a positive integer,
while that less than zero (-1,-2,-3,-4,-5…….) is called a negative integer.
 A numeral consists of digits; defined as one of the numerals 0, 1, 2, 3, 4, 5, 6,
7, 8, 9. Some numerals consist of one digit, some of two digits and others
three or more.

3.  A one-digit number lies between 1 and 9, a two-digit number (e.g. 54) lies
between 10 and 99; a three-digit lies between (e.g. 751) lies between 100
and 999; a four-digit (e.g. 5671) lies between 1000 and 9999, and so on.
 The number system we use is the decimal system because it is based on the
number 10. The value of a digit depends on its position within the numeral
(see Table 1.1)

Position of digit
(Counting from the
right)
Value Name
First 1 Ones (Units)
Second 10 Tens
Third 100 Hundreds
Fourth 1 000 Thousands
Fifth 10 000 Tens of thousands
Sixth 100 000 Hundreds of thousands

4. EXAMPLE 1.1
 Write in words the number presented by each of the following numerals:
a) 56
b) 368
c) 1679
d) 367 297
e) 1 428
f) 21 349
g) 66 554 321
h) 54 333 456 900 812

5. Solution
a) 56 consists of 6 ones/units and 5 tens. It represents the number fifty-six.
b) 368 consists of 8 units, 6 tens and 3 hundreds. It represents the number
‘three hundred and sixty-eight’
c) 1679
d) 367 297
e) 1428
f) 21 349
g) 66 554 321
h) 54 333 456 900 812

6. FOUR BASIC MATHEMATICAL OPERATIONS
 THERE ARE FOUR BASIC MATHEMATICAL OPERATIONS THAT CAN BE DONE ON
NUMBERS.
1. Multiplication (often represented by X)
2. Division (often represented by ÷ or /)
3. Addition (represented by +)
4. Subtraction (represented by -)
The order in which these operations should be performed in an expression is:
a) Multiplications and divisions first
b) Then additions and substractions

7. For example: 3 + 2 × 5 = 3 + 10 = 13
 To avoid any ambiguity, we can make use of parentheses (or brackets), which
take precedence over all four basic operations. In the above case we could
have written 3 + (2 × 5) to make the expression more clearer, although we
would still obtain the same answer of 13.
 Suppose, however, that we wanted the 3 and the 2 to be added together
before multiplication by 5. In this case we could write (3 + 2) × 5 = 5 × 5 = 25.
Now let us look at each of the operations.

8. MULTIPLICATION
There are several ways of indicating that two numbers are to be multiplied. For example,
suppose we want to multiply the number 3 by the number 4. Some common notations for this
are:
1) 3 × 4
2) 3 . 4
3) (3)(4)
4) 3(4) or (3)4
All these expressions mean the product of 3 and 4.
An expression of the type 3 × 4 = 12 is a mathematical statement that the product ‘3 × 4’ and the
number 12 are really the same. This type of expression is called an equation, since it equates
two quantities.
An important property of multiplication is symmetry. For example, we could reverse the order of
3 and 4 in the above equation and we would get exactly the same result.
Also, if the two numbers to be multiplied have the same sign, the result will be positive; if they
have different signs, the result will be negative.
Remember that when multiplying numbers in expressions that contain parentheses (brackets),
you must perform the calculations within the parentheses first.

9. EXAMPLE 1.2
 Calculate:
a) 3 × (6 + 7)
b) (5 + 9) × 4
c) (4 + 3 – 1) × (8 – 5 + 6)
d) 4 × -3
e) -5 × 6
f) -7 × -9
Solution:
a) 3 × (13) = 39
b) (14) × 4 = 56
c) (6) × (9) = 54

10. DIVISION
 There are three main ways of indicating that two numbers are to be divided.
Suppose we want to divide 12 by 4:
1. 12 ÷ 2
2. 12/4
3.
12
4
The number to be divided (12) is called the numerator or dividend, and the
number that is to be divided by (4) is called the denominator or divisor. The
answer to the division (in this case 3) is called the quotient.
Division has no symmetry. Tampering with the numbers changes the entire
expression. Also, if the signs for both the denominator and numerator are the
same, the quotient will be positive and if different, the quotient will be negative

11. EXAMPLE 1.3
a) -3/6
b) 5 ÷ -15
c)
−20
−4
Solution:
a) -3/6 = -(3/6) = -(1/2)

12. ADDITION
Addition has symmetry; the
order in which the numbers
appear does not affect the
result. For example, the
expressions ‘2 + 3’ and ‘3 + 2’
are the same as they are both
equal to 5.

13. SUBSTRACTION
 Unlike addition, subtraction does not have symmetry; the order in which the
numbers appear does affect the result. For example, the expression ‘8 - 3’ is
equal to 5, but ‘3 - 8’ is equal to -5, not 5.
 When performing additions and subtractions, you should treat a + and – signs
as simply a – sign. Treat two – signs as a + sign
EXAMPLE 1.4
Evaluate:
a) 8 + (-5)
b) 12 – (+3)
c) 18 – (-5)

14. 1.2 FRACTIONS
A fraction comes in a form:
𝑎
𝑏
As discussed previously, a is called the numerator and b is the denominator. The
denominator cannot be zero, because if it is, the result is undefined.
A proper fraction is one in which the numerator is less than the denominator
(e.g. 4/6, 12/34); an improper fraction has the numerator greater than the
denominator (e.g. 10/4, 50/6).
Addition of fractions with the same denominator
To add fractions with the same denominator, add together the numerators to
obtain the new numerator. The denominator remains the same.
e.g. 1/5 and 3/5 = (1+3)/5 = 4/5
4/9 , 3/9 and 2/9 = (4+3+2)/9 = 9/9 = 1

15. Addition of fractions with different
denominators
Change the denominators to be the same before adding the numerators. The
same denominator is called the Lowest Common Denominator and is the smallest
number into which the denominators will all divide. This smallest number is
called the Lowest Common Multiple. It may be found by multiplying together
common factors of the numbers.
Example 1.6: Find the lowest common multiple of
a) 2, 5, 6, 8
2 x 5 x 3 x 4 = 120
a) 5, 12, 15, 20
5 x 3 x 4 = 60

16. Example 1.7
Evaluate:
a) 1/3 + 2/9 + 5/6
b) 2/5 + 4/15 + 1/12
Solution
a) The LCM of 3, 9 and 6 is 18, so each fraction must be converted to an
equivalent one with a denominator of 18
(6 + 4 + 15)/18 = 25/15
a) The LCM of 5, 15 and 12 is 60.
(24 + 16 + 5)/60 = 45/60 = 3/4

17. Subtraction of fractions
The rules for addition also apply to subtraction. This time the numerators are
subtracted.
Example:1.8
Evaluate:
a) 11/15 + 7/15 – 14/15
(11 + 7 - 14)/15
= 4/15
b) 5/16 – 1/8 + 1/12. the LCM for 16, 8 and 12 is 48
(15 – 6 + 4)/48
13/48

18. Multiplication of Fractions
To multiply fractions we multiply the numerators to get the new numerator and
multiply the denominators to get the new denominator. If there are common
factors in the resulting fraction, they should be divided.
Example 1.9
Evaluate:
a) 2/3 × 5/6 = 2×5/3×6 = 10/18= 5/9
b) 4/5 × 3/4 × 7/12 = (4×3×7)/(5×4×12) = 84/240 = 7/20 (12 is the lowest
number that divides both 84 and 240)

19. Division of Fractions
To divide one fraction by a second fraction, invert (i.e. turn upside down) the
second fraction, then multiply it by the first. Every division problem can be
changed into one of multiplication.
Example 1.10
a) ½ ÷ 5/12 = ½ × 12/5 = 12/10 = 6/5
b) 6 ÷ 2/3 = 6 × 3/2 = 18/2 = 9
c) 5/14 × 1/8 ÷ ¾ = 5/15 × 1/8 × 4/3 = 20/336 = 5/842

20. 1.3 DECIMALS
Since the number system is based on the number 10, it is possible to express any
fraction as a decimal. For example, we use a decimal system in our currency and
in metric measurement. A decimal is really a fraction in which the numerator has
been divided by the denominator to yield an equivalent decimal expression. A
decimal consists of three components: an integer, followed by a decimal point,
followed by another integer. The values of the digits that occur after the decimal
point depend on how many positions they are after the point.
Position of digit after
decimal point
Value Name
Fist 0.1 Tenths
Second 0.01 Hundreds
Third 0.001 Thousandths
Fourth 0.0001 Ten-thousandths
Fifth 0.00001 Hundred-thousands
Sixth 0.000001 Millionths

21. CONT….
If a number is expressed in decimal form, any zeros on the right-hand end after
the decimal point and after the last digit do not change the number’s value. For
example, the decimals 0.5, 0.50, 0.500 and 0.5000 all represent the same
number. Also, if a number that is expressed in decimal form is less than 1, it is
not uncommon for the 0 before the decimal point to be omitted. For example,
0.4 can be written as just .4.
Example:
a) 0.3 is three tenths or 3/10
b) 0.47 is forty-seven hundredths or four tenths or 47/100
c) O.763 is seven hundred and sixty-three thousandths or 763/1000
d) 0.25 is twenty-five hundredths or 25/100 or ¼.

22. Example 1.12
Express as decimals: These may be found by expressing the numerators in
decimal form.
a) 1/5 = 0.2
b) ¾ = 0.75
c) 7/8 = 0.875

23. Addition, Subtraction, multiplication and
division of decimals
Apply the same rules as those of integers for this module but there are advanced
rules for multiplication and division of decimals.
Example 1.13
a) 2.3 + 0.34 + 1.672 = 4.312
b) 5.70 – 2.49 = 3.21
c) 245.8794 + 356.37621 = 602.25561
d) 3.12 × 2.7 = 8424
e) 5.542 × 6.78 = 37 57446
f) 3.248 ÷ 0.04 = 81.2
g) 82.5 ÷ 4.125 = 20