INTEGERS
In this topic, we will learn more about:
INTEGERS
A)
Understanding Integers1. An integer is a whole number with the positive
sign or negative sign, including zero which is
neither positive nor negative.
2. Positive and negative numbers include decimals
and fractions.
For example :-
(a) Positive numbers : +12, 8.5, + 11
2
(b) Negetive numbers : -15, - 3.6, - 33
4
3. Positive integers can be written without the '+' sign.
For example :-
+4 is usually written as 4.
Worked Example 1
From the numbers above, list all the
(a) integers,
(b) positive integers,
(c) negative integers.
Solution:
(a) - 5, -2, 0, +8 and 18
(b) +8 and 18
(c) -5 and -2
B) Representing Integers on Number Lines
1. Integers can be represented on either a horizontal
or a vertical number line.
2. Normally, a horizontal number line is used.
For example :-
Worked Example 2
Draw a number line and mark the following numbers :-
- 15 , - 5, 10, 15
Solution :
C) Comparing the Value of Two Integers
We can use the number line to compare the values
of integers. A given integer, x, is greater than all the
integers on its left but smalller than all the integers
on its right.
Worked Example 3
In each of the following pairs of integers, state the
integer that is greater in value.
(a) - 3, 2 (b) - 6, - 3
Solution:
Worked Example 4
State the integer with the smaller value : - 5 or 0
D) Arranging Integers in Order
1. In increasing order, the integers are arranged
from the smallest to the largest.
For example :-
2. In decreasing order, the integers are arranged
from the largest to the smallest.
For example :-
Worked Example 5
(a) Arrange the integers -6, 2, -1, +4, 0, - 2 in
increasing order.
(b) Arrange the integers 3, 0, 5, - 5, 6, - 4 in
decreasing order.
Solution:
(a) - 6, - 2, -1, 0, 2, +4
(b) 6, 5, 3, 0, - 4, - 5
E) Determining the Largest Integer or the Smallest Integer
Worked Example 6
Determine the largest integer and the smallest
integer in the following : 8, - 3, - 14, - 9, - 4, 7
Solution:
F) Completing a Sequence of Integers
To complete a sequence of integers, it is necessary
to identify the pattern of the sequence.
Worked Example 7
State the integers that are represented by the letters
P, Q and R on the number line above.
Solution:
Therefore, P = -8, Q = 0, R= 4
Worked Example 8
Complete the following sequence of integers.
- 12, - 9, - 6, __, __, __
Solution:
G) Using Positive and Negative Integers in Real- life Situations
Positive and negative integers can be used in
the following contexts in real-life situations.
(a) An increase or a decrease in value
For example :-
(i) +RM100 represents an increase of Rm100
in price whereas -RM50 represents a decrease
of RM50 in price.
(ii) +60 sen represents a profit of 60 sen whereas
- 30 sen represents a loss of 30 sen.
(iii) +4 kg represents a mass gain of 4 kg whereas
- 6 kg represents a mass loss of 6 kg.
(b) A value which is either greater than or less than
zero.
For example :-
(i) +5oC represents 50C more than 00C whereas
- 20C represents 20C less than 00C.
(ii) As 0 m is the sea level, +3 m represents 3 m
above sea level whereas -10 m below sea
level.
(c) A positive or a negative direction (i.e.opposite
directions)
For example :-
(i) If +8 km represents 8 km to the east, then -7 km
represents 7 km to the west.
(ii) If a lift which moves 4 floors upwards is repre-
sented by +4, then a lift which moves downwards
is represented by - 3.
Worked Example 9
Use positive or negative integers to represent the
following.
(a) A temperature rise of 180C
(b) 800 m below sea level
(c) A loss of RM70
Solution:
(a) +180C (c) -RM70
(b) - 800 m
ADDITION AND SUBTRACTION OF INTEGERS
A) Addition of integers1. For a horizontal number line, the direction from left
to right indicates the positive direction while the
direction from right to left indicates the negative direction.
For example :-
2. For a vertical number line, and upward direction
indicates the positive direction indicates the negative
direction.
For example :-
3. We can add two or more integers using a number line.
(a) When adding a positive integer, we move in the
positive direction on the number line.
(b) When adding a negative integer, we move in the
negative direction on the number line.
(c) Zero is usually taken as the starting point.
Worked Example 10
Solve the following by using the number lines.
(a) 3 + (+5) (c) - 2+ (- 5)
(b) - 6 + 4 (d) 3 + (- 5)+(+8)
Solution:
Worked Example 11
Solve each of the following.
(a) - 9 + (+5) (c) - 6 + (- 2)
(b) + 7 + (- 5)
Solution:
B) Problem Solving involving Addition of Integers
Worked Example 12
A submarine which is 9 km below sea level
moves up 3 km. What is the position of the
submarine now ?
Solution:
1. Understand the problem
Given information :
The submarine is 9 km below sea level.
It moves up 3 km.
Find : Final position of the submarine
2. Devise a plan
Use addition.
3. Carry out the plan
- 9 + 4 = - 5
Therefore, the submarine is 5 km below sea
level now.
4. Check
C) Subtraction of Integers
1. Subtraction of integers is to find the difference
between two integers.
2. We can use a number line to find the difference
between two integers.
3. The difference can be determined by counting the
number of units to be moved from the second integer
to the first integer. The direction of the movement will
determine the '+' or '_' sign in the answer.
Worked Example 13
Solve the following by using number line.
(a) 3 - 5 (c) - 2 - 4
(b) 8 - (+4) (d) 0 - (- 7)
Solution:
Worked Example 14
Find the value of each of the following.
(a) 8 - (+5)
(b) 6 - (- 3)
(c) - 9 - (- 9)
Solution:
Worked Example 15
Find the value of each of the following.
(a) 30 - (+18) - (- 6)
(b) - 21 - (- 7) - (- 13)
Solution:
(a) 30 - (+18) - (- 6) = 30 - 18 + 6
= 12 + 6
= 18
(b) - 21 - (- 7) - ( - 13) = - 21 + 7 + 13
= - 14 + 13
= -1
D) Problem Solving involving Subtraction of Integers
A submarine is 8 m below sea level. An eagle is 3 m
above sea level directly above the submarine. What
is the distance between the submarine and the eagle?
Solution :
1. Understand the problem
Given information :
A submarine is 8 m below sea level.
An eagle is 3 m above sea level.
Find : Distance between the submarine
and the eagle
2. Devise a plan
Use subtraction.
3. Carry out the plan
3 - (- 8) = 3 + 8 = 11
Therefore, the distance between the submarine
and the eagle is 11 m.
4. Check
E) Combined Operations of Addition and Subtraction of Integers
Worked Example 17
Find the value of each of the following.
(a) 25 - (+30) + (- 12)
(b) - 12 - (- 26) + (- 15)
Solution:
(a) 25 - (+30) + (- 12)
= 25 - 30 - 12
= - 5 - 12
= - 17
(b) - 12 - (- 26) + (- 15)
= - 12 + 26 - 15
= 14 - 15
= - 1
F) Problem Solving involving Addition and Subtraction of Integers
Worked Example 18
A diver at 7 m below sea level swam up 3 m. A
turtle at that moment was 2 m below sea level.
What was the distance between them when the
turtle was directly above the diver?
Solution:
1. Understand the problem
Given information :
The diver was 7 m below sea level at first.
He swam up 3 m.
The turtle is 2 m below sea level.
Find : Distance between the diver and the turtle.
2. Devise a plan
Use addition and subtraction.
3. Carry out the plan
(- 7 + 3) - (-5) = (- 7 + 3) + 5
= - 4 + 5
= 1
Therefore, the distance between the diver and
the turtle was 1 m.
4. Check
GLOSSARY
- Integers- integer
- Positive Integers- integer positif
- Negative Integers- integer negatif
- Positive Number- nombor positif
- Negative Number- nombor negatif
- Inverse- songsangan
- Like signs- tanda serupa
- Unlike signs- tanda tak serupa
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