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4
Directed Numbers
After studying this lesson, you can get a good understanding of
 subtracting, multiplying and dividing integers.
 adding, subtracting, multiplying and dividing directed numbers.
All the numbers written with a positive or negative sign ( i.e. with a direction)
are called directed numbers.
You have learned before that the positive and negative whole numbers including
zero, are known as integers. Do the following exercise using the knowledge you
have gained in Grade 7 about the addition of integers.
Exercise 4.1
(i)
(iv)
(vii)
(x)
(+5) + (+3)
(+5) + (_3)
(_3) + 0
(_5) + (+7) + (_2)
(ii)
(v)
(viii)
(xi)
(_7) + (_5)
(_3) + (_2)
(_3) + (+2) + (_4)
(_5) + (_2) + (_1)
(iii)
(vi)
(ix)
(xii)
(_1) + (+4)
(+5) + (-5)
(+3) + (_5) + (+7)
0 + (+8) + (_6)
4.1 Subtraction of integers
Study the following relations well.


(+7) + (+2) = (+9)
(+1) + (+5) = (+6)
(+9)
− - (+2) = (+7)
(+9)
− - (+7) = (+2)
(+6)
− - (+5) = (+1)
(+6)
− - (+1) = (+5)
Engage in the following activity using the above relation.
For Free Distribution only
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Activity 4.1
Copy the following expressions and fill in the blanks.
(+7) −
(a) (+5) + (+2) = (+7)
−
(+6) −
(b) (+5) + (+1) = (+6)
(c) (+5) + 0
=
= (+2)
=
(+6) − (+5) = (+1)
(+5) − 0 = (+5)
= (+5)
(+5) −
=0
(+4) − (−1) = (+5)
(d) (+5) + (−1) = (+4)
(+4) −
=
− (−2) = (+5)
(e) (+5) + (−2) =
(+3) −
=
It is shown below that the answers of the above subtractions can be obtained by
considering them as additions. See also the differences that happen with signs.
(+6) − (+5) = (+1)
(+6) + (−5) = (+1)
 (+4) − (−1) = (+5)
(+4) + (+1) = (+5)

As a difference (a subtraction)
As total (an addition)
 (+5) − 0 = (+5)

(+5) + 0 = (+5)
(+5) − (+5) = 0
(+5) + (−5) = 0
4.2 Obtaining the answer by arranging a subtraction as an addition
Example 1
Examine how the answers of the following subtractions have been obtained
by arranging them as additions.
1. (+5) − (+2)
(+5) + (−2) = (+3)
2.
(+3) − (+7)
(+3) + (−7) = (−4)
3. (−2) − (−3)
(−2) + (+3) = (+1)
4.
(−8) − (−5)
(−8) + (+5) = (−3)
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For Free Distribution only
Exercise 4.2
(1)
Arrange as additions and obtain the answers.
(i)
(iv)
(vii)
(x)
(+2) − (+3)
(+1) − (−6)
(−5) − (+5)
(−7) − (−4) − (+3)
(ii)
(v)
(viii)
(xi)
(−5) − (+2)
0 − (+3)
(−3) − (−3)
(−5) − (+3)
(iii)
(vi)
(ix)
(xii)
(−7) − (−1)
0 − (−2)
(+5) − (+2) − (−1)
(−10) − (−10) − (−3)
4.3 Subtraction of integers using the number line
You may remember how integers were added using the number line. Similarly
let us subtract integers using the number line.
Positive direction
−3 −2 −1 0
Example 2
+1 +2 +3
(+5) − (+2)
-3
-2
-1
0
+1
+2
+3
+4
+5
+6
From the number (+2) which is the number to be subtracted, 3 units have to
be moved to the positive direction to go to (+5).
Accordingly, (+5) − (+2) = (+3)
Example 3
(+3) − (+7)
-1
0
+1
+2
+3
+4
+5
+6
+7
+8
From the number (+7) which has to be subtracted, 4 units have to be
moved to the negative direction to go to (+3).
Accordingly, (+3) − (+7) = (−4)
Example 4
(− 2) − (− 3)
−7
−6
−5
−4 −3
−2
−1
0
+1
+2
One unit has to be moved to the positive direction from (−3) to reach (−2).
Accordingly, (−2) − (−3) = (+1)
For Free Distribution only
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Exercise 4.3
(1)
Obtain the answers using the number line.
(i) (+2) − (+5)
(iv) (−3) − (+3)
(2)
(ii)
(v)
(−1) − (−5)
0 − (−4)
(+2) − (−3)
(−6) − (−4)
From the followings, select the pairs with the diffrence of (-3).
(i) (+1) − (+4)
(ii) (−1) − (−2) (iii) (−2) − (+1) (iv) (−7) − (−4)
(v) (+2) − (−4) (vi) 0 − (+3)
(3)
(iii)
(vi)
(vii) (−3) − 0
(viii) (−5) − (+2)
Obtain the answers for the boxes using the number line.
(−2)
(−1)
(0)
(�� )
(�� )
(�� )
(�� )
(+3)
(+1)
(0)
<
< < < <
(+3)
(+3)
(+2)
(�� )
<
(+3)
(+3)
(+3)
(+3)
(+3)
(+3)
(�� )
(+3)
(−3)
(+3)
(−4)
(+3)
4.4 Multiplication of integers
Study the folowing patterns.
(+2) × (+3) = (+6)
(+2) × (+2) = (+4)
(+2) × (+1) = (+2)
(+2) × (0) = 0
(+2) × (−1) = (−2)
(+2) × (−2) = (−4)
(+2) × (−3) = (−6)
−2
−2
−2
−2
−2
-6 -5 -4 -3 -2 -1
0 1 2 3 4 5 6
−2
By studying the above pattern it can be understood that when a positive number
is multiplied by a negative number you get a negative number as the result.
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For Free Distribution only
Next study the following patterns.
+2
< <
+2
+2
+2
+2
+2
< <
-6 -5 -4 -3 -2 -1
+2
<
(−2) × (+3) = (−6)
(−2) × (+2) = (−4)
(−2) × (+1) = (−2)
(−2) × 0 = 0
(−2) × (−1) = (+2)
(−2) × (−2) = (+4)
(−2) × (−3) = (+6)
0 1 2 3 4 5 6
By studying the above pattern it can be understood that when a negative number
is multiplied by a negative number you get a positive number as the result.
Activity 4.2
Fill in the cage ‘A’ first. Then by considering the pattern of the results fill in the
empty cages of ‘B’,‘C’,‘D’.
X
−3
−2
−1
0
1
2
3
+3
D
+2
+1
0
−1
−2
C
A
 What is the sign of the answers you
got for cages ‘A’ and ‘C’ ?
 What is the sign of the answers you
got for cages ‘B’ and ‘D’ ?
B
−3
Accordingly, when two numbers with the same sign are multiplied you
get a positive number as the result and when two numbers with different
signs are multiplied you get a negative number as the result.
For Free Distribution only
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Exercise 4.4
(1)
Multiply.
(i)
(iv)
(vii)
(x)
(−3) ×
(−6) ×
(−3) ×
(−7) ×
(2)
Fill in the empty boxes.
(ii) (−5) × (−1)
(v) (−4) × 0
(viii) (−4) × (−3) × (−2)
(xi) (+10) × (+10)
(iii) (+7) × (−2)
(vi)
0 × (+3)
(ix) (+5) × (+3) × (−2)
(xii) (−8) × (+9)
(i)
×
= (+6)
(ii)
×
= (+6)
(iii)
×
= (+6)
(iv)
×
= (+6)
= (+10)
(vi)
(+5) ×
(v)
(vii)
(3)
(+5)
(+6)
(+2) × (−1)
(−30) × 0
(+5) ×
× (−3) = (+3)
= (−20)
(viii) (−2) × (−2) ×
= (−8)
For what products out of the following do you get (−12) as the answer?
Write those serial numbers.
(i) (+6) × (−6)
(ii) (−4) × (−3)
(iii) (+3) × (+4) × (−1)
(iv) (−2) × (−2) × (−3)
(v) (−6) × (−2)
(vi) (+12) × (−1)
(vii) (−1) × (−2) × (−2) × (−3) (viii) (−6) × (+2) × 0 (ix) (−2) × (−5) × (+9)
4.5 Division of Integers.
Study the following patterns.

(+3) × (+2) = (+6)

(+3) × (−2) = (−6)

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(−3) × (−2) = (+6)
(+6) ÷ (+2) = (+3)
(+6) ÷ (+3) = (+2)
(−6) ÷ (−2) = (+3)
(−6) ÷ (+3) = (−2)
(+6) ÷ (−2) = (−3)
(+6) ÷ (−3) = (−2)
For Free Distribution only
According to the relations given above,
 when one number is divided by another with the same sign, you get a
positive number as the result.
 when one number is divided by another with a different sign, you get a
negative number as the result.
Exercise 4.5
(1)
Do the following divisions.
(i)
(+15) ÷ (+3) (ii)
(−14)
(+2)
(−5) × (+2)
(+8)
(ix)
(x)
(−10)
(−2) × (+2)
(v) (−24) ÷ (−6) (vi)
(2)
(vii) (−3) × (−4)
(+2)
(−5)
(+5)
(viii) (−6) × (+3)
(−2)
(−5) × (−4)
(−2) × (+2)
(xii) (+9) × (−8)
(−4) × (+3)
(−12) ÷ (+4) (iii) (+12) ÷ (−3)
(xi)
(iv)
Fill in the blanks.
(i)
(−24)
(iii)
(v)
(vii)
3
(−10)
=
(−40)
(+8) ×
=4
(ii)
= −1
(iv)
(−12)
= (−5) (vi)
(−4) ×
(+14)
(−2)
= −1
(viii)
(−6)
= −3
=6
× (−7)
(−2) ×
= (−2)
=
−28
= (+7)
4.6 Mathematical operations related to Directed Numbers
Applying the knowledge you have gained through this lesson about addition,
subtraction, multiplication and division of integers and the knowledge you
have gained earlier about fractions and decimals do the exercise on the next
page.
For Free Distribution only
29
Exercise 4.6
(1) Simplify.
(i)
(−3.5) + (+5.2)
(ii) (−7) + (−1.3)
(iii) (−3.5) + (−5.2)
(iv) (−2.3) + (+8.12)
 1  1
(vi)  +3  +  +5 
 2  2
 1
(viii) (−4) +  −7 
2
(v) (+5.1) + (+3.24) + (−0.7)
 1
(vii) (−7) +  − 
 2
 1
 1
(ix)  −  +  +5 
 4
4
(x)
 1
(+15) +  +7 
 2
(2) Simplify.
(i)
(−5) − (−7.3)
(ii)
(+5.3) − (+1.5)
(iii) (−4.2) − (−4.2)
(iv) (+7) − (+3.25)
(v)
 1
(vi) (+5) −  −3 
 2
(−3) − (−7.2) − (−10)
 1  4
(vii) +  −  − 
 5  5
 1  1
(viii)  −3  −  −7 
 2  2
 1
(ix) (−4) −  −8 
 4
 1
 1
(x) (+5)−  −  −  +3 
 2
 2
Summary
 All the numbers written with a positive sign or a negative sign are directed
numbers.
 Positive and negative whole numbers including zero are called integers.
 Subtraction of integers can be done easily by arranging them as additions.
 Terms with equal signs when multiplied or divided, the answers will be
positive numbers and terms with different signs when multiplied or divided,
the answers will be negative numbers.
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