CC Maths
RATIONAL NUMBERS AND THE FOUR OPERATIONS
Note
Representations of a rational numbers in the form
a
, where b ≠ 0, are not unique. Why?
b
a na
Because =
for all integer n (equivalent fractions).
b nb
Addition and Subtraction of Rational Numbers
Check understanding
3 − 3 3 + (−3) 0
− 3 3 (−3) + 3 0
= = 0 and
= = 0.
+
=
+ =
5
5
5
5
5
5
5
5
−
 3 − 3
So   =
.
5
5
 
1
2
and
(ii) (a)
3
4
1 5 1(12) + 4(5) 12 + 20 32 2
=
=
=
+ =
48 3
48
4(12)
4 12
5
2 1
So – = .
3 4 12
(i)
(a)
1
−2
and
4
3
1 − 11 1(12) + 4(−11) 12 − 48 − 36 − 2
+
=
=
=
=
4 12
4(12)
48
48
3
− 2 1 − 11
So
– =
.
4
3
12
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CC Maths
Practice Ongoing Assessment 5-2 p. 266
Question 19
1 3 5 3 1
+ = = +
(b)
2 4 8 4 2
Question 26
1
1
+
=
(a)
4 3× 4
1
1
+
=
5 4×5
3
1
+
=
3× 4 3× 4
4
1
+
=
4×5 4×5
3 +1
4
1
=
=
3× 4 3× 4 3
4 +1
5
1
=
=
4×5 4×5 4
Multiplication and Division of Rational Numbers
Check understanding
− 2 − 5 (−2) × (−5) 10
×
=
=
= 1 and
5
2
5× 2
10
− 5 − 2 (−5) × (−2) 10
×
=
=
= 1.
2
5
2×5
10
−5
−2
So the multiplicative inverse of
is
.
2
5
(ii) 0 has no multiplicative inverse because for all rational numbers
x, x 0 = 0.
(iii) The multiplicative inverse of the identity 1 is 1 itself.
3 10 3 × 10 30 2 × 15 2
2 3 10
=
=
= . So ÷ = .
(iv) (a) × =
5 9
5 × 9 45 3 × 15 3
3 5 9
3 − 10 3 × (−10) − 30 (−2) × 15 − 2
(b) ×
=
=
=
=
.
5
9
5×9
45
3 × 15
3
2 3 10
So ÷ = .
3 5 9
(i)
Practice Ongoing Assessment 5-3 pp. 278-280
Question 5
10 3
1 10
3 10
and × = 1 = × .
(b) 3 =
3 10
3 9
10 3
-2-
CC Maths
1
3
So the multiplicative inverse of 3 is .
3 10
− 7 −1
−1 − 7
−7
×
=1=
×
and
.
(d) − 7 =
1
7
7
1
1
−1
.
So the multiplicative inverse of – 7 is
7
Question 49
1 2 3
1 2 3 3 3
1 8 9
but ( ÷ ) ÷ = ÷ = 1
(b) e.g. ÷ ( ÷ ) = ÷ =
2 3 4
2 3 4 4 4
2 9 16
(d) Inverse property does not hold for division of rational numbers
because the identity property does not hold. OR simply say
that 0 does not have an inverse over the set of rational numbers.
Chapter Review p. 290
(b)
(d)
1
7
is
.
7 22
4
3
The multiplicative inverse of – is – .
3
4
The multiplicative inverse of 3
Summary of Sections on Rational Number Operations
Property
Closure
Commutativity
Associativity
Identity
Inverses
On the set Q of rational numbers
Addition
Subtraction Multiplicati
Division
on
Y
Y
Y
N
Y
N
Y
N
Y
N
Y
N
Y
N
Y
N
Y
N
N
N
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