Natural Numbers, Integers, Rational Numbers Worksheet With Answers - Mathematics Secondary Course Page 7
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36MODULE -
1
Number Systems
Algebra
−
−
7
6
12
16
,
,
,
Thus,
are all negaive rationals.
−
−
2
5
4
3
1.3.2 Standard form of a Rational Number
Notes
We know that numbers of the form
−
−
p
p
p
p
,
,
and
−
−
q
q
q
q
are all rational numbers, where p and q are positive integers
We can see that
(
)
(
)
−
⎛
⎞
−
−
−
−
−
p
p
p
p
p
p
p
p
=
−
⎜ ⎜
⎟ ⎟
=
=
=
=
,
,
,
( )
( )
−
−
−
−
−
−
q
⎝
q
⎠
q
q
q
q
q
q
In each of the above cases, we have made the denominator q as positive.
p
, where p and q are integers and q ≠ 0, in which q is positive (or
A rational number
q
made positive) and p and q are co-prime (i.e. when they do not have a common factor
other than 1 and –1) is said to be in standard form.
−
−
−
2
2
5
3
Thus the standard form of the rational number
is
. Similarly,
and
are
−
3
3
6
5
rational numbers in standard form.
Note: “A rational number in standard form is also referred to as “a rational number in its
lowest form”. In this lesson, we will be using these two terms interchangably.
18
2
For example, rational number
can be written as
in the standard form (or the lowest
27
3
form) .
−
25
5
Similarly,
, in standard form (or in lowest form) can be written as
(cancelling out
−
35
7
5 from both numerator and denominator).
Some Important Results
(i) Every natural number is a rational number but the vice-versa is not always true.
(ii) Every whole number and integer is a rational number but vice-versa is not always true.
Mathematics Secondary Course
9
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