Page 77 - Start Up Mathematics_7

At this point if the division is continued the quotient will emerge as 0.181818 ... because the
remainder 2 is equal to the original dividend and the division process will keep on repeating.
2 · ·
∴ = 0.181818 ... = 0.18 or 0.18 (the bar or dot above 18 means 18 is repeating)
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On the other hand decimals like 1.732050807 ... neither terminate nor have any repeating part. In
fact these decimals are not equal to any rational number. So they are called irrational numbers.

The decimal 1.732050807 ... represents an irrational number √3 .
A rational number p where q ≠ 0 is represented either Extension
q
by a terminating decimal or a non-terminating repeating π is another irrational number.
Hence, its decimal representation
1
decimal. For example, 3 = 1.625, = 0.142857. is non-terminating, non-recurring.
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Exploring Extended Concepts

Let a, b, c and d be rational numbers which lie in the following intervals.

0 1 a is greater than 1, i.e., a > 1
b lies between 0 and 1, i.e., 0 < b < 1
0 1
c lies between –2 and –1, i.e., –2 < c < –1
–2 –1
d lies between 0 and 3, i.e., 0 < d < 3
0 3
Verify by taking at least 5 examples and fill the appropriate symbol, <, > or = in the boxes
given below:
a d
(a) b . d d (b) b a (c) a d (d) a . c b . d

3 2
a b c b 2
(e) 12 (f) c ÷ c
d b

At a Glance
p
1. A number that can be expressed in the form , where p and q are integers and q ≠ 0 is called
a rational number. q

2. All fractions and integers are rational numbers but the converse is not true.
a
3. A rational number is positive if both a and b are positive or both are negative. If a and b
b
have different signs then a is negative. 0 is a rational number which is neither positive nor
b
negative.
p p p n
4. For any rational number and a non-zero number n we have, = .
q q q n
5. Equality of rational numbers: p = r iff p × s = r × q.
q
s
p
6. A rational number is in standard form if p, q are coprime and q is a positive integer.
q
7. Comparison of rational numbers

(a) A positive rational number is always greater than a negative rational number.

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