This document discusses how to express repeating decimals as rational numbers in the form p/q, where p and q are integers and q ≠ 0. It provides solutions to express 0.333..., 1.272727..., 0.2353535..., and 0.3142678 as rational numbers. It also explains how to locate square roots such as √2 and √3 on a number line using the Pythagorean theorem.

1. MATHEMATIC
NUMBER SYSTEM
Question and solution
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EXERCISE 1.1
 Q1.SHOW THAT 0.333… = 0.3 can be expressed in , where p and q
are integers and q ≠ 0.
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Solution : Since we do not know what 0.333… is, let us call it ‘𝜒’ and so
𝜒 = 0.333…
Now here is where the trick comes in. look out
10𝜒 = 10𝜒 (0.333…) = 3.333…
Now, 3. 333… = 3 + 𝜒, since 𝜒 = 0.333…
Therefore, 10𝜒 = 3 + 𝜒
Solving for 𝜒, we get
9𝜒 = 3 i.e., 𝜒 =
1
3
𝑝
𝑞

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Q2. show that 1.272727… = 1.27 can be expressed in
𝑝
𝑞
, where p and
q are integers and q ≠ 0.
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Solution : let 𝜒 =1.272727… since two digits are repeating, we multiply
𝜒 by 100 to get
100𝜒 = 127.2727…
So, 100𝜒 = 126 + 1.272727… = 126 + 𝜒
Therefore, 100𝜒 - 𝜒 = 126, i.e., 99𝜒 = 126
i.e., 𝜒 =
126
99
=
14
11
You can check the reverse that
14
11
= 1.272727…

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Q3. Show that 0.2353535… = 0.235 can be expressed in the form of
𝑝
𝑞
where p and q are integers and q ≠ 0.
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Solution : let 𝜒 = 0.235 . Over here, note that 2 does not repeat, but
the block 35 repeats. Since two digits are repeating, we multiply 𝜒
by 100.
1000𝜒 = 235.3535…
So, 1000𝜒 = 235 + 0.2353535… = 235 + 𝜒
Therefore, - 10𝜒 = 0 + 0.2353535…
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i.e., 990𝜒 =
233
1
, which gives 𝜒 =
233
990
You can also check the reverse that
233
990
= 0.235.

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Q4. show that 0.3142678 is a rational number. In other words, express
3.142678 in the form
𝑝
𝑞
, where p and q are integers and q ≠ 0.
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Solution : We have,
3.142678 =
3142678
1000000
and hence it is a rational number.

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Q5. locate √2 on the number line.
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Solution : to locate √2 on number line follow these steps :-
1) draw a line A and B such that AB = 1cm.
2) Construct a perpendicular line on point B of 1cm and mark that
point C.
3) Join A and C and you will get AC = √2.
4) Now, open your compass and take the radius AC and after
taking the radius from point A cut an arc between point B and D.
5) And mark the A to point where the arc has cut the number line
√2.
1 cm
1 cm√2
√2
A B
C
D
√2 cm

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NOW YOU WHERE THINKING HOW AC = √2.
AC = √2 because of the formula of Pythagoras theorem.
𝑎2 + 𝑏2 = 𝑐2
So, 1 + 1 = √2
2 2

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HOW TO LOCATE OTHER √ ’S ON NUMBER
LINE.
You can locate other √ ’s with the help of the Pythagoras theorem.
For example : to locate √3 on number line.
√2 + 1 (note : any √ multiply by the same √ give an integer.)
2 + 1 = 3
3 = √3
Now from this we know we have to take AB = √2cm and BC = 1cm
because we have to take AB = √2 we have to first make √2 on
number line and then √3.
2 2

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THANKS FOR VIEWING
MADE BY : ADITYA MATHUR