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Number Systems

Short Answer Type

51. Find one irrational number between the numbers a and b given below:

a = 0.1111 ........ =

0. top enclose 1

b = 0.1101.

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52. Find two rational numbers in the form

straight p over straight q

between 0.34 344 3444 34444 3 ...... and 0.36 366 3666 36666 3......

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53. Express

5. top enclose 347

in the form

straight p over straight q

where p and q are integers and q ≠ 0.

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54.
Express $0. top enclose 001$ in the form of $straight p over straight q$ where p and q are integers and q ≠ 0.

Let $straight x equals 0. top enclose 001$

Then, x = 0.001 001 001 001...... ...(1)
⇒ 1000x = 1.001 001 001 001...... ...(2)
Subtracting (1) from (2), we get

99x = 1
$rightwards double arrow space space space space space space space space straight x equals 1 over 99$
$rightwards double arrow space space space space space space space space straight p equals 1 comma space space space straight q space equals space 999 space left parenthesis not equal to 0 right parenthesis$