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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.

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55. Express

5. top enclose 2

in the form of

straight p over straight q comma

where p and q are integers and q ≠ 0.

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56. How many irrational numbers lie between $square root of 2$ and $square root of 3 ?$ Find any three irrational numbers lying between $square root of 2$ and $square root of 3.$

Infinitely many irrational numbers lie between $square root of 2$ and $square root of 3.$

One irrational number between $square root of 2$ and $square root of 3$ is

$square root of square root of 2 space square root of 3 end root space equals square root of square root of 6 end root space equals space 1 over 6 to the power of 4$
Another irrational number between $square root of 2$ and $square root of 3$ is $square root of square root of 2 space 1 over 6 to the power of 4 end root$
$equals 1 over 2 to the power of 4.1 over 6 to the power of 8 equals fraction numerator 1 over denominator 2 to the power of 4. end fraction fraction numerator 1 over denominator 2 to the power of 8. end fraction 1 over 3 to the power of 8 equals space 1 over 2 to the power of 4 plus 1 over 8.1 over 3 to the power of 8 equals 2 to the power of 3 over 8 end exponent.3 to the power of 1 over 8 end exponent$

Third irrational number between $square root of 2$ and $square root of 3$ is

$square root of square root of 2.2 to the power of 3 over 8 end exponent.3 to the power of 1 over 8 end exponent end root equals 2 to the power of 1 fourth end exponent.2 to the power of 3 over 16 end exponent.3 to the power of 1 over 16 end exponent equals 2 to the power of 7 over 16 end exponent.3 to the power of 1 over 16 end exponent$