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Real Numbers

Short Answer Type

21.

Prove that $square root of 5$ is irrational.

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22.

Prove that $space 3 plus 2 square root of 5$ is irrational.

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23. Prove that the following are irrational:

(i) $fraction numerator 1 over denominator square root of 2 end fraction$

(i) $fraction numerator 1 over denominator square root of 2 end fraction$

Let us assume, to the contrary, that $fraction numerator 1 over denominator square root of 2 end fraction$ is rational.

So, we can find coprime integers a and b (≠ 0) such that

$rightwards double arrow space space space space space space fraction numerator 1 over denominator square root of 2 end fraction equals straight a over straight b rightwards double arrow square root of 2 equals straight b over straight a$

Since, a and b are integers, $straight a over straight b$ si rational, and so $square root of 2$ is irrational
So, we conclude that $fraction numerator 1 over denominator square root of 2 end fraction$ is irrational.

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24. Prove that the following are irrational:

(ii)

7 square root of 5

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25. Prove that the following are irrational:

(iii)

6 plus square root of 2

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26.

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i) $13 over 3125$

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27. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(ii)

17 over 8

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28. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

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29. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: