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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

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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) $6 plus square root of 2$

Let us assume to the contrary, that

square root of 2

is rational.

Then,

6 plus square root of 2

is rational.

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

space space space space space space space space 6 plus square root of 2 equals straight a over straight b rightwards double arrow space space space space 6 minus straight a over straight b equals square root of 2

Since, a and b are integers, we get

straight a over straight b

is rational and so,

6 minus straight a over straight b

is rational and so,

square root of 2

is rational.

But this contradicts the fact that

square root of 2

is irrational.

So, we conclude that

6 plus square root of 2

is irrational.

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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: