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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.

Let us assume, to the contrary that

space 3 plus 2 square root of 5

is rational.

That is, we can find coprime a and b (b ≠ 0)

$such space that space space space space space space space space space space space 3 plus 2 square root of 5 equals straight a over straight b$

$Therefore comma space space straight a over straight b space equals space 3 space plus 2 square root of 5 space$

$rightwards double arrow space space space space space space fraction numerator straight a minus 3 straight b over denominator straight b end fraction equals 2 square root of 5 rightwards double arrow space space space space space space space space fraction numerator straight a minus 3 straight b over denominator 2 straight b end fraction equals square root of 5 rightwards double arrow fraction numerator straight a over denominator 2 straight b end fraction minus 3 over 2$

Since a and b are integers, we get $fraction numerator a over denominator 2 b end fraction minus 3 over 2$ is rational, and so $square root of 5$ is rational.

But this contradicts the fact that $square root of 5$ is irrational.

This contradiction has arisen because of our incorrect assumption that $space space space 3 plus 2 square root of 5$ is rational

So, we conclude that $space space 3 plus 2 square root of 5$ 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

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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: