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21.
Prove that $square root of 5$ is irrational.

Let us assume, to the contrary, that

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is rational.

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

space space space space space space space space space space space space square root of 5 equals straight a over straight b space space space space rightwards double arrow space space space space space space square root of 5 space straight b equals straight a

Squaring on both sides, we get

5b² = a²

Therefore, 5 divides a²

Therefore, 5 divides a

So, we can write

a = 5c for some integer c.

Substituting for a, we get

5b² = 25c²

⇒ b² = 5c²

This means that 5 divides b², and so 5 divides b.

Therefore, a and b have at least 5 ts a common factor.

But this contradicts the fact that a and b have no common factors other than 1.

This contradiction has arisen because of our incorrect assumption that $square root of 5$ is rational.

So, we conclude that $square root of 5$ is irrational.

148 Views

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

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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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141 Views

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: