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Question

Prove that 52 is irrational.

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Solution

Let √2 = ab where a,b are integers b ≠ 0
We also let that ab is written in the simplest form.
⇒ √2 = ab

⇒ 2 = a2b2

⇒ 2b2=a2
2b2 is divisible by 2
a2 is divisible by 2
⇒ a is divisible by 2
∴ Let a = 2c
a2 = 4 c2

⇒ 2 b2 = 4 c2b22 = 2 c2
∴ 2c2 is divisible by 2
b2 is divisible by 2
∴ b is divisible by 2
∴a are b are divisible by 2 .
This contradicts our notion that a/b is written in the simplest form as our notion is wrong
∴ √2 is irrational number.
If we multiply 5 in this so 52 is also irrational.


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