1.

Prove that √5 is irrational no., by using the result prove that 6+√5 is irration​

Answer»

ANSWER: Let √5 be a rational number.

This means √5= a/B (where a and b are coprime and q is not equal to 0)

{SQUARING both sides}

5=a^2 ÷ b^2

Then b^2=a^2÷5

This means that 5 DIVIDES a^2

Which ALSO means 5 divides a.

[Let a=5c]

now. b^2=(5c)^2 ÷ 5

b^2=5c^2

c^2=b^2 ÷5

This means 5 divides b^2 which means 5 divides b.

Thus, a and b have 5 as their common factor. But this contradicts

our assumption that a and b are coprime. So our assumption is wrong and √5 is irrational.

Let 6+√5 be rational.

This means 6+√5= a/b

√5=a/b -6

√5=(a-6b)/b is rational because a,b,6 are rational.

This means √5 is also rational.

But this contradicts the fact that√5 is irrational. So our assumption is wrong and 6+√5 is irrational.


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