Question

prove that root 5 +root 6 is an irrational number

Answer 1

This question can be solved in two ways.
1)Let us assume, to the contrary, that √5+√6 is irrational.
So we can find two integers numbers a and b(≠0), in the following way,
√5+√6 = a/b
Rearranging,
√5 = a/b - √6
= integer/integer - rational ...( as √6 is assume as rational)
So, this means that √5 is rational.
But this contradict the fact that √5 is irrational.
Our assumption is wrong.
Hence, √5 + √6 is irrational.
2) Let us assume, to the contrary, that √5+√6 is irrational.
So we can find two integers numbers a and b(≠0), in the following way,
√5+√6 = a/b
Rearranging,
√6 = a/b - √5
= integer/integer - rational ...( as √5 is assume as rational)
So, this means that √6 is rational.
But this contradict the fact that √6 is irrational.
Our assumption is wrong.
Hence, √5 + √6 is irrational.
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