Maths-
General
Easy

Question

Prove that 5 minus square root of 5 is an irrational number.

The correct answer is: YES an irrational number.


    Hint:
    The real numbers which cannot be expressed in the form of p/q(where p and q are integers and q ≠ 0) are known as irrational numbers. In the given question we are asked to prove if 5-square root of 5  is an irrational number. To do so we use the method of contradiction.
    Solution
    Let's assume that 5-square root of 5 is a rational number.
    Step 1 of 2:
    If 5-square root of 5 is rational, that means it can be written in the form of p/q, where p and q integers q ≠ 0.
    SO 5 - square root of 5 = p over q
    5 - p over q = square root of 5
    fraction numerator 5 q minus p over denominator q end fraction = square root of 5
    Step 2 of 2:
    Here, we see that fraction numerator 5 q minus p over denominator q end fraction is a rational number. But square root of 5 is an irrational number. So our assumption is wrong.
    Final Answer:
    As square root of 5 is an irrational number. So fraction numerator 5 q minus p over denominator q end fraction  must also be an irrational number. Thus, our assumption is wrong. Hence, 5-square root of 5 is an irrational number

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