prove that √5 is irrational
Answers
Answer:
Here we go with the proof;
Let us assume that √5 is rational
So, √5 is a rational number
All rational numbers are in p/q form
So, √5 = p/q , where p≠0 and p&q are integers
Squaring on both sides
(√5)²=(p/q)²
5=p²/q²
5q²=p²–(1)
5 divides p [p is any prime number and a is any positive integer and p divides
a² then p divides a]
So, p is a positive integer—(2)
Then, p=5r, r is a positive integer
Squaring on both sides
p²=(5r)²—(3)
By 1&3
(5r)²=5q²
25r²=5q²
5r²=q²
So, 5 divides q²
Then, 5 divides q [by above reason]
Then,q is a positive integer—(4)
By 2&4
p and q have common factor 5.
This is a contradiction since p and q are co-primes
So, our assumption is wrong
Therefore √5 is an irrational number.
Step-by-step explanation:
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