32. Prove that root 5 is irrational
Answers
Answer:
Step-by-step explanation:
If √5 is rational, then it can be expressed by some number a/b (in lowest terms). This would mean:
(a/b)² = 5. Squaring,
a² / b² = 5. Multiplying by b²,
a² = 5b².
If a and b are in lowest terms (as supposed), their squares would each have an even number of prime factors. 5b² has one more prime factor than b², meaning it would have an odd number of prime factors.
Every composite has a unique prime factorization and can't have both an even and odd number of prime factors. This contradiction forces the supposition wrong, so √5 cannot be rational. It is, therefore, irrational.
TO PROVE :- Root of 5 is irrational
PROOF
We can easily prove this by using the cobtradiction method :-
Let root 5 be x/y where x and y are co-primes.
Hence both x and y are divisible by 5 so x and y aren't co-primes.
Therefore it contradicts our assumption so root 5 is irrational.