Prove that √5 is an irrational number.?

Let us assume that √5 is a rational number.
So √5 = p/q, where p, q are co-prime integers and q ≠ 0

Squaring both the sides:
5 = p²/q²
p² = 5q² …… [i]

p²/5= q²
* So 5 divides p², and thus p is a multiple of 5 …… (*)
p = 5m
p² = 25m² …… [ii]

[i] = [ii], we get
5q² = 25m²
q²=5m²
* q² is a multiple of 5, and thus q is a multiple of 5 …... (#)

From (*) and (#):
p and q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number.
√5 is an irrational number.

Hence proved.

====
* By theorem:
If p is a prime number and p divides a², then p divides a where a is a positive number.

Assume that sqr(5) is a rational number, then
it can be expressed as a fraction p/q where
(p,q)=1 & q=/=0. Thus
p/q=sqr(5)
=>
(p/q)^2=5
=>
(p/q)(p/q)=5
The LHS is an irreducible fraction &
is impossible^* an integer. However,
the RHS is 5, an integer. This conflict
proves that the assumption is false,
hence sqr(5) is an irrational number.
^*
Further details: p,q are integers, p,q>0 & (p,q)=1 (relative prime).
(1) q=/=1 for otherwise,p^2=5=>p=5/p(p=/=5 of course)=>5 is not a prime that is false. Thus LHS=/= an integer.
(2) If p<q, then (p/q)^2 is a proper fraction & is not an integer. If p=0, then 0=5!
(3) If p=q, then (p/q)^2=1=>1=5 is false.
(4) If p>q, then (p/q)^2 is an improper fraction & if p/q=n, an integer=/=1, then it contradicts to (p,q)=1.
So, (p/q)^2 is impossible to be an integer.

The square root of 5 is a never ending and never repeating decimal so thus is irrational.
I do believe they want you to show your work with a number line but I don't know. Good luck

p/q = sqrt(5)

p and q are integers and they share no common factors

p^2 / q^2 = 5
p^2 = 5q^2

q^2 will have an even number of prime factors
p^2 will also have an even number of prime factors

So we have an even number of prime factors on the left hand side and an odd number of prime factors on the right hand side.  That means that we have a contradiction.  Either p or q can't be an integer, which means that our initial assumptions don't hold.  Therefore, sqrt(5) is irrational.