Prove that Root 5 is an Irrational Number

Prove that root 5 is an irrational number

Let us demonstrate that √5 is an irrational number by proving the theorem.

Using the method of contradiction, it is possible to demonstrate the answer to this question. Let’s take it for granted that the number √5 is a rational number. If √5 is rational, then it may be expressed as an equation of the type a/b, here integers are a and b. 

√5/1 = a/b

√5b = a

Bringing both sides into balance,

5b² = a²

b² = a²/5 —- (1)

This indicates that a² is divided by 5.

That indicates that it can also divide a.

a/5 = c

a = 5c

On squaring, we get

a² = 25c²

Replace a² in the equation with its value (1).

5b² = 25c²

b² = 5c²

b²/5 = c²

This indicates that b² may be divided by 5, and hence, b can likewise be divided by 5. Since this is the case, a and b share the factor 5 in common. However, this goes against the fact that a and b are in the coprime position. The fact that we made the assumption that √5 is a rational number led to the formation of this contradiction. As a result, we are forced to the conclusion that √5 is irrational.

Therefore √5 is an irrational number.