Proof: 22 is an irrational number


Let’s assume the definition of rational numbers is known. Real numbers that are not rational are called irrational numbers.

Here, we will prove that 22 is an irrational number.

Proof

We will prove by contradiction.

Assume that 22 is rational, then there exist coprime natural numbers m,nm,n such that it can be written as:

2=mn2=mn

From this, we get:

2n2=m22n2=m2

The left side ( 2n22n2 ) is even, so is the right side ( m2m2 ). In other words, we can express mm as m=2km=2k for some natural number kk, and:

2n2=(2k)2=4k2n2=2k22n2=(2k)2=4k2n2=2k2

From this, we can infer that nn is also even. However, this contradicts the assumption that m,nm,n are coprime. Therefore, 22 is irrational.

Similarly, it can be proven that nn is irrational when nn is a prime number.

Generally, when bb is not a perfect square, ab(a,bN) becomes irrational. The product of a rational and an irrational is irrational, so it’s sufficient to prove that b is irrational. Also, if b is the product of prime factors raised to the power of 2, then it’s sufficient to prove the case where b is a prime number.

Proof

Assume b is factorized into k different prime numbers b1,b2,,bk as follows:

\[ b = b_1 b_2 \cdots b_k ]

Assume b is rational, then there exist coprime natural numbers m,n such that it can be written as:

b=mn

From this, we get:

bn2=m2

The left side bn2 includes b1,b2,,bk as prime factors, so does the right side m2. In other words, m can be expressed as:

m=cb1b2bk=cb

Similarly to m, we have:

bn2=(cb)2=c2b2n2=c2b

From this, we can infer that n is also a multiple of b, but this contradicts the assumption that m,n are coprime.

Thus, n becomes rational (an integer) only when n is a perfect square.

In this case, we have proved the property for square roots, but the same proof can be applied to mn(m,nN).

Additional Proposition

The product of a rational number and an irrational number is irrational.

Proof

We will prove by contradiction.

Let p,q,r,s be integers, and x be an irrational number. Assume that the product of a rational number pq and an irrational number x is a rational number rs. Then, we have:

pq=xrs

Rearranging the terms:

x=psqr

This implies that the right side is rational, which contradicts the fact that x is irrational.

Therefore, the product of a rational and an irrational is irrational.