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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 √2√2 is an irrational number.
Proof
We will prove by contradiction.
Assume that √2√2 is rational, then there exist coprime natural numbers m,nm,n such that it can be written as:
√2=mn√2=mnFrom this, we get:
2n2=m22n2=m2The 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=4k2⇔n2=2k22n2=(2k)2=4k2⇔n2=2k2From this, we can infer that nn is also even. However, this contradicts the assumption that m,nm,n are coprime. Therefore, √2√2 is irrational.
Similarly, it can be proven that √n√n is irrational when nn is a prime number.
Generally, when bb is not a perfect square, a√b(a,b∈N) 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=mnFrom this, we get:
bn2=m2The 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=cb1b2⋯bk=cbSimilarly to m, we have:
bn2=(cb)2=c2b2⇔n2=c2bFrom 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 m√n(m,n∈N).
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=x⋅rsRearranging the terms:
x=psqrThis 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.