Recall structures made from legos. We do not see these as just one lego brick af

Question

Recall structures made from legos. We do not see these as just one lego brick after another, we see substructure. Try to find some substructure in the following lines of proof.

Assume r is in Q.

Assume r*r = 2

choose m,n in Z such that the greatest common divisor(gcd) of m and n is 1

choose p,q in Z such that q does not equal 0 and r=p/q

let m, by definition, be p/gcd(p,q)

let n, by definition, be q/gcd(p,q)

m/n=r

2 divides m

m*m = 2*n*n

2 divides n

n*n = 2 p*p

2=(m/n)(m/n)

2=(2p/n)(2p/n)

Therefore, the square root of 2 is irrational.

The substructure is that a lemma is used in the overall proof. The lemma is, if a square number y-x^2 is even, then so is x even.

Explanation / Answer

The above problem proving that square root of 2 is irrational, can be broken into three lego steps

First Step:

Assume r is in Q.

Assume r*r = 2

choose m,n in Z such that the greatest common divisor(gcd) of m and n is 1

choose p,q in Z such that q does not equal 0 and r=p/q

let m, by definition, be p/gcd(p,q)

let n, by definition, be q/gcd(p,q)

Second Step:

m/n=r

2 divides m

m*m = 2*n*n

2 divides n

n*n = 2 p*p

2=(m/n)(m/n)

2=(2p/n)(2p/n)

Third Step: (Conductive/Hypothesis step)

In this step, we will contradict the above statement that square root of 2 is rational and hence the number square root of 2 is irrational

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