Recall structures made from legos We do not see these as jus

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.

Solution

let m=2 and n = sqrt2

r= m/n = 2/sqrt2= sqrt2

m = p/gcd(p,q) = 2

=> m=p=2

similarly

n= q/gcd(p,q)=sqrt2

=> q = sqrt2