Proof By Contradiction | Number Theory | Chegg Tutors

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A lot of students have asked me in the past about proofs, and about how scared they are of them. To be totally honest with you, when I was an undergraduate, I was initially terrified of proofs! I thought that it was some sort of wizard skill that couldn’t be learned. The interesting thing is that, for a lot of higher-math (modern algebra, modern analysis, number theory), the actual math you will use to solve the proofs is very basic! In this exercise, we prove that the square root of 3 is an irrational number. We use a 'Proof by Contradiction' and assume that the square root of 3 is actually a rational number, and we set it up as (sqrt)3 = a/b. We understand that a and b have to be integers, that they have to have a greatest common divisor of 1 (in other words, they can’t be reduced any further), and b cannot be 0. From there, we square both sides and perform some basic algebraic operations to discover that a^2 = 3b^2… which we discover means that 3 divides a. A bit later, we discover a similar truth about b… 3 divides b! Realizing that 3 divides both a and b, we can see that the greatest common divisor (gcd) is greater than one… and we have our contradiction! This is a fun proof that can really introduce you to the creativity that exists in mathematics beyond the rote memorization to which you may have grown accustomed.

-- From Augie K., Number Theory tutor on Chegg Tutors


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