Prove that sqrt(2) is irrational

First, let's assume that sqrt(2) is rational. That is, it can be expressed in the form a/b, where a and b are integers and the fraction is simplified as far as possible.
So we have sqrt(2) = a/b --> 2 = a2/b2 --> a2=2b2. Since b2 must also be an integer, doubling it to a2 must be an even number. Only even numbers square to give even numbers, so a is also even. Let a = 2n, then a2 = 2b2 = 4n2 --> b2 = 2n2. From this we can see that b2 is even, so b must be too. We've now established that both a and b are even, but this means that the original fraction wasn't simplified as far as possible. This is a contradiction, so we can conclude that the assumption that sqrt(2) is rational is incorrect.

Answered by Mark W. Maths tutor

4062 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

A curve has parametric equations: x = 3t +8, y = t^3 - 5t^2 + 7t. Find the co-ordinates of the stationary points.


A Polynomial is defined as X^3-6X^2+11X-6. a)i Use the factor theorem to show that X-3 is a factor. ii Express as a linear and quadratic b)Find the first and second derivative c) Prove there is a maximum at y=0.385 to 3DP


Find the inverse of the function g(x)=(4+3x)/(5-x)


solve 2cos^2(x) - cos(x) = 0 on the interval 0<=x < 180


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences