Prove by contradiction that sqrt(3) is irrational. (5 marks)

To the contrary assume that sqrt(3) is a rational number. Thus we can write sqrt(3) = a/b where a and b are coprime integers and b is non-zero. (1 mark for this or equivalent statement - may have that this fraction is irreducible)
By rearrangement we get that b * sqrt(3) = a, and therefore that 3b2 = a2. (1 mark)
Therefore 3 divides a2 and, as 3 is prime, 3 divides a, so we can write a = 3k where k is an integer. (1 mark)
By substitution we then get that 3b2=(3k)2=9k2 so b2 = 3k2. (1 mark)
Therefore 3 divides b2 and, as 3 is prime, 3 divides b. However, 3 also divides a so a and b are not coprime (alternatively may have that a/b is not irreducible). Hence we have a contradiction, so sqrt(3) can not be rational and must be irrational. (1 mark for this or equivalent statement - must use either irreducible or coprime consistently throughout)

DS
Answered by Dan S. Maths tutor

9913 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Relative to a fixed origin O, the point A has position vector (8i+13j-2k), the point B has position vector (10i+14j-4k). A line l passes through points A and B. Find the vector equation of this line.


Given the points P(-1,1) and S(2,2), give the equation of the line passing through P and perpendicular to PS.


What is the coefficient of x^2 in the expansion of (5+2x)^0.5?


Solve the equation: 2x+3y=8 & 3x-y=23


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences