answersLogoWhite

0

By way of contradiction, suppose sqrt(2) is rational. Then there exist integers n and d such that sqrt(2) = n/d. We can also assume (without loss of generality) that n and d have no common factors, like writing the fraction in lowest terms. Multiplying both sides by d, and then squaring both sides, gives 2d2 = n2. Every integer can be written as a power of 2 times an odd number, so write n = 2ia and d = 2jb, where a and b are odd. Plugging into the previous equation gives 22j+1b2 = 22ia2. Since a2 and b2 must be odd, they must be equal, and hence 2j+1 = 2i. This is impossible; an odd integer cannot equal an even integer. Therefore the original assumption must be false, namely sqrt(2) is an irrational number.

User Avatar

Wiki User

15y ago

Still curious? Ask our experts.

Chat with our AI personalities

RafaRafa
There's no fun in playing it safe. Why not try something a little unhinged?
Chat with Rafa
ProfessorProfessor
I will give you the most educated answer.
Chat with Professor
MaxineMaxine
I respect you enough to keep it real.
Chat with Maxine

Add your answer:

Earn +20 pts
Q: Why is the square root of 2 an irrational number?
Write your answer...
Submit
Still have questions?
magnify glass
imp