Is √3 rational or irrational?

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Answer 1 · 2018-01-26T00:26:34

#sqrt(3)# is irrational

In common with the square root of any prime number, #sqrt(3)# is irrational.

Suppose #x > 0# satisfies:

#x = 1+1/(1+1/(1+x))#

#color(white)(x) = 1+(1+x)/(2+x)#

#color(white)(x) = (3+2x)/(2+x)#

Then multiplying both ends by #(2+x)# we find:

#x^2+2x = 3+2x#

Subtracting #2x# from both sides:

#x^2 = 3#

and hence #x = sqrt(3)#

So:

#sqrt(3) = 1+1/(1+1/(1+sqrt(3)))#

#color(white)(sqrt(3)) = 1+1/(1+1/(2+1/(1+1/(2+1/(1+1/(2+...))))))#

Since this is a non-terminating continued fraction, #sqrt(3)# is not expressible with a teminating fraction. That is, #sqrt(3)# is irrational.