Is #sqrt3# rational or irrational?

2 Answers
Feb 22, 2018

Irrational

Explanation:

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)#

Multiplying both ends by #(2+x)# we get:

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

Subtracting #2x# from both sides we get:

#x^2 = 3#

and hence:

#x = sqrt(3)#

So we have found:

#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 continued fraction does not terminate, #sqrt(3)# is not rational.

Bonus

Here's a fun method to calculate rational approximations to #sqrt(3)#:

First note that:

#7^2 = 49 = 48+1 = 3(4^2)+1#

Hence an efficient approximation to #sqrt(3)# is #7/4#

Consider the quadratic with zeros #7+4sqrt(3)# and #7-4sqrt(3)#:

#(x-7-4sqrt(3))(x-7+4sqrt(3)) = (x-7)^2-48#

#color(white)((x-7-4sqrt(3))(x-7+4sqrt(3))) = x^2-14x+49-48#

#color(white)((x-7-4sqrt(3))(x-7+4sqrt(3))) = x^2-14x+1#

Now if #x^2-14x+1 = 0# then #x^2 = 14x-1#

Define a related sequence of integers recursively by:

#{ (a_0 = 0), (a_1 = 1), (a_(n+2) = 14a_(n+1)-a_n) :}#

Then the ratio between successive terms will tend to #7+4sqrt(3)#

The first few terms are:

#0, 1, 14, 195, 2716, 37829, 526890#

Let's stop there to find:

#7+4sqrt(3) ~~ 526890/37829#

So:

#sqrt(3) ~~ 1/4(526890/37829-7) = 1/4(262087/37829) = 262087/151316 ~~ 1.7320508076#

Feb 23, 2018

Irrational

Explanation:

#sqrt3# is irrational because its decimal expansion is an infinite number of digits that never repeat in a particular pattern, and we cannot express it as a ratio. Think about this:

#sqrt36# is rational, because it yields a finite value, #+-6#. Also, we can express #sqrt36# as the ratio of two numbers. We know #sqrt36# (which is #+-6#), can be expressed as:

  • #6/1#
  • #18/3#
  • #30/5#

We cannot express #sqrt3# exactly as the ratio of two numbers. If we evaluate #sqrt3# on a calculator, we get #1.73205081#. Now, your calculator gives you a finite amount of digits, but in reality, like #pi#, #sqrt3# has an infinite number of digits that don't repeat in a pattern, making it irrational.

NOTE: #sqrt3# is approximately equal to #97/56#.