Question

Is `sqrt3` rational or irrational?

Answer 1

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`

Answer 2

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`.