Question

Is the number `2.645751311` irrational?

Answer 1

No, see an explanation below:

Explanation:

We can rewrite this number as:

`2.645751311/1`

We can now multiply it by `1000000000/1000000000` which is a form of `1` and therefore does not change the value of the fraction:

`2.645751311/1 xx 1000000000/1000000000 = 2645751311/1000000000`

Because this number is a fraction of two integers it is a rational number.

Answer 2

No, but we can identify it as a rational number approximating `sqrt(7)`

Explanation:

If you take the square root of `7` on a calculator, it may well display `2.645751311`, but this is only a rational approximation to the square root of `7`.

Any decimal representation that terminates or repeats is rational and any that does not is irrational. So the number `2.645751311` is itself rational.

So if you only see a finite number of digits, how can you tell that the expression that it comes from is supposed to be irrational?

We can try spotting a pattern in the continued fraction.

Starting with `2.645751311`...

• Write down the integer part `color(red)(2)` and subtract it to get:

  `0.645751311`

• Take the reciprocal to get (approximately):

  `1.548583771`

• Write down the integer part `color(red)(1)` and subtract it to get:

  `0.548583771`

• Take the reciprocal to get (approximately):

  `1.822875653`

• Write down the integer part `color(red)(1)` and subtract it to get:

  `0.822875653`

• Take the reciprocal to get (approximately):

  `1.215250441`

• Write down the integer part `color(red)(1)` and subtract it to get:

  `0.215250441`

• Take the reciprocal to get (approximately):

  `4.645751225`

• Write down the integer part `color(red)(4)` and subtract it to get:

  `0.645751225`

Note that this is very close to a number we saw after step `1`, namely `0.645751311`

In fact it is close enough that we can put the difference down to rounding errors and identify the continued fraction for our number as:

`[2;bar(1,1,1,4)] = 2+1/(1+1/(1+1/(1+1/(4+1/(1+1/(1+1/(1+1/(4+...))))))))`

Repeating continued fractions like this are always irrational numbers of the form `p+sqrt(q)` for some rational numbers `p` and `q`.

To find out what expression this is a continued fraction for we can solve:

`x = 2+1/(1+1/(1+1/(1+1/(2+x))))`

`color(white)(x) = 2+1/(1+1/(1+(2+x)/(3+x)))`

`color(white)(x) = 2+1/(1+(3+x)/(5+2x))`

`color(white)(x) = 2+(5+2x)/(8+3x)`

`color(white)(x) = (21+8x)/(8+3x)`

So multiplying both ends by `(8+3x)` we get:

`8x+3x^2=21+8x`

Subtract `8x` from both sides to get:

`3x^2=21`

Divide both sides by `3` to find:

`x^2=7`

Since `x > 0` we have:

`x = sqrt(7)`