Question

Is 3.567 rational?

Answer 1

Yes. Irrational values, like `pi` or `sqrt2`, have a never ending decimal value that does not repeat. But `3.567` does have an end, and we can also write it as a fraction and have it remain accurate: `3567/1000`

There is no decimal or fractional value of `pi` that is accurate (not an approximation but a representation of the exact value of `pi`), so `pi` is irrational

Answer 2

Yes! The decimal equivalent of a `ul("rational number")` is either

Terminating decimal `-> "example " 0.125`
Has a repeating cycle of values for ever`->" example " 0.33333....`

Another example: `" "0.125125125125...`

Explanation:

`color(red)("Just for the hell of it")` lets determine the fractional equivalent of
0.125125125....

Set `x=0.125125125...." "Equation(1)`

We need to 'get rid' of the decimal part so to do this

Multiply both sides by 1000 giving

`1000x=125.125125125...." "Equation(2)`

Subtract Eqn(1) from Eqn(2) to 'get rid' if the decimal

`1000x =125.125125125.....`
`ul(color(white)(1000)x=color(white)(12)0.125125125...larr" Subtract")`
`color(white)(1)999x=125`

Divide both sides by 999

`x=125/999 color(white)("d") =color(white)("d")0.125125125....`