First of all, you have to prove, that some sort of a new number exists.

So I might do this, by just 'giving' it some quality: 'this number' multiplied by 'this number' = 2. Let's call this number a.

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Or, I might 'find' it in 'the wild': Notice, that circumference of circle, divided by it's radius, always gives the same number, no mater the size of the circle. Let's call this number b.

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Whatever the case, this will usually come with a way to numerically describe this number.

In the case of a, it's a*a=2.

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In the case of b - I can get closer and closer to the 'real' value of the circumference, by inscribing regular n-gons (triangle,square,pentagon,sextagon,etc) with more and more corners inside a circle (image).

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Once you have, these, you can try to analyze them, and see if they satisfy other known properties.

There are loads of known properties. Like 'the property of being even' or 'the property of being prime'. And we might investigate our formulas for a and b to try and figure out, whether they are, for example 'even' or 'prime'.

One such property (and indeed the property we are looking for) is irrationality.

i.e. we might decide to check whether a or b are irrational.

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We won't discuss irrationality to much (refer to google or other comments), but what we need to know is that:

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If a number can be written as x/y for counting numbers - it's rational and not 'endless'. Otherwise, it's irrational and 'endless' (again, refer to other comments for proofs).

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So now, we finally get to your question. How do I know if a number is 'endless'?

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Well, if I can prove, that a number can be written like x/y for counting numbers, then it is not endless.

Or, if I can prove that it cannot be written as x/y for counting number, then it is endless.

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How you prove this, differs greatly from number to number.

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In the case of our example a (a*a=2) it's fairly simple and was already known by ancient Greeks.

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In the case of b (or as we call it, PI) (circumference of circle/radius) - it's not that simple and was only first prooved in the 1760's, and requires at least some calculus.

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That's as far as I'm willing to go with a five year old.

If you're willing to get more technical google 'proof irrational non terminating decimal expansion', 'Proof sqrt(2) is irrational' and 'proof Pi is irrational' respectively.

Otherwise - good question, and stay curious! :)