The mystery of pi.

I don't know the answer, but here are some comments. Thanks for the great question.

Exact division

10 / 2 means distributing 10 items in 2 groups, such as 10 candies between 2 people.

Division with reminder

11 / 2 is similar, but we get 5 candies each, and 1 candy is left as reminder. When we have more people than candies left, we perform a trick. We imagine multiplying it by ten: 1*10 = 10 candies again, yay!

This trick is specified as a number after decimal point. We can do the example above, in two steps:

1. With 11 / 2 we distribute 10 (5 each), and are left with 1.
2. So we do our trick, multiply by 10, and re-distribute between two: that is 10 / 2 = 5.

So the result of 11 / 2 = 5.5, the first 5 comes from the first step, the second five (after the decimal point) from the second step.

Periodic Reminder

This can be continued forever when there is a reminder. For example, 1/3 is:
1. 0 from our first step (because 1 is less than 3),
2. With the trick, we get 10 / 3, distribute 3 to each, and get 1 left. (As a partial result we then have 0.3),
3. With the trick, we get 10 / 3 again! And again distribute 3 to each, and get 1 left. (We have 0.33),

This process continues forever. (We have 0.333...). Some numbers, such as the beautiful 1/7 repeat not one digit but a set of them. In this case, if we perform the steps above, every six steps we go back to the same numbers.

Rational Numbers

Those are rational numbers because we can express them as a ratio of integers (a fraction). As we saw, a rational number can still repeat forever (1/3, 1/7,..).

• If a number terminates, it is rational, since it can always be represented as a fraction (multiply by many thousands until we get an integer).

Imagine we have 0.124134643250982432159 (my random typing). Then we can just do 124134643250982432159 / 1000...

And we got the fraction.

We could ask a few questions: Do all infinite numbers with a repeating motif (such as 0.333...) have a corresponding fraction? (I assume so, but can't prove it)

Irrational Numbers

What if it does not repeat and continues forever (does not have an end)? How do we even know if those don't have an end? (Usually by some proof).

• So we may hypothesise here (I may be wrong) that, if a number does terminates (previous bullet) or repeats a motif it's rational. Irrational numbers then repeat forever without a motif.

Examples of such numbers are the square root of 2 or of 5, continue forever without a repeating pattern. Pi and Euler's number e are cousins of those.

The proof for "the square root of 2 being irrational" is more accessible than the proof for pi (I don't understand the one for pi).

How do we prove the case for the square root of 2 being irrational?

The simplest way is by contradiction, and can be found here.

Thanks, emanningis, I think I'm out of my depth here among such bright people.

Apologies if it contains too much detail.

The question is great, and the last link is an example of how hard it is to prove, even for a simpler case (square root of two, which is also irrational).

I tried to justify simpler cases, as examples, before getting to the link.

If I can help further or there are parts unclear feel free to ask.

It may be coincidence at one level, since there are infinite irrational numbers.

The connections are more at the conceptual level, and some of the simplest proofs for each are by contradiction.

Pi and e do have some connections on important formulas, but not sure whether their irrationality proofs are similar.

It may be coincidence at one level, since there are infinite irrational numbers.

The connections are more at the conceptual level, and some of the simplest proofs for each are by contradiction.

Pi and e do have some connections on important formulas, but not sure whether their irrationality proofs are similar.