# How come that pi (π) extends its decimal numbers infinitely?

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Is there any explanation behind this?

In: Mathematics

Also I am curious, Is there any data of the longest Pi decimal place that has been recorded?

All irrational numbers, and most rational numbers have an infinite decimal expansion. The only ones that don’t are a small subset of rational numbers (like 1/2), and even those can be interpreted to have an infinite decimal expansion: all rational numbers have repeating decimals. For such rationals that repeating decimal happens to be equal to zero, since 1/2=0.5000000… we usually just don’t write those zeros down.

Basically, not a special property of pi.

Pi is an irrational number. This means that it can’t be written as the ratio between two integers. This is not a special property of pi in any way – many numbers are irrational, for example the square roots of 2, 3, 5 (and of any number that isn’t a square of a whole number), and others. In fact, there are more irrational numbers than rational!

Anyway, if you try to write an irrational numbers – any irrational number – as a decimal fraction, you’ll end up with an infinite and non repeating sequence of digits.

The proof that pi is irrational however is a bit too complicated for ELI5.

Note: there is a hypothesis that pi is a *normal* number. If pi is a normal number, then it means that every finite sequence of digits appears in pi. However there is no proof yet that pi is normal.

How come? Because it just is. There’s no fundamental reason why it is this way.

It was shown in the 1700s that pi is irrational (cannot be written down as a fraction of integers), and this proved that its decimal expansion was infinite and non-repeating.

We don’t actually know it’s an infinite number, it’s just that we haven’t counted to the end yet, which is always the problem with “infinity” – really it’s a case of “a number so big (or small) we haven’t counted to yet”

To be honest, you have to look it the other way: if you pick any (!) number from 0 to 100 or from 0 to 10 or from 0 to 1 etc , randomly (!), chances are 100% (not 99.9%, not 99.99999999999%, but 100%) that the decimal expansion of that random number is infinite. The number of numbers were finite decimal expansion is so little compared to all numbers, your question like asking: “why is not every particle in the universe a sand grain of a certain color?” times infinity.

The fact that we humans learn about numbers with finite decimal before we learn about any other numbers, is frankly kind of random.

Unfortunately there isn’t a “simple” proof that pi is irrational (ie it has infinite non-repeating decimals), in fact it’s such a hard problem that despite pi being studied since ancient Greece we only proved it was irrational in 1760.

A lot of the proofs involve sin and cos (remember your trigonometry class), which stop working properly if there was a way to write pi as a/b (ie if pi was a rational number).