Edit: Thanks for the answers everyone! From what I can gather, pi just do be like that, and other irrational numbers be like that too.
In: 5168
It just be like that.
Pi is an *irrational* number, which means that it cannot be (fully and accurately) expressed as a ratio of two integers. That means that, as a decimal expression, the digits will just go on and on without any clear pattern.
By contrast, *rational* numbers (which can all be expressed as a ratio of two integers) have decimal expressions that either terminate (like 3/4 = 0.75 exactly) or repeat (like 1/3 = 0.33333…).
The real numbers are far more dense in the irrationals, tho.
The question “Why do the decimals of pi go on forever without repeating?” is the wrong question. From our perspective it can seem like this is a miraculous and unique thing. But this cannot be further from the truth. *Almost all* numbers have this property. It is, actually, an innately boring and unspecial property that most numbers have. In fact, it is so rare for this NOT to be the case that if you choose a random real number between 0 and 1 then there is a 100% chance that its digits go on forever, without repeating, and contain infinite copies of every finite sequence of digits.
(Note: 0% does not mean “impossible” in math and 100% does not mean “guaranteed to happen”, see [Almost All](https://en.wikipedia.org/wiki/Almost_all) for a technical discussion. The gist is if you have infinitely many equally possible outcomes, then an individual outcome can’t have a positive probability since you could add enough of the probabilities together to get something over 100%, which can’t happen.)
The real question, when you have a number, is: Why *wouldn’t* the decimals go on forever without repeating? That is, you need a specific reason to make the number special like with its decimals eventually repeating or something. This is usually a special *arithmetic* property or relationship. For pi, there is no such relationship.
Moreover, we have already proved that pi’s digits go on forever without repeating. So we know it as a fact.
Irrational numbers be like that. No exact ratio using integers is possible.
This is slightly different from endless decimals, rational numbers like !/3 or 3/7 that just never can be expressed in terms of a ratio over 10 (in base ten). 3/7 in base 7 would be 3, and 1/3 in base 3 would be 1, but then 3/10 would be an endless number in either of those bases.
It just so happens that in our universe, squares and circles are incommensurable with each other (they cannot be used to measure each other exactly). If you take the countable integers to represent the length of the sides of a given square, then there will never be a number that can be represented by any combination of countable integers to represent the length of the sides of a corresponding circle which is inscribed within that square (having the same diameter as the side of that square). The ratio between these sides can be approximated as 3.14159…. But if we want to speak about it directly we have to use the term pi and acknowledge that any decimal description of the numbers we are manipulating will be an approximation.
Many other things in our universe are also incommensurable with each other— for example, the distance between two opposite corners of a square is incommensurable with the sides of that square in a different way than the sides of the square and the inscribed circle are. We therefore speak of this ratio as “the square root of two”
Take a single pixel in Microsoft paint, or a single Block in Minecraft. That’s a super small, not very accurate circle right? Now make a bigger circle out of those blocks, and the bigger the circle you make the more accurate a circle it is right?
Imagine making a circle the circumference of the whole universe, but you’re still making it out of atom sized pixels. It’s super accurate, literally can’t get any closer to a circle… And yet, it still has points and corners that prevent it from being perfect.
The “ideal image” of a circle cannot ever exist truly in a world built of smaller things. It’ll always be bumpy, so there’s always room mathematically to make such a circle bigger and more accurate. But no matter how accurate you get, it’s still not a perfect circle, so the measurement of Pi gets closer and closer to “true” but never actually reaches “perfect” or “finished”.
It’s hard to answer the “is there a reason” question, because it is something of a mistaken way of looking at it. You might ask, is there a reason why 4 is between 5 and 3 on the natural number line? That is just what it is to be the number 4. There is not much sense to be made of the question of why it is the the number between 5 and 3. There is no causal or historical explanation that can be appealed to.
There are, however, adjacent questions that can be asked, which might help you better understand what is going on. The number line consisting in whole numbers in sequence, like 1,2,3,4… and so on, is the natural numbers. When we add decimals into the mix, we get numbers between the natural numbers. The line of numbers called “The Reals” includes all of the numbers that are between the whole numbers, like 1,2,3,4. The Reals have a property that is sometimes called “density.” What this means is that for every two numbers, there is another number in between them. Between 1 and 2, there is 1.1. Between 1 and 1.1, there is 1.01. Between 1 and 1.01, there is 1.001. Etc. If you do this enough, you’ll realize very quickly that this process can go on forever, creating an infinite expansion of decimal places for each new number you generate using this method. So, it is just a property of real numbers that they go on forever like this, with an infinite decimal expansion that is the result of this density property. Pi is just one of the numbers on the line of the Reals and it goes on forever.
What makes Pi unique among numbers that go on forever is that it is an irrational number. This means that it cannot be expressed as a fraction. This also means that the numbers in the line don’t repeat. Contrast this with a number like 1/3. This is .33333…(3’s repeating infinitely). Pi and the number picked out by 1/3 both go on forever. But Pi does not repeat.
So, what is the reason why Pi goes on forever? Because it is a real number and Real numbers have infinite decimal expansions creating a dense number line. Why are they like that? That’s just what it is to be a Real number. Might as well ask why does a square have 4 sides. That’s just part of what it is to be a square.
In short, it just be like that.
I see a lot of answers explaining that pi is irrational, but not many answers for WHY that means the decimals go on forever. I’ll try my best here:
In a decimal like 3.1415… we have 3 + 1/10 + 4/100 +…. We KNOW pi is irrational, and you can do some research into that later if you’re curious why. That means that it can’t be written as a fraction of two whole numbers. If there was some end to the digits, then you could simply do all that fraction addition and, while it would be a pain to do, you would end up with a fraction of two whole numbers, making it a rational number. (E.g. 3.14=3+1/10+4/100=314/100)
That’s the contradiction. If the digits ever stopped, then pi is rational. We know pi is irrational, so the digits can never stop. (You can also look into the logic in that last step if you’re curious! A lot of math can be pretty simple, the notation can just seem scary because mathematicians like to be precise)
TLDR: if the decimals stop then pi would be rational, which it’s not.
Fun fact, the typical number (one you pull out randomly from the real number line) is almost guaranteed to be a decimal that goes on forever without repeating. It’s not a special property of pi specifically, the vast vast vast vast majority of numbers are irrational.
It’s the numbers that DON’T do this that are the strange unusual ones.
While answers like “it just be like that” are certainly true, I find it helpful to think of irrational numbers like pi as infinite series rather than specific magic numbers.
If you think of pi as the Leibniz formula i.e 4(1-1/3+1/5-1/7+…), you can sort of see how its computation may lead to infinite decimals as you add up more terms to infinity. Infinite series won’t always converge or converge to irrational numbers, but in this case it does.
Latest Answers