how or why does pi have those specific digits?



3.14159…. so on and so forth, but my question is *why* and *how* did we decide or learn that those digits were of pi if it’s an irrational number? if it never ends couldn’t you technically just make it up? like i saw a news article, a woman/her team calculated TRILLIONS of digits of pi—how is that possible?

In: Mathematics


pi can be written as the result of an infinite sum of rational numbers. While it’s impossible to compute an infinite sum, what you can do is use an appriximation of the formula.

By summing only the first 1000 terms of the sum, you will already have a result very close to pi. And if you want an even higher precision with more decimals, you can sum the first 10000 terms instead.

That’s also the way your calculator will compute the value of sin(2) or e^2.4. These irrational numbers can also be written as an infinite sum of rational numbers.

The first thing to keep in mind is that we as modern humans tend to use base-10 (although that is not always the case), but other bases exist. For example, minutes and seconds are base 60, hours are base 24, and computers work in base 2 (0 and 1).

Pi is defined as the ratio between the circumference and the diameter of a circle. As any number it can be represented in any base, although in different ways. [Here]( you can find some examples of pi represented in other numeral systems. So even if you found a special pattern in the digits of pi (in base 10), it wouldn’t be the same in other bases.

That said,

>why and how did we decide or learn that those digits were of pi if it’s an irrational number

we didn’t *decide* those were the digits of pi. Putting aside [some absurd proposal to arbitrarily set a different value to pi](, once you define a numeral system (base 10? base 60? base 2?), you don’t *decide* the digits of pi. You simply try to *learn* them, to discover them, especially since pi has actually a meaning. It’s not like someone suddenly said “hey, the number 3.1415926…. doesn’t represent anything, but let’s call it pi” or “hey I want to call some number pi. What number? Well, 3.1415926… can do the trick”.

>if it never ends couldn’t you technically just make it up

No. It never ends, so you’ll never be able to compute its *exact* value but that doesn’t mean you can make it up. You can set a threshold where you stop caring, for example after the second or fifth decimal digits, you can approximate, but you can’t make up the *exact* value of pi.

Firstly to clarify we didn’t decide the digits of pi, we discovered them in the same way we discovered that the square root of 2 is 1.414213…. . You can calculate pi to a couple of significant figures by constructing a circle (e.g with a compass) and actually measuring the ratio of the diameter to the circumference but this quickly becomes inaccurate.

One step up from that (and the method used for many years) was by drawing polygons (shapes with straight lines) which approximated a circle and working out their areas but again this is very slow and inefficient.
Then around the time of Newton we made some very important discoveries that there were many infinite summations which converged to some multiple of pi.
For example 1 – 1/3 + 1/5 – 1/7 + 1/9 -… = pi/4. This allows you to very quickly calculate pi to many decimal places. As time went on we discovered more series which converged to pi even quicker and using computers sped this up by an incredible amount.

Pi is defined as the ratio of a circle’s circumference and its diameter. So you can take a circle and you can draw a shape, say a hexagon, around the circle like this:


The center of the circle and the center of the hexagon are the same. The radius of the circle is therefore also the distance from the center of the hexagon to the midpoint of one of its sides (where it touches the circle). We can also calculate the perimeter of the hexagon, which is more than the perimeter of the circle. We can take the ratio of the perimeter of the hexagon and the radius of the circle and we know we’ll get an answer that’s more than pi.

Similarly, we can draw a hexagon *inside* the circle like this:


And do the same thing and get an answer that’s less than pi.

Combine both and we have an upper and lower boundary for pi. We can improve this boundary by doing the same thing with shapes with more sides, as their perimeters will be closer to the circumference of the circle.

Now, this is a very crude and slow way of doing it, but illustrates the principle of how we can nail down a number like pi. Today we have some clever mathematical formulae which have been proven to produce pi, and do so much more quickly than the above method.

This video taught me how to instinctually understand pi and how we calculate it better than any teacher ever did for me. This visual will explain WHY the number is 3.14, and as others have said, the “calculating to the X number” is getting it more and more precise to that degree. Reddit – Damnthatsinteresting – Explaining Pi With Pizza Pies