Question #c82fe

Mar 30, 2015

No, the infinite amount of non-repeating digits of $\pi$ has nothing to do with geometrical questions. There are simply three kind of numbers:

1. Numbers with a finite amount of decimal digits. This set of numbers includes the ones with no decimal at all (i.e. $2$, $5$, $- 10$, and so on), and those with a finite amount (i.e. $2.2$, $8.65$, $0.2659$, and so on). This numbers can alyaws be expressed in a fraction form, since you can take the whole number and divide by the right power of ten: for example, $28.659 = \frac{28659}{1000}$.
2. Numbers with repeating decimals. These kind of numbers have a block of digits which goes on forever. Examples can be $0.33333 \ldots$, or $2.1826262626 \ldots$, and so on. These numbers can be written as a fraction, too, but the algorithm is a bit more complicated.
3. Finally, there are numbers with an infinite amount of decimal digits, with no pattern. Examples can be $e$ or $\pi$, and most log, sine and cosine values, except for some lucky cases. These numbers cannot be expressed as a fraction, and this means that you only can obtain its decimal digits by approximation. But even if you found the first $n$-th digits, with a very, very, very big $n$, you'd have no clue for the $n + 1$-th digit.

Some of these numbers, like $e$ or $\pi$, are trascendental, which means that they cannot be solution of a polynomial with whole coefficients. There is a variety of studies and results about this subject, and many things are quite hard or technical, so mine was only an introduction and an attempt to help you "focusing" on the right direction. Hope it helped!

Mar 30, 2015

The number $\pi$ cannot be expressed in our base 10 number system as a terminating decimal. (A decimal that ends.)

This has nothing to do with "number of corners" in a circle. (A circle has no corners.)
It has to do with the fact that $\pi$ cannot be written as a ratio (fraction) of integers (whole numbers).

$\frac{3}{25}$ can be expressed as a terminating decimal, $0.12$

So can $\frac{317}{50}$ (it's $6.34$).

$\frac{1}{3}$ cannot be expressed with finitely many digits in our base 10 number system (our decimal system). In a base 6 system, we can express $\frac{1}{3}$ Using a "decimal" (better to call it a "dot")

In our base 10 system, digits to the right of the 'dot' count ${\left(\frac{1}{10}\right)}^{\text{ths}}$ then ${\left(\frac{1}{100}\right)}^{\text{ths}}$, then ${\left(\frac{1}{1000}\right)}^{\text{ths}}$ and so on.

In a base 6 system, digits to the right of the 'dot' count ${\left(\frac{1}{6}\right)}^{\text{ths}}$ then ${\left(\frac{1}{36}\right)}^{\text{ths}}$, then ${\left(\frac{1}{216}\right)}^{\text{ths}}$ and so on.

So in a base 6 system, we write $\frac{1}{3} = {0.2}_{6}$ (The subscript is a politeness to warn people that we're using a base other than 10.)

Back to $\pi$. It can be proven (first proved in the 1700's) that $\pi$ cannot be written as a ratio (fraction) of two integers (whole numbers). We say that $\pi$ is irrational (not a ratio)

It can also be shown that an irrational number cannot be written with finitely many digits in any whole number base.

$\pi$ cannot be written using a finite number of digits to the right of the 'dot' in any whole number base.

The proof given in the 1700's relies on a continued fraction expression for the trigonometric function , "tangent".
Other proofs were given later. There is now a proof using ideas from calculus.
Another irrational number is $\sqrt{2}$ It also has a decimal representation that goes in infinitely. (And the same is true of $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$ and so on.)

Another example or an irrational number is $0.101001000100001000001 . . .$ (Each step puts an additional zero before the next $1$.

Mar 30, 2015

Others have given good answers as to why $\pi$ is irrational, so your only question that remains is why we only know 6.4 billion digits of $\pi$.

Put simply, computer power (or lack thereof). It takes ages to find the next digit, because you need to factor in 6.4 billion other calculations first to make sure you're getting the right one. Also, if you're being strict about it, you should also have to prove beyond any doubt that the digit you've found is actually correct. All this takes time, and currently our computing power is not at a level where we can accurately find more digits (and prove our result) with any great kind of speed.

In reality, though, finding more and more digits of $\pi$ is more useless than it seems. In higher level mathematics, you don't need to know the exact value of $\pi$ at all - only what it represents. In fields where mathematics is applied, such as physics and engineering, usually you won't need more than about twenty digits even for the most precise answers.