# Question #c82fe

##### 3 Answers

No, the infinite amount of non-repeating digits of

- 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 = 28659/1000# . - Numbers with repeating decimals. These kind of numbers have a block of digits which goes on forever. Examples can be
#0.33333...# , or#2.1826262626...# , and so on. These numbers can be written as a fraction, too, but the algorithm is a bit more complicated. - 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

The number

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

So can

In our base 10 system, digits to the right of the 'dot' count

In a base 6 system, digits to the right of the 'dot' count

So in a base 6 system, we write

Back to

It can also be shown that an irrational number cannot be written with finitely many digits 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

Another example or an irrational number is

Others have given good answers as to why

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