What if pi was not an irrational number?

Question

What if at the 1000000th digit it started to repeat then it would be rational!

Answer 1

It doesn't.

The simple fact is that we didn't prove pi to be rational by computing it out to however many digits and showing it didn't repeat.  But this ISN'T the standard definition of being rational.  Rational means it can be written as m/n for two integers m,n.  (for the hardcore math guys, yes, I know that they're interchangeable, but would you really argue that the repeating digits is semantically more simple?)

So all the proofs I've seen start with the two lines:

Assume that pi is rational.  Then pi = m/n for integers m,n. [...]

Then we just have to show that something doesn't work anymore... that is, our assumption that pi is rational contradicts the mathematical truths we've created.

This proof isn't quite easy to see, so to illustrate, here's a much easier proof that sqrt(2) is irrational (however, it is a proof, so if you don't care, you may want to skip it)

Assume that sqrt(2) is rational.  Then sqrt(2) = m/n for integers m,n.  

Note that if m,n have a common divisor, then we can simplify it until they have no common divisor, so we may as well assume the only whole number that divides both is 1.

Well, then 2 = m^2 / n^2, and 2n^2 = m^2.

But since 2 divides 2n^2, it must divide m^2.  This can be shown to mean that 2 divides m, so we can write m = 2k.

2n^2 = (2k)^2 = 4k^2

n^2 = 2k^2

Now, similar to the above logic, 2 must divide n.  But then 2 divides m and 2 divides n, contrary to my requirement that only 1 divides both.  My assumption that sqrt(2) is rational must be false, so that sqrt(2) is irrational.

Answer 2

There are mathematical proofs that demonstrate that this is not so.

Answer 3

What if pi was actually equal to pie and you could eat it?

Just joking. Read up on real analysis, there are some proofs that show that pi is an irrational number.

Answer 4

While some of these answers correctly state that it has been proven

that pi is irrational (and also transcendental, a subset of the rrationals),

none of them address your actual question, "What if ?"

pi comes up in many places other than circle geometry,

more than a few of them rather surprisingly.

If it were rational, it would rock the foundations of much of mathematics,

although at the same time it would probably make a lot of things

simpler.

It would be something like non-Euclidean geometry I suppose.

However interesting it might be to contemplate, it would just

be an exercise or thought experiment, since, in fact, the case

is quite closed, and pi has long be proved to be irrational.

Answer 5

decimal value is never ending and actually it is =22/7

An Irrational Number is a number that cannot be written as a simple fraction - the decimal goes on forever without repeating.

Example: Pi is an irrational number. The value of Pi is

3.1415926535897932384626433832795 (and more...)

There is no pattern to the decimals, and you cannot write down a simple fraction that equals Pi.

Values like 22/7 = 3.1428571428571... get close but are not right.

please see

http://www.mathsisfun.com/irrational-num...

Answer 6

Interesting topic, but pi is just...irrational.

Answer 7

The value of pi has been computed to more than a trillion digits

Answer 8

I have thought about that to Its not possible to prove ANY number to be Irrational! what if it repeats on the Trillionth or Quadrillionth number!

its just not POSSIBLE to prove it!  

Answer 9

Sound like a very innocent question.

But, if that is true, the whole Mathematical field and sciences we ever developed, going to be turned upside down.

That will be a google times more impact than Y2K problem.

Answer 10

you have the idea right, but it was actually proven to be irrational about 200 years ago

Answer 11

pi is not an irrational number at all

pi, epsilon, and other such numbers are classified under another set of numbers called 'TRANSCENDENTAL numbers'.

transcendental numbers are defined as numbers which cannot, by any algebraic methodology, be determined, i.e. NO 'finite value' algebraic function can be used to derive these numbers.

Thus there is no question of  the 1000000th digit repeating because the number is defined as, a number that cannot be determined accurately.  

Answer 12

Well, it doesn't.