# Why is the value of PI not absolute?

1. Jul 27, 2013

### webboffin

Why is the value of PI not absolute mathematically?

2. Jul 27, 2013

### 1MileCrash

The value of pi is 100% absolute.

EDIT: nugatory brings up a good point. I interpreted your question to be asking why pi doesn't have an exact, known value. It does. If this is not what you were asking, please clarify.

Last edited: Jul 27, 2013
3. Jul 27, 2013

### Staff: Mentor

Can you be a bit more precise about what you mean by "absolute" in this context?

4. Jul 27, 2013

### webboffin

The value of PI is an irrational number so it's value is infinitely long.

5. Jul 27, 2013

### 1MileCrash

No, its decimal representation is infinitely long. Its value is not infinitely long, nor do I know what that would mean.

6. Jul 27, 2013

### webboffin

I don't know what you mean either

7. Jul 27, 2013

### 1MileCrash

Just because the decimal representation of pi is infinitely long doesn't mean we know anything less about its value.

You asked why the value of Pi isn't absolute, but Pi is a number and it is well defined, so the value of Pi is absolute. That a number can't be expressed in a specific way in our decimal system doesn't make its value unknowable.

Last edited: Jul 27, 2013
8. Jul 27, 2013

### Staff: Mentor

OK, now that you've clarified what you're asking....

Not only is pi an irrational number, it is a transcendental number (all transcendentals are irrational, but not all irrationals are transcendental). Knowing that, you can google for "pi transcendental proof" to find the standard proofs for why pi must be transcendental... But be aware that some of this material is going to be fairly heavy going, and if you have follow-up questions you might want to try the General Mathematics forum on this site.

And don't let yourself get too hung up on this "infinitely long value". Pi still corresponds to a single point on the number line, just like any other number. True, we can't write its value down as a decimal without using an infinite number of digits, but that's doesn't mean it doesn't have an exact value, just that we can't write that value down as decimal number.

BTW, would you accept $\pi=(\Gamma(\frac{1}{2}))^2$ as an exact value for pi?

9. Jul 27, 2013

### micromass

Staff Emeritus
Like Nugatory already said, saying that $\pi$ is infinitely long makes no real sense mathematically. I can't blame you though, people say this all the time. It's just not correct.

What you mean is of course that $\pi$ has an infinite decimal representation. This is true. But $1/3$ also has an infinite decimal representation, indeed, it is

$$0.33333333333333333...$$

The difference with $\pi$ is that we know each digit in $1/3$, but we don't easily know all digits in $\pi$ without some heavy calculations. This is because $\pi$ is irrational. Proofs of irrationality of $\pi$ (and even transcendence) can be found in a lot of places, but are very difficult.

10. Jul 27, 2013

### verty

Pi is the area of the circle with radius 1. Perhaps area just doesn't apply that well to circles, I don't know any better reason for why π is like it is. Squares and circles are just not like each other.

11. Jul 27, 2013

### Elroch

The simplicity of Pi

Often people think of pi as being far more complex than say 1/3 or the fifth root of 2. In a sense that is true as it is not a fraction, a surd expression or even the root of a polynomial. But it happens that there are representation which are as simple to express as some of these numbers.

Perhaps the simplest (and also one of the slowest converging) is this:

pi = 4*(1 - 1/3 + 1/5 - 1/7 + 1/9 - ... )

[ If I was famous, it would probably be a great anecdote to say that I typed this program into an earlier personal computer at a computer fair in 1978, having only learnt how to send programs by post to a mainframe from school, and was worried about the harm I might have done by sending it into an infinite loop. But I'm not. :) ]

12. Jul 27, 2013

### Seydlitz

If the value of $π$ is not absolute, then each time one draws a circle it will look different every time.

But then the fact that you can always draw a circle shows that you can, so to say crudely, represent $π$ in an exact and simple way, notwithstanding its decimal representation.

13. Jul 28, 2013

### webboffin

Thanks guys, you are now making sense to me and my apologies if I didn't seem clear initially to the question I was asking but you anyway got the gist of what I was meaning. PI does have an exact value but just not able to decimally write is own its entire value... I got the point good with the 1/3 demonstration. Again, thanks.