Why is Pi never going to be precisely calculated?
It has been precisely calculated many, many times. The level of precision has varied with calculation models, but the values we have right now are actually a lot more precise than what’s needed for any practical purposes.
If what you mean is “why won’t we ever know it with all its figures”, that’s because it has an infinite number of non-repeating decimals.
Because it doesn’t have a precise value.
Think about a third. Its decimal value is .3333333… and onward. You’re never going to calculate the final three in the sequence because there is no final three.
It does have a precise value. But the value cannot be expressed as a finite sequence of digits in decimal notation.
Same for a third.
There are even reasons for computing π to high precision. If you calculate, say, a billion or two decimal digits, and the results are correct, it amounts to a basic test that your CPU is functioning properly.
Now, there are plenty of real numbers that, unlike π, cannot be precisely calculated.
One famous example is Chaitin’s Ω.
Okay, but I thought with recurring decimals, such as a third, you can predict that it will go on forever. But what I’m asking is how can you predict pi will go on forever since you never know what comes next. Or do you? I may have seriously misunderstood something here and maths isn’t my strong suit so please bear with me. I do recall them saying they have a mathematical proof that pi isn’t going to end but that’s way beyond me. 
So what you mean is you can have a precise value such as a third (as discussed in a previous post) but in terms of the decimal system cannot ever be precisely expressed. Yes? But wait a minute…they say as a fraction, pi is roughly equal to 22/7, right? But I have been told that is only an approximation and now I’m even more confused. 
So are we here really saying there is a fundamental incompatibility between fractions, which consist of a numerator and denominator and can be expressed as one number divided by another, and decimals, which try to force natural fractions to ‘fit’ the decimal system? So would I be correct in saying that the decimal system is flawed?
huh, is this a troll ? the decimal system is what the decimal system is.
irrational numbers are irrational in any base, but repeating decimals may be finite length in another base. (eg a third is 0.1 in base three.)
Also, if you do have a repeating decimal , say 0. 0 142857 142857 142857 142857 … repeating 142856 … you can easily convert it to a fraction.
thats a 10 th of 0. “142857” repeated.
Now for the repetated decimal that is directly after the . , just put the repeated digits as a fraction over a number made of as many 9’s as the denominator has digits. If its a six digit repeating decimal, put it over 6 digits made of 9’s. (of course, thats 10^n - 1 , if you want to state it more carefully. ) 142 857 / 999999 … then simplify that… thats 1/7.
So 0. 0 142857 142857 142857 … is 1/70. No real problem to convert repeating digits back to fractions if you want to go back.
pi can be precisely ‘calculated’ for any fundamentally meaningful definition of ‘calculated’, i.e. given any other number whose precise value is known it is possible to calculate whether pi is greater or less than that number. If you want you can also calculate arbitrary many digits of pi, what you can’t do with pi is:
a) express it in a terminating decimal sequence
b) express it is a fraction of two integers (i.e. it is irrational)
c) express it is a root of a polynomial with rational coefficients (i.e. it is transcendental)
Let’s say I handed you a bag with three marbles in it. Then I asked you to remove precisely a third of the marbles from the bag.
Could you do it?
We could all look at the results and say whether you removed precisely a third of the marbles. Yet we cannot write “a third” as a finite decimal.
Pi is not equal to 22/7. 22/7 is a number that is close enough to Pi for many practical purposes, but it is not Pi. If I owed you $1000, and handed you a bag with $999.99 in it, you would probably say, OK, that’s good enough. 22/7, to most people is good enough. To those for whom it isn’t, well they can come up with a better approximation.
There are not enough drops of ink in the entire universe to give each number a unique name, regardless of the naming system you choose.
If you don’t like decimal fractions, you can always calculate a sequence of general fractions that converge to pi, e.g.
22/7, 333/106, 355/113, 103993/33102, etc.
Like Archimedes said, it is computable.
Yes, I just remove one marble.
Sure, that is my point. It’s as if the decimal system is incompatible with regular fractions because sometimes there isn’t an exact correspondence. Yes, in the real world it works, but from a purely mathematical point of view it’s two different systems. That is all I’m saying. No big deal but it needs clarification.
I did say this. But if you check back one of my respondents stated pi was an exact value.
I’m not quite sure what you mean here.
I said that. It is an exact value, just like 1/3 is. That value is not 22/7. And just like 1/3, it cannot be expressed as a finite sequence of digits in the decimal system.
Yes I see, but are you any better off because you still never come to an end? I do see the value of it though, I’m just being forensic.
Okay but I’m confused. You say it’s an exact value like 1/3, right? What is that value please?
We call the value pi. There’s no finite representation of pi in the decimal system, just like there’s no finite representation of 1/3 in the decimal system. There are many formulas for pi (some of which can be found here Pi Formulas -- from Wolfram MathWorld) - these formulas can give you as many digits of pi in the decimal system as you want, but if you want to refer to the exact value, just saying “pi” is exact as possible.
Abashed, 22/7 is not the value of Pi. It is only a value introduced for practical purposes, useful in school for teaching principles of Pi. For that matter, 3.14 is not the value of Pi, but a decimal value used for the same purpose. Whether to use a decimal or a fractional value is a decision made based on the way the problem is presented.
Sometimes, using an exact value of Pi, or a very accurate value, isn’t as important as teaching the concept. Quick, now – if you were taking a math test and had to multiply something by Pi, which would you prefer to use: 22/7, or a more accurate value with a million digits?