It's exactly 'transcendental'.

The classical Greeks used to slaughter each other over just such heresies. But these days Ph.D's do not blink at irrational, transcendental, imaginary, and just fucking ridiculous numbers.

Pi is irrational in the sense that it cannot be expressed as a ratio between two whole numbers, (known to the thought police under the alias of a vulgar fraction). But the square root of 2 is also irrational, and to the extent that 2 is an exact number, the number that, multiplied by itself, comes to 2 must also be exact. Therefore the accusation of congenital vagueness against irrational numbers fails, and the thought police are liable to investigate you on suspicion of irrationalist prejudice.

I asked a math PH.D. and they said Pi is an exact number. How can an irrational number be exact if we can't even reach the last digit ever? — TiredThinker

It is exact in the a priori intensional sense of being defined as an equation or algorithm with instantly recognizable form.

It is inexact in the a posterori extensional sense of being a sequence of rational numbers, for the reason you point out; pi as a constant is ambiguous - just ask Matlab.

In practical calculations, Pi is never exact. It's is just computed to a given precision. In C++, the value of Pi is 3.14159265358979323846, which is sufficient for most calculations.

Every number is an exact number. If you approximate pi with 3.14, the number 3.14 is exactly itself. The number pi is exactly itself, and the difference between them is exactly (3.14 - pi). Those are numbers as mathematical objects.

When you represent a number using another number- eg pi as 3.14 - that representation is called an approximation. You know that 3.14 is within 0.01 of the true value of Pi. Approximations have errors, that's what makes them approximations. Nevertheless, the number you use to approximate another number is still exactly itself.

If Pi had a last digit, say it were 3.141... Then you would be able to write it as 3141/1000, and it would be a fraction. But you can prove that Pi isn't a fraction - it can't be written as one integer divided by another -, so it doesn't have a last digit.

Pi not having a last digit, and further that it can't be written as one integer divided by another -, means that any way of writing digits down for Pi will be an approximation of Pi. So long as you're just writing digits, there will only be finitely many, and Pi has no last digit, so you'll always be off by the part of Pi you don't write down.

3.141 is off by 0.00057...

When you tell a computer to represent a number, it behaves like the kind of approximations above. It will only be able to represent a certain number of digits at once, because infinity doesn't fit inside the computer.

π is accurate, no need to start typing the decimal (base-10) representation.

π has no last decimal digit, just like 1/3 and 1/7, for example.



π = C/d

Not bad. :chin:

It is exactly a number. Like 0000000000000000000000000000000000000000000000.

pi as a constant is ambiguous - just ask Matlab — sime

Is Matlab binary based?

Yes. I mean that different implementations of the constant of pi will yield different values. Orthodox convention says that those values are 'truncations' of some ideal value. The problem is, if one asks what that ideal value is, one can only be referred back to the intensional definition of pi, which isn't the same thing as a value. Hence the only conclusion that can be reached, is that an ideal value of pi doesn't exist, and that so-called 'approximations' of pi aren't approximations of anything specific that is external to them.

Hence the value of pi is ambiguous in the same sense that 'one metre' is ambiguous, in appealing to the uncertain contigencies of practical experiments with finite resolution.

Are any numbers exact? What is meant by 'exact'?

It is exact in the a priori intensional sense of being defined as an equation or algorithm with instantly recognizable form. — sime

Set theoretically, it is not defined "as an equation or algorithm". Rather the constant symbol is defined by the ordinary definitional method of mathematics, which is to state a given property that is had by a certain object and no other object.

a sequence of rational numbers — sime

It is not a sequence of rational numbers. It is an equivalence class of Cauchy sequences of rational numbers.

pi as a constant is ambiguous — sime

It is not ambiguous. The constant refers to exactly one object.

intensional definition of pi — sime

The definition specifies a certain definite object.

appealing to the uncertain contigencies of practical experiments with finite resolution. — sime

That is not at all how 'pi' is defined.

1 atom of hydrogen is exact, 2 atoms of hydrogen is exact. 4.6 atoms of hydrogen isn't exact because it no longer makes sense. An exact perfect circle can't be represented by an incomplete value of Pi?

$\pi$ is irrational i.e. its decimal expansion is infinite & nonrepeating. However, each digit that appears in $\pi$ is specified (if the 1000,000,000,000, 000,000^th digit is 9, it is/has to be 9). In that sense, $\pi$ is exact.

Cum grano salis, I'm bad at math.

Are any numbers exact? What is meant by 'exact'? — emancipate

"Having one value and no more than one value"

So the number "gazillions" is not exact: there are gazillions of people in America and gazillions of people in China but there is not the same number of people in America as in China. "Integer less than five and greater than two" is not exact because it could be three or four.

Pi is an exact number because it has one value: it is the length of a circle's circumference divided by its diamter. Pi does not have more that one value. Any shape whose circumference and diameter are not related by pi is not a circle.

An exact perfect circle can't be represented by an incomplete value of Pi? — TiredThinker

The value of pi is exact, as above. The expression of the number pi cannot be made exact using finite decimal expansion. However, decimal expansion is not the only way we have of expressing numbers. An exact expression of the number pi is the Greek letter pi.

There is nothing special about pi in this. We cannot express the number "one third" exactly by finite decimal expansion. But we can express it exactly using the fraction 1 / 3 or by using notation in decimal that means '...and so on for ever.'

There is not even anything special about numbers. We can refer to PF members exactly at a given moment in time as "the latest person to post", "the second last person to post", "the third..." etc., but the reference will apply to several different people over the course of a day. If we want an exact expression to refer successfully and uniquely over time then we are better off using names, "Cuthbert" etc. The fact that we cannot refer to Cuthbert reliably as "the n-th person to post" does not imply that Cuthbert has a vague or inexact identity. It merely shows that we are using the wrong tool in the box to try to make a successful, unique and exact reference. It's the same with pi.

Pi is an exact number because it has one value: it is the length of a circle's circumference divided by its diamter. — Cuthbert

Pi defined this way will never give a consistent concrete value. It is only exact in an ideal sense.

It is only exact in an ideal sense. — emancipate

I explained one unideal sense, which is 'having one and no more than one value.' Pi is an exact number in that sense. What is the ideal sense that you have in mind?

There is a consistent value for pi. When a class does a geometry problem they don't all get different answers depending on how they drew their circles. If the question is 'What's the area of a quarter circle?' then they will all end up with pi-r^2 / 4. Either that, or they've made a mistake. No inconsistency.

I am not sure whether there is concrete value for pi. Perhaps you mean, for example, that you can count out exactly three apples but you can't measure out exactly pi kilograms of sugar. That's true.

In practical calculations, Pi is never exact. It's is just computed to a given precision. In C++, the value of Pi is 3.14159265358979323846, which is sufficient for most calculations. — pfirefry

I find 3.14 is sufficient for my practical purposes. I suppose if you're a cosmologist, or someone who is multiplying pi by a googolplex or something like that, it might be worth while to take pi to a few more decimals, or your conclusion might not be very exact.

I find 3.14 is sufficient for my practical purposes. — Metaphysician Undercover

Well that settles it then. Next time Elon needs some calculations to land a craft, he should just call you for your results rounded to two decimal points.

It is only exact in an ideal sense. — emancipate

Like mathematics generally.

intensional definition of pi — sime

The definition specifies a certain definite object. — TonesInDeepFreeze

Set theory is extensional. A set is determined solely by what are its members (what its extension is). But you are using 'extensional' in the sense of there being a finite listing. Well, in that sense no infinite set has an extensional definition. The very definition of 'is infinite' is 'does not have a finite listing'. There's nothing special about pi in that way. Even a rational number (which is an equivalence class of integer pairs) does not have a finite extension.

Next time Elon needs some calculations to land a craft, he should just call you for your results rounded to two decimal points. — TonesInDeepFreeze

Judging by the news of a whole fleet of crashed crafts a week or so ago, calling me couldn't have hurt.

I'll let him know you're available. And cheap - just two bits a digit and two digits max.

At least with 1/3 you poses all the information even if you can't write 3s forever. Or if you have a number system based on 6 instead of 10 it wouldn't need to go forever. Pi however can't be fully known. It is limited by our means to measure a circle physically or within a computer? It must be more of a concept than a certain thing?

with 1/3 you poses all the information even if you can't write 3s forever — TiredThinker

With pi you have all the information you need to write its digits for as long as you want to write them.

Pi however can't be fully known. — TiredThinker

What is the definition of "fully known"?

Pi is fully known to be a unique real number with an exact definition.

It must be more of a concept than a certain thing? — TiredThinker

Every mathematical object is an abstract concept and not a physical object.

Pi however can't be fully known. It is limited by our means to measure a circle physically or within a computer? It must be more of a concept than a certain thing? — TiredThinker

I would say we can know what pi is because we learn what it is - the ratio between circumference and diameter of a circle. That's all we need to know and so we fully know it.

We can also express the number pi exactly: the Greek letter pi does just that.

What we cannot do is to measure pi exactly in the same way that we can count exactly. You can pick up exactly three apples and put back exactly two of them, leaving you with exactly one. But you can't measure out exactly pi kilos of sugar. If you happen to be holding exactly pi kilos of sugar then you can never know that is what you are holding.

That observation is not unique to pi or to irrational numbers. We cannot measure exactly half a kilo of sugar or, if we do, we cannot check or know that is what we have done. Our measurement might be correct to a million decimal places and incorrect at the million and first place.

At least with 1/3 you poses all the information even if you can't write 3s forever. Or if you have a number system based on 6 instead of 10 it wouldn't need to go forever. — TiredThinker

Yes, that's right. And just as we have base 6 as a good alternative for expressing thirds, so we have a notation for pi that similarly does not require infinite expansion. It's the Greek letter pi and it can used in normal arithmetic to refer exactly to pi.

Every mathematical object is an abstract concept and not a physical object. — TonesInDeepFreeze

The problem of the OP has turned out to be not specifically about pi. It is about the relationship between 'mathematical objects' and 'physical objects'.

What we cannot do is to measure pi exactly in the same way that we can count exactly. You can pick up exactly three apples and put back exactly two of them, leaving you with exactly one. But you can't measure out exactly pi kilos of sugar. If you happen to be holding exactly pi kilos of sugar then you can never know that is what you are holding. — Cuthbert

The issue with pi being an irrational ratio is not a measurement problem. It is a logical problem with the defined (mathematical) object, the circle. It is actually an impossible object. Simply put, a circle cannot have a centre point. But since the circle is defined as a circumference equidistance from the centre, a circle is actually impossible. This should not be a surprise to anyone, it's commonly stated that a perfect circle is impossible. What we have is approximations.

It seems like some people want to dissolve the distinction between theory and practice, and describe what is a problem with the theory as a problem of practice. There is no problem making circles in practice, there is no problem measuring them, and there is no problem employing pi to determine the area. The problem is that the circles employed in practice cannot obtain the degree of precision and accuracy which we request in theory. And the irrational nature of pi demonstrates that we will never ever get that degree of precision because it is impossible. A true circle, as defined, is an impossible object to create.

A true circle, as defined, is an impossible object to create — Metaphysician Undercover

The problem of the OP has turned out to be not specifically about pi. It is about the relationship between 'mathematical objects' and 'physical objects'. — Cuthbert

Every mathematical object is an abstract concept and not a physical object. — TonesInDeepFreeze

Agreement on this point seems to be breaking out! I would only add that a true half is equally impossible to create (physically) - or, if created, impossible to know that it has been created.

Posting again, not an annoyance hopefully.

What does the OP mean by "is $\pi$ an exact number?"

First things first, $\pi$ is a number! That's that!

What's exactness in re $\pi$?

The first mathematician to use an algorithm to calculate $\pi$ was Archimedes and he used a rational approximation (vide infra).


$\frac{223}{71} \lt \pi \lt \frac{22}{7}$ (Inexact, to be exact)

So, yeah $\pi$ isn't exact if you use rational numbers (something like that).