# Can pi be valued mathematicly?

In summary: This is all quite confusing, but I think it's beyond the scope of this topic.In summary, the conversation discusses the concept of pi being irrational and the methods for calculating it. It is also mentioned that pi is not a pure number and that there are different ways to define numbers. The conversation also touches on the idea of infinite decimal expansion and the relationship between circumference and diameter in a circle. Finally, it is noted that pi is both irrational and transcendental.

in calculus class i have been told that pi is irational, so then i realized that pi is not valued by actually measuring by hand the diameter and perimeter, since it would not be a pure number...
1. so how the hack is pi measured?
2. i want a proof of it being irational.crap, sorry wrong forum, admin, could you please move it to general math? thank you

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first off its Pi and its the ratio of a circles circumferance to its diameter, i think.

http://www.math.clemson.edu/~simms/neat/math/pi/piproof.html [Broken]

Currently the most popular method for calculating approximations of pi is probably the Taylor series:

http://mathworld.wolfram.com/PiFormulas.html

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And if you want a way to solve it by using math that may seem more obvious to you: take the integral of a quarter of a unit circle. With a known relationship between the radius and area, you can then use this to determine the ratio between the area and its circumference. Voila! Pi!

This obviously isn't the normal/best way, but it may seem like a more obvious solution to you.

Eeh, what is impure about pi??
Is purity of a number related somehow to repeating decimal sequences?

2. Euler did a clever proof of that back in the 18th century, whereas Lindemann furnished a rather trivial of the transcendence of pi a century later.

The band-width on this site is too small to contain both or anyone of those proofs..
(or possibly not..)

in calculus class i have been told that pi is irational, so then i realized that pi is not valued by actually measuring by hand the diameter and perimeter, since it would not be a pure number...

pure? impure? what is that? i think you have a misapprehension about real numbers here. in particular you seem to have the idea that we find numbers by 'measuring' things in real life. real life and mathematical abstractions are not the same thing.

1. so how the hack is pi measured?
2. i want a proof of it being irational.

get a number theory textbook then. I think that le veque proves the irrationality of pi. transcendence is harder.

Also remember that defining numbers in terms of the denary/decimal system is not the only way to define a number.

For example, a linear equation of one variable with one solution may be regarded as defining the so-called "x".
We could, for example, use the equation x-2=3 as the definition of the number "5".

arildno said:
Also remember that defining numbers in terms of the denary/decimal system is not the only way to define a number.

It's actually a crappy way of defining numbers, since you need infinite accuracy to add any two (rational even) numbers and be sure you got the right answer

Office_Shredder said:
It's actually a crappy way of defining numbers, since you need infinite accuracy to add any two (rational even) numbers and be sure you got the right answer

Don't fall into the same trap. It makes no sense to talk of the accuracy of adding two decimal expansions, unless you decide to terminate the sum at some point, which could only be for practical reasons. The sum of two decimal numbers, even irrational ones, is perfectly well defined and exact.

status, that's what I mean. The OP seems to think that pi isn't a real number because of the inability to write it down.

Obviously if you know the full number, you can add the decimals together and know what the new decimal is supposed to be. But since we're talking about pi here, good luck knowing the decimal expansion in the first place ;)

Office_Shredder said:
It's actually a crappy way of defining numbers, since you need infinite accuracy to add any two (rational even) numbers and be sure you got the right answer
Eeh?
I can define numbers any way I like.
For example, the number Johnny is defined as follows:
$$Johnny=\sum_{i=1}^{\infty}\frac{1}{n^{2}}$$

Office_Shredder said:
.

Obviously if you know the full number,
What does that mean?
you can add the decimals together
Why would you do that?
and know what the new decimal is supposed to be.
Signifying what, exactly?
But since we're talking about pi here, good luck knowing the decimal expansion in the first place ;)
Who cares about the decimal expansion of $\pi$?
$\pi$ would remain perfectly well defined, even if hadn't had the intelligence to deduce that it had to be slightly greater than the denary number 3.

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arildno, the original poster claimed that pi, by being irrational, is somehow less pure. I extrapolated that he was comparing irrationals to rationals in general (certainly not an unreasonable leap), and thinking rational numbers were more pure than irrationals somehow. The main reason for this line of thinking is that you can't write down the decimal expansion of an irrational number, so it doesn't feel as real.

So I responded by saying defining a number by its decimal expansion is a bad way of doing things, since decimal truncation (which is absolutely necessary to write the number down) means you can't accurately add two numbers together unless you have knowledge from elsewhere as to what the two numbers are intended to be in their entirety

arildno said:
Eeh?
I can define numbers any way I like.
For example, the number Johnny is defined as follows:
$$Johnny=\sum_{i=1}^{\infty}\frac{1}{n^{2}}$$

This gives another (very slowly converging) method for approximating pi:

$$\pi=\sqrt{6\sum_{n=1}^\infty\frac{1}{n^2}}$$

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There's one famous product series... To prove that Pi is irrational you should look into these type of series.

Hrrm... I haven't yet gotten to infinite sequences and series, so I cannot contribute calculations.

What I do know (or think I know) is that $$\pi$$ is an irrational (trancendental, too) number with an infinite decimal expansion.

The fact that for a circle of diameter 1 that the circumference is:

$$Circumference = 2{\pi}r$$
$$= {\pi}D$$
$$= {\pi}(1)$$
$$= \pi$$

Means that the circumference is finite, but pi is still infinite.

Though I guess it depends on your definition of what finite is. If you're using a physical circle, I suppose the circumference is actually an infinite connection of dimensionless points in space. Likewise, you could consider it to be a finite number of atoms.

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calcnd said:
Means that the circumference is finite, but pi is still infinite.

Pi is finite... it has an infinite decimal expansion, but that certainly doesn't make the number itself infinite.

Thank-you for clarifying that.

Hi Tuvia,

As far as I know, and proved:
1. There are many ways to calculate it. (Hint: If Taylor Series is to be used, it must be expanded in a certain way to ensure convergence.)
2. Pi has an infinite number of decimals. Examine some of the ways of finding its value and you can check this. This means that it is irrational.
3. 22/7 is a historical value used as an approximation to the Measured value. It is of course irrational. (Hint: When you say an irrational number; this means that there is some way to describe this number exactly, but, Only, it has a decimal form that contains and infinite 'number' of digits.)
4. Pi is an arc angle, i.e. it is angle in radians that corresponds to 180 degrees. It's main definition is: "The ration between the circumference and the diameter of a (any) circle."

But, what I can find an astonishing question:
Is the value of Pi is the origin in geometry (may be in algebra)? Or, it is a consequence of another geometrical property?..;)... It deserves to check!

Welcome for any (request for) clarification.

Amr Morsi.

Amr Morsi said:
Hi Tuvia,

As far as I know, and proved:
1. There are many ways to calculate it. (Hint: If Taylor Series is to be used, it must be expanded in a certain way to ensure convergence.)
2. Pi has an infinite number of decimals. Examine some of the ways of finding its value and you can check this. This means that it is irrational.
No, THAT does not mean it is irrational. 0.33333... has an infinite number of decimal places but it is rational.

3. 22/7 is a historical value used as an approximation to the Measured value. It is of course irrational.(Hint: When you say an irrational number; this means that there is some way to describe this number exactly, but, Only, it has a decimal form that contains and infinite 'number' of digits.)
Again, no. There exist many (in fact, "almost all") rational numbers that have infinite decimal expansions. All such are eventually repeating where irrational numbers are not.

4. Pi is an arc angle, i.e. it is angle in radians that corresponds to 180 degrees. It's main definition is: "The ratio between the circumference and the diameter of a (any) circle."

But, what I can find an astonishing question:
Is the value of Pi is the origin in geometry (may be in algebra)? Or, it is a consequence of another geometrical property?..;)... It deserves to check!
I don't follow this. Of course, the "origin" of Pi was in geometry. It's definition was just what you said in "4". Now, of course, it is just a number that has many different applications.

Welcome for any (request for) clarification.

Amr Morsi.
You mean you won't accept any clarification by someone else?

Office_Shredder said:
arildno, the original poster claimed that pi, by being irrational, is somehow less pure. I extrapolated that he was comparing irrationals to rationals in general (certainly not an unreasonable leap), and thinking rational numbers were more pure than irrationals somehow. The main reason for this line of thinking is that you can't write down the decimal expansion of an irrational number, so it doesn't feel as real.

So I responded by saying defining a number by its decimal expansion is a bad way of doing things, since decimal truncation (which is absolutely necessary to write the number down) means you can't accurately add two numbers together unless you have knowledge from elsewhere as to what the two numbers are intended to be in their entirety
Hmm..yesterday, I was positive that what you meant was "crappy" was my definition of 5 as the number solving the equation x-2=3.
That's what I basically opposed against (although I won't extol the virtues of this particular definition of 5).

Today, however, it seems clear that what you regarded as "crappy" was definition by decimal expansion, a sentiment I fully agree with.

I don't believe that the original post said that pi itself was "less pure". I believe he said that MEASURING a circle to determine pi (measure circumference and diameter and dividing) was not a "pure" (i.e. mathematical) way of determining pi.

arildno said:
Eeh, what is impure about pi??
Is purity of a number related somehow to repeating decimal sequences?

2. Euler did a clever proof of that back in the 18th century, whereas Lindemann furnished a rather trivial of the transcendence of pi a century later.

The band-width on this site is too small to contain both or anyone of those proofs..
(or possibly not..)

till now i didnt know that pai can mathematicly measured... so when i said impure number, i was talking about the impure number you will have by physicly measuring pai(measuring perimiter, and diameter, and them deviding them)...

by the way, i was thinking about the problem, and i measued a proximity of pi by using geometry, i got pi=3.064. i meaured the area of triangles within the circle. i think that if ill find a connection between the triangles, i will be able to make an infinite serie of pi..

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What do you mean by "mathematically measured"??
The expression seems meaningless:
1. Do you mean: "Mathematically well-defined"
2. "Exactly measurable" physically speaking?
3. Its decimal expansion cannot be calculated in a finite number of steps?

arildno said:
What do you mean by "mathematically measured"??
The expression seems meaningless:
1. Do you mean: "Mathematically well-defined"
2. "Exactly measurable" physically speaking?
3. Its decimal expansion cannot be calculated in a finite number of steps?

i meant to the approximation of pi. at first i thought that pi was used as a number that was physicly measured, until one day i was told that pi is irratonal, and at that poit it was obvious that pi has a mathematical defenition... since you can't measure by hand an irrational number...
thats all to it...

by the way, is there a defenition of pie in the form of a serie that was made by using geometry?

... since you can't measure by hand an irrational number...
Why do you say that?

by the way, i was thinking about the problem, and i measued a proximity of pi by using geometry, i got pi=3.064. i meaured the area of triangles within the circle. i think that if ill find a connection between the triangles, i will be able to make an infinite serie of pi..

This is exactly the most traditional method of measuring pi in mathematics... you draw a polygon on the inside, and a polygon on the outside of the circle (circumscribed and inscribed, of course), and find the areas of each. Assuming the radius of the circle is one to make calculations easier, pi is bounded by the two areas. You can keep making polygons with more and more sides

Halls,

(Requests for) means: "You can put or remove the words between brackets.". This is the meaning of brackets .....

You are right, Halls. But, I have a question: Talking from a decimal point of view, by what do we define that we have a number? Is it only by knowing the digits, or may also be through an equation, series, relations? {By the way, the difference, in science, between rationals and irrationals is not the repetition of certain set of digits up to infinity, but it is the ability of writing it as a ratio of two integers or not, and there is no equivalence between the two "differences". This debate had been settled down, by mathematicians, long ago.}

Another question please, if you wouldn't mind: "Is Pi a consequence of Pythagoras Theorem or the opposite? Or, is there an equivalence? And, if so, is it an exact equivalence? Or, there is another property needed to complete equivalence.

I am sorry that I will not be able to respond immediately (if there is a need to reply once more). I will try to do in a week or two.

Sorry for any inconveniences caused.

Amr Morsi.

Amr Morsi said:
Another question please, if you wouldn't mind: "Is Pi a consequence of Pythagoras Theorem or the opposite? Or, is there an equivalence? And, if so, is it an exact equivalence? Or, there is another property needed to complete equivalence.

Amr Morsi.

you can prove pythagoras without using pi, therefor pythagoras is not a sonsequence of pi.
about pi, i don't know, maybe there is a way to find it without pythagoras... and if there isnt, they are both independent from each other..

Werg22 said:
There's one famous product series... To prove that Pi is irrational you should look into these type of series.
i can't seem to remember the name of the series although i remember that there was an infinite product. is the name of that infinite product something like wallis formula or wallis equation?