MHB Why is Pi an irrational number?

[Kruidnootje]

Pir² (I am looking in the greek alphabet and geometry symbols and can not find the symbol for pi that looks anything like pi when in preview mode) Sorry.

If Pi is the ratio of a circumference to the diameter of a circle and geometrically this gives the perfect measurement of a circumference with a finite measurement, why is Pi an infinite number? This means that the physical reality and the geometry are never in agreement. Or am I wrong somewhere?

 

[topsquark]

Pir² (I am looking in the greek alphabet and geometry symbols and can not find the symbol for pi that looks anything like pi when in preview mode) Sorry.

Quick tip: You can type [math]\pi[/math] to get a symbol for pi. The Forum here uses LaTeX (see https://mathhelpboards.com/math-formulas-mathjax-62/mhb-latex-guide-pdf-1142.html) and is very quick to learn for simple uses like this one.

The pdf uses symbols like \ [ and \ ]. Use the text [math] and [/math] instead. So instead of \ [ \pi \ ] use [math]\pi[/math]. This gives [math]\pi[/math]. There are other ways but this will get you started. And it is very easy to learn to do basic equations like [math]x = \sum _{n = 0}^{N} e^{-n}[/math] ... [math]x = \sum _{n = 0}^{N} e^{-n}[/math]

Technically speaking, we use MathJax but any LaTeX coding will work for the examples I used above.

-Dan

 

[Evgeny.Makarov]

If Pi is the ratio of a circumference to the diameter of a circle and geometrically this gives the perfect measurement of a circumference with a finite measurement, why is Pi an infinite number?

In mathematics errors are often hidden in the use of concepts that are not defined precisely enough. To me it is unclear what you mean by a finite measurement, and how the fact that the finite measurement gives the circumference implies that $\pi$ is rational. Once you start using more and more precise language hopefully it becomes clearer whether some text is or is not a proof. Formally to prove that $\pi$ is rational you probably need to provide natural numbers $m$ and $n$ such that $m/n=\pi$.

Also, $\pi$ is not infinite. It has an infinite decimal expansion.

 

[Kruidnootje]

Quick tip: You can type [math]\pi[/math] to get a symbol for pi. The Forum here uses LaTeX (see https://mathhelpboards.com/math-formulas-mathjax-62/mhb-latex-guide-pdf-1142.html) and is very quick to learn for simple uses like this one.
-Dan

Much appreciated Topsquark thankyou.

 

[Kruidnootje]

In mathematics errors are often hidden in the use of concepts that are not defined precisely enough. To me it is unclear what you mean by a finite measurement, and how the fact that the finite measurement gives the circumference implies that $\pi$ is rational. Once you start using more and more precise language hopefully it becomes clearer whether some text is or is not a proof. Formally to prove that $\pi$ is rational you probably need to provide natural numbers $m$ and $n$ such that $m/n=\pi$.

Also, $\pi$ is not infinite. It has an infinite decimal expansion.

First you are right, I have to learn the Math 'Talk' language, unfortunately I am not coming from a maths back ground at all, but I will do my best.
Secondly what I meant by finite measurement is that the circumference of a circle is finite, has an end. So my Logic suggested (seems reasonable logic to me at the moment) that if one makes a calculation between a finite number and an irrational number how can that calculated result ever end? I am not trying to be difficult or pedantic, just wondering if there is an explanation for the fact that the Irrational number has to be arbitrarily rounded up to nth decimal places for any equation to make geometric sense. The alternative is that the answer has no end either? So a circle never ends- the circumference has no end? I'm happy to be shot down here if the question is baseless but I needed to ask.

 

[topsquark]

First you are right, I have to learn the Math 'Talk' language, unfortunately I am not coming from a maths back ground at all, but I will do my best.
Secondly what I meant by finite measurement is that the circumference of a circle is finite, has an end. So my Logic suggested (seems reasonable logic to me at the moment) that if one makes a calculation between a finite number and an irrational number how can that calculated result ever end? I am not trying to be difficult or pedantic, just wondering if there is an explanation for the fact that the Irrational number has to be arbitrarily rounded up to nth decimal places for any equation to make geometric sense. The alternative is that the answer has no end either? So a circle never ends- the circumference has no end? I'm happy to be shot down here if the question is baseless but I needed to ask.

I looked around the internet a bit and proved what I had suspected... There seems to be no proof of the irrationality of pi that does not involve integral calculus. If you do a search you can find any number of proofs of the irrationality of pi using Calculus. As far as measurements are concerned you will not be able to prove irrationality due to the limitations of any physical measuring device. All you can do is say that pi can be approximated to any level you like using better and better measuring devices.

By the way, pi is a very special number in Math and Physics as it is not only irrational, but is also transcendental. (All transcendental numbers are irrational but not all irrational numbers are transcendental. For example, [math]\sqrt{2}[/math] is irrational but is not transcendental since it is a solution of the equation [math]x^2 - 2 =0[/math] .)

-Dan

 

[HOI]

I remember seeing, long ago, a proof of a very interesting theorem:

"Given a number, c, if there exists a function, f, such that all of its "iterated anti-derivatives" can be taken to be integer valued at 0 and c, then c is irrational."

I don't remember the proof unfortunately but it is clear that all anti-derivatives of sin(x) can be taken (by taking the added constant to be 0) to be sin(x), -sin(x), cos(x), and -cos(x). Their values, at 0 and [math]\pi[/math], are 0, 1, or -1, all integers. So by this theorem, [math]\pi[/math] is irrational.

 

[Evgeny.Makarov]

Secondly what I meant by finite measurement is that the circumference of a circle is finite, has an end.

It seems that your use of "finite" and "infinite" are not just typos, but revel and important issue that has to be cleared.

All real numbers are finite. They are located a finite distance from zero on the number line. One my consider an infinite sequence $a_1,a_2,\ldots$ of numbers. In addition to being infinite in length, this sequence can be unbounded in magnitude. This means that for every $B$ there exists an $i$ such that $a_i>B$. But a single number cannot be unbounded.

A real number can have finite or infinite decimal expansion. Note that rational numbers can also have infinite expansion. What separates them from irrational is that expansions of rational numbers are periodic.

Finally, circumference can be arbitrary. From the idealized standpoint of mathematics it can be any real number (therefore finite) with either finite or infinite expansion. Of course, real physical round objects cannot be measured with infinite precision and are not perfectly round.

So my Logic suggested (seems reasonable logic to me at the moment) that if one makes a calculation between a finite number and an irrational number how can that calculated result ever end?

For example, $0.3333\ldots\cdot 3=1$. Here you have an arithmetic operation that takes a number with infinite expansion and another with a finite one, and the result is a number with a finite expansion. Granted, the number $0.3333\ldots=1/3$ is rational, but it is not clear in your reasoning where you use the fact that one of your numbers is irrational rather than rational with infinite expansion. But ultimately this depends on the nature of the calculation you are talking about.

So a circle never ends- the circumference has no end?

Here again we have to distinguish between the magnitude and the length of expansion.

 

[Kruidnootje]

It seems that your use of "finite" and "infinite" are not just typos, but revel and important issue that has to be cleared.

All real numbers are finite. They are located a finite distance from zero on the number line. One my consider an infinite sequence $a_1,a_2,\ldots$ of numbers. In addition to being infinite in length, this sequence can be unbounded in magnitude. This means that for every $B$ there exists an $i$ such that $a_i>B$. But a single number cannot be unbounded.

A real number can have finite or infinite decimal expansion. Note that rational numbers can also have infinite expansion. What separates them from irrational is that expansions of rational numbers are periodic.

Finally, circumference can be arbitrary. From the idealized standpoint of mathematics it can be any real number (therefore finite) with either finite or infinite expansion. Of course, real physical round objects cannot be measured with infinite precision and are not perfectly round.

For example, $0.3333\ldots\cdot 3=1$. Here you have an arithmetic operation that takes a number with infinite expansion and another with a finite one, and the result is a number with a finite expansion. Granted, the number $0.3333\ldots=1/3$ is rational, but it is not clear in your reasoning where you use the fact that one of your numbers is irrational rather than rational with infinite expansion. But ultimately this depends on the nature of the calculation you are talking about.

Here again we have to distinguish between the magnitude and the length of expansion.

It is difficult to digest and understand the way you are writing, to a mathematician I am certain that all what you said makes sense. But I am not used to such phraseology when I am wanting to grasp a certain concept. I read your post at least 5 times and bits make sense. But the fundamental query I had seems to have been dissected a little too much where I am using basic words to ask a simple question. I know I am at fault, so I will try now to ask again using more correct language in more detail.

3.1415926...is always referred to as an irrational number in literature and videos. That is why I called it infinite, in this case irrational. The radius or diameter such as 4 or 10 units is a finite number a rational number. My silly question, which was rather a thought really after considering these things was this: Theoretically one can never multiply a rational number by an irrational number and arrive at a rational result. 4*3.1415926... is impossible. Hence, the next seemingly logical deduction was, if this formula gives the circumference of a circle then we have a geometrical nightmare! The circle could never end, one would need an electron microscope to actually see how the circle's line still continues ad infinitum. Of course in the real world we just round up the 3.1459... to 3.142 and have done with it. But I was thinking just a little ahead and musing over this situation and wondering how this even came about as a mathematical piece of precision?

 

[Evgeny.Makarov]

I read your post at least 5 times and bits make sense.

You are encouraged to ask specific questions about what I wrote. Instead of two people exchanging long statements that contain the whole argument again and again, it is more productive to ask questions about specific words and sentences. For example, "What do you mean by the decimal expansion?", "Why is $0.3333\ldots\cdot 3=1$?", "What is the difference between a rational number with infinite expansion and an irrational number?". Then we can reasonably clarify at least small portions of the other person's statement instead of understanding it as a whole, but only vaguely.

3.1415926...is always referred to as an irrational number in literature and videos.

Yes, $\pi$ is irrational.

That is why I called it infinite, in this case irrational.

I have said already that I don't like using "infinite" in this sense. "Infinite" means "larger than 1, 2, 3 and all other natural numbers". It is not the case that $\pi>100$. But it is true that $\pi$ has infinite number of decimal digits after the point. Note, however, that many rational numbers also have infinite number of decimal digits.

The radius or diameter such as 4 or 10 units is a finite number a rational number.

Yes, 4 and 10 are rational. But what prevents the diameter from having an infinite decimal expansion? Can't you imagine a circle whole diameter is $1/3$? Or suppose you have a cylinder with diameter 1. Therefore, its circumference is $\pi$. Suppose you wrap a thread around the cylinder so that the length of the thread equals the circumference. Then you straighten the thread and draw a circle with that radius, i.e., radius $\pi$. In the physical world it's impossible to do with infinite precision, but as a thought experiment it is possible.

Theoretically one can never multiply a rational number by an irrational number and arrive at a rational result.

This is absolutely correct. If $r_1$ and $r_2$ are rational and $x$ is irrational, then $r_1x=r_2$ is impossible unless $r_1=0$. Indeed $r_1x=r_2$ implies $x=r_2/r_1$, and it is easy to show that the ratio of rational numbers is again rational.

Hence, the next seemingly logical deduction was, if this formula gives the circumference of a circle then we have a geometrical nightmare! The circle could never end

Here you need to clarify what you mean by "The circle could never end". If you mean that a point could move an a fixed rate along the circumference and never reach the original point, this is not true. A point moving at a fixed rate requires only a finite amount of time to go around the circle and come back to the original point regardless of whether the circumference is rational or irrational.

one would need an electron microscope to actually see how the circle's line still continues ad infinitum.

But if you mean that some other process (rather than moving uniformly) is infinite, it is possible. We could measure how many segments of length 1 cm fit in the circumference in full. Then we take the remainder (the part of the circumference not covered by 1 cm segments) and cover it with segments 0.1 cm in length. Then again we take the remainder and cover it with segments 0.01 cm in length and so on. Very soon this process will indeed require an electron microscope, and it is indeed infinite if the diameter of the circle is 1 cm and the circumference, correspondingly, is $\pi$ cm.

So you need to clarify what is meant by finite or infinite segment. Otherwise it first looks like the segment's length has infinite number of decimal digits, which is indeed the case if the length is irrational. But you rather unfortunately call the segment itself infinite, and then this seems to be interpreted as though the segment is infinitely long. These are different things.

However, it is true that real numbers are strange, and in some sense essentially infinite objects. Specifying some real numbers on the computer is impossible, for example, because it would require infinite amount of information. A real number is like a pointer that says, "Go to 3 cm mark. Then go right by 0.2 cm. Then go left by 0.06 cm. Then go right by 0.002 cm, then left by 0.0005 cm, etc.". This sequence of directions can indeed be infinite. I guess some paradoxes like Zeno's are related to this.

 

[Alaskan Son]

Pir²
If Pi is the ratio of a circumference to the diameter of a circle and geometrically this gives the perfect measurement of a circumference with a finite measurement, why is Pi an infinite number? This means that the physical reality and the geometry are never in agreement. Or am I wrong somewhere?

I'm not exactly a mathematician but I do enjoy a good exercises in basic logic so I'll take a crack at this one that may be a little more in layman's terms than some of the other answers you've received.

Saying a number is irrational is just saying that it cannot be expressed as a ratio of two rational numbers (i.e. a simple fraction). This does not however make it inaccurate. As someone else pointed out, even a rational number like 1/3 cannot be precisely expressed in decimal form without going on for infinity.

The square root of 2 is another irrational number and cannot be precisely expressed in decimal form without going on for infinity. All this really means is that if you're forced to use the decimal equivalent, your level of accuracy is going to be limited by how many decimal places you use. Luckily, those before us have come up with a number of other methods for expressing these things though. Here's an example...

Draw a right triangle where the 2 legs are each 1 unit long. Now find the length of the hypotenuse. The hypotenuse of course is figured by adding the lengths of the 2 legs together and finding the square root of that total. What would the answer be then?

If you wanted to display the answer in decimal form it would be something like 1.41421356237 which of course is not completely accurate. That does NOT mean "that the physical reality and the geometry are never in agreement". It just means that the language and logic we're using to represent it has it's limits. Now the other answer we could give is this...

√2

Completely accurate but still irrational.

 

[Olinguito]

As someone else pointed out, even a rational number like 1/3 cannot be precisely expressed in decimal form without going on for infinity.

This is due to the base system chosen (in this case base 10) and not to any special property of the rational number itself. In base 10, only fractions with a denominator of the form $2^m5^n$ ($m,n$ non-negative integers) have a decimal expansion that terminates. If we use base 12 instead, for example, the fraction $\frac13$ will have a terminating decimal (or duodecimal) expansion: it will be $0.4$ ($\because\ \frac13=\frac4{12}$). However the fraction $\frac1{10}$ (in base 10) will not be have a terminating expansion in base 12 – it will be $0.1249724972497...$ as
$$\frac1{10}\ =\ \frac1{12}+\left(\frac2{12^2}+\frac4{12^3}+\frac9{12^4}+\frac7{12^5}\right)\left(1+\frac1{12^4}+\frac1{12^8}+\cdots\right).$$
In the case of irrational numbers, their decimal expansions will never terminate in any base.

There seems to be no proof of the irrationality of pi that does not involve integral calculus.

Here is a visual (non-rigorous) proof of the irrationality of $\pi$ based on Lambert:

[YOUTUBE]Lk_QF_hcM8A[/YOUTUBE]

 

[I like Serena]

In the case of irrational numbers, their decimal expansions will never terminate in any base.

Just nitpicking. Suppose we pick base $\pi$.
Then $\pi, \pi^2, 2\pi$ will all have number representations of finite length won't they?
It's just that e.g. the base $10$ number $2$ will have a representation of infinite length.

 

[Olinguito]

Well, maybe I should have said any positive-integer base. (Wink)