What is wrong with pi?

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Circles can be considered rational objects, shapes or stimuli. The diameter, radius and circumference of a circle can also be considered rational. Then why is it that the number which is used to calculate the diameter of a circle from a measurement of the circumference is irrational?

Everyday my mind is boggled by this question, there are so many different possible answers one might come up with. It could be the fact that human measurement systems still have a degree of error, maybe humans lack the fundamental understanding of our universe required to fully understand the circle, or maybe circles do not even exist.

One might say that any understanding of our universe formulated from the circle is radically incorrect, until pi is no longer a transcending number.

Lots of mathematicians prefer Tau, which is 2 * pi

I understand what you're saying but the square root of 2 is also irrational. I don't think much of it.

Pi, the ratio of the circumference of a circle to its diameter is provably transcendental. It is not the root of a polynomial with rational coefficient.

By the way, in the mathematical context rational means the ratio of two integers. It does NOT mean logical or reasonable.

ruveyn

I've always the following story amusing. Apparently, irrational numbers were "fightin' words" to Pythagoras. Who knew that math could be so dramatic?

The drowning of Hippasus is a just-so story. There is really no hard proof that any such thing really happened. For folks who did not like Pythagoras' "Hippie Commune" this is just the story they would like to hear and repeat.

ruveyn

I sometimes wonder if some of these things share properties with that they describe. I mean a circle is like a continuous surface with no edge or end and Pi seems to not end either.

The meme "the decimal expansion of pi never ends" is misleading.

Firstly, every decimal expansion "never ends" - if we don't write all of it, that just means that all of the remaining digits are 0. So instead of saying "the decimal expansion never ends", we should say "the decimal expansion does not end in recurring 0".

Secondly, some rational numbers have decimal expansions which do not end in recurring 0. For example, 1/3 has this property.

Thirdly, decimal expansions are not always unique. In fact, if a real number has a decimal expansion which ends in recurring 0, then it also has a decimal expansion which ends in recurring 9.

Fourthly, the decimal notation is entirely arbitrary - it is based on the fact that we have ten fingers, which has nothing to do with mathematics. For example, one of the ternary expansions for 1/3 ends in recurring 0, but the unique decimal expansion for 1/3 does not.

I think I know what you are trying to say, though. Here is a more precise statement of what I think you are getting at:



In particular, since pi is irrational, its unique decimal expansion does not end in a certain finite string of digits repeated over and over forever.

However, this is not a special property of pi. It is a property of all irrational real numbers. In fact, most real numbers are irrational and therefore have this property! (Using a technical definition of the word "most".)

There are two ways in mathematics in which we can make sense of the idea that "almost all" members of an infinite set A are also members of an infinite set B.

One is called cardinality. If we say that "almost all" things which are in A are also in B in the sense of cardinality, we are basically saying that the number of things in A which are not in B is tiny when compared to the number of things in A.

The other is called measure. If we say that "almost all" things which are in A are also in B in the sense of measure, we are basically saying that the collection of things which are in A but not in B takes up practically no space at all when compared to the collection of things which are in A.

Almost all real numbers are irrational, both in the sense of cardinality and in the sense of measure.

When you look at it like this, you realise that the fact that pi is irrational is nothing special. Any "randomly chosen" real number will almost certainly be irrational. In fact, what is really surprising is when a real number is rational.

The rational numbers are a handful of well-behaved weirdos scattered on the vast ocean of the real numbers.

Since there are only a countable set of algebraic (non-transcendental) numbers almost all real numbers are transcendental.

ruveyn

There's nothing wrong with pi. Pi is perfect. Especially when it's filled with apples and served hot! Mmmmmm....

If the nature of Pi is giving the OP an existential crisis I wonder how he'll cope if he learns about relativity and quantum physics? The bottom will really fall out of his neat impression of the world then.

@kxmode: Pervert! You Americans and Apple pie. I saw the movie.

Oh Ericys, you're just being irrational!! !

I was actually referring to the dessert. Apple Pie a la Mode is awesome!

PS I've never seen the movie so I have no idea what you're referring to.

I haven't seen the movie and I get the referrence.
Let's just say he loves apple pie

Besides rational and irrational numbers, you can define an even larger class of numbers called "surreal" numbers that contains all the real numbers plus different infinite and infinitesimal numbers.

pi is irrational because a perfect circle (a theoretical construct in my opinion, because it's quite hard to create a real, physical perfect circle) is perfectly round. if you have a picture of a circle on a computer screen, and you zoom in, eventually you'll get square pixels. However, teh theoretical perfect circle will always be circular, even if you zoom in infinitely. That's my guess anyway. It reminds me of quantum mechanics...