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Actually, there are an infinite number of rational numbers that contain an infinite number of digits in their decimal representation. 1/3, for example, expands to 0.333333.... for as many 3's as you'd care to write down, and then some.
What's special about pi is that its digits never repeat, no matter how long you look. That's what makes it "irrational," that is, not the ratio of two whole numbers divided by one another.
Ok pi is irrational, but it is also trancendental as is e.
if you want a more detailed explanation on why it is irrational and
trancendental then check out
I think I can understand what is going on about why it's irrational
and also trancendental.
Read the article - it never says that pi is irrational because it has an infinite number of decimal places - it says the reverse.
The current record of agreed digits is 206,158,430,000.
Q. There are more irrational numbers than rational numbers. Prove it.
(Cambridge Maths Tripos part 2, about any year from 1950 onwards)
Just don't expect me to prove it... I did physics.
P.S. e^(i*pi) = -1
Now THAT is neat.
Cool, I'd forgotten about that.
As an engineer (electronics) we had to do lots of maths with pi,e
and i in it, shame the lecturer wasn't that good at the time
Oh yes, can you prove that e^(i*pi) = -1 or is it just one of those
things that just is with the `standard' number theory in use.
It is provable and you study it at 'A' level on the London board. One of its concequences is that we can now define the natural log of negative numbers (ln -1 = pi i) and so a whole family of hyperbolic graphs (related strongly to hyperbolic functions) comes to light based around 2 mutually perpendicular Argand diagrams sharing no axii. This was fundemental to Taniyama-Shimura conjecture and so the Langlands Programme as well as being the basis of the proof of Fermat's Last Theorum.
It is one of the most elegant, beautiful and awe-inspiring formula in maths as it shows a glimmer of how it all fits together. It is known as the Euler equation after the mathematician who found it. Its full form is:
e^(theta i) = cos theta + i sin theta
Where theta is measured in radians.
I believe it is provable via DeMoives theorum but I am not sure...
On another note, the use of the letter Pi to represent the ratio of the circumference to diameter was first invented by an English mathematician in the 18th century and popularised by Euler.
Joe aka Arnia
Oh and the proof for the fact that there are more irrationals than rationals has something to do with Dedekind's cut, if I remember right.
Joe aka Arnia
I have just written an article on the Euler Equation at:
Please tell me what you think
Joe aka Arnia
I looked at the article. It looks nice...........but it is not a proof. First of all, Euler's equation is a definition rather than a theorem. Secondly, you try to prove a basic property of the exponential function by using the inverse (log). That is like eating bread you are planning to bake tomorrow. An intuitive argument to show that the Euler equation fits with our usual definition of exp(x), see my page.
It is not difficult to prove that there are more irrational numbers than rational ones. If someone is interested..... On the other hand, it is quite easy to prove that between every two irrational numbers is a rational one and vice versa.
The final statement about ln -1 = pi i wasn't intended as a proof but merely as a statement of consequence.
I am intrigued by the proof for irrational to rational ratio. Could you point me towards a place with it?
Joe aka Arnia
The proof on my home page. Go check it out and tell me what you think. So long....
There are as many rational numers as there are natural numbers, because you can get them in a one-to-one corresponence:
1/1 2/1 3/1 4/1 ...
1/2 2/2 3/2 4/2 ...
1/3 2/3 3/3 4/3 ...
1/4 2/4 3/4 4/4 ...
Where the dots mean 'and so on'. Give each of those rationals a number in the following way:
1/1->1, 2/1->2, 1/2->3, 1/3->4, 2/2->5, 3/1->6, 4/1->7, 3/2->8, 2/3->9, 1/4->10, 2/4->11,...
This way you can give each rational a number from the naturals.
Now let's try such a thing with the irrationals:
Now you see the obvious problem: if you try such a thing with irrationals you can always find ones that you have "forgotten" to give a number, in this case 0.4123426453543453454345435354454... would be one. So there are more irrationals than natural numbers. We have already seen that there are as many rationals as natural numbers we can conclude that there are more irrational numbers than rational ones.
(answer to a question from Jan)
Actually, now I have more time... that is a very nice proof
Who's is it and from what year?
Joe aka Arnia
I read about a more comprehensive version of the above proof in the brilliant book 'Godel, Escher, Bach: an Eternal Golden Braid'. Basically, if you start writing a list of all the irrational numbers:
You could eventually get to an infinitely long list of infinitely long numbers. However, even if the list is infinitely long, you've still missed some numbers out. Here's how to find a number that's missing:
Take the first digit of your new number from the first digit of the first number in the list, plus one. Whatever else you add to your number now, it's different from the first number in the list. Take the second digit of your new number from the second digit of the second number in the list, plus one - it's now certain to be different from that number.
Carry on down the list, diagonally, adding one to each digit (and going back to zero if you get a nine). You end up with an infinitely long number which is different from every number in the list. You can do this as many times as you like, and there will always be more irrational numbers that you haven't written down...
QED, you can't write down all the irrational numbers in an infinitely long list (you can't even write down all the irrational numbers between zero and one in an infinitely long list). You can, however, write down all the rational ones in an infinitely long list. I think
Basically, all the proofs above are the same (and similar to the one on my page). They are all based on Cantor's diagonalization process. This means the proof is made before 1872. In 1872 Cantor published his results of the construction from the rational to irrational numbers with Cauchy sequences.
You can indeed list the rational numbers in an infinitely long list. First you make the two-dimensional list as above. Then you read the list diagonally to make it one-dimensional. Finally, you remove the double numbers (since 1=2/2=3/3=... etc).
The greeks didn't actually use the character pi. Matheamticians used to use it to mean perimeter and when they meant pi they said pi/diameter. However it somehow changed meaning.
I read this in a book called `The Joy of Pi'