## A Conversation for Pi

### Nit for the day

Researcher 93445 Started conversation Dec 2, 1999

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.

### Nit for the day

Phil Posted Dec 2, 1999

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

http://www.torget.se/users/m/mauritz/math/num/irat.htm

I think I can understand what is going on about why it's irrational

and also trancendental.

### Nit for the day

Jan^ Posted Dec 3, 1999

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.

### Nit for the day

Jan^ Posted Dec 3, 1999

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.

### Nit for the day

Phil Posted Dec 3, 1999

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

### Nit for the day

Phil Posted Dec 3, 1999

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.

### Nit for the day

Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming) Posted Mar 11, 2000

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

Post-lecture fishies

### Nit for the day

Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming) Posted Mar 11, 2000

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

Irrational fishies

### Nit for the day

Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming) Posted Mar 11, 2000

I have just written an article on the Euler Equation at:

http://www.h2g2.com/A278408

Please tell me what you think

Joe aka Arnia

Awe-inspired fishies

### Nit for the day

Researcher 114836 Posted Mar 21, 2000

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.

### Nit for the day

Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming) Posted Mar 22, 2000

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

Plain Fishies

### Nit for the day

Researcher 114836 Posted Mar 23, 2000

The proof on my home page. Go check it out and tell me what you think. So long....

### Nit for the day

Researcher 31283 Posted Apr 18, 2000

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:

0 ->1

0.12341413352534542523542354253424523234676798... ->2

0.41234258475892357490502745981257810578701987... ->3

0.48908282358847583787387132234252345423545235... ->4

.

.

.

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.

### Nit for the day

Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming) Posted Apr 20, 2000

Actually, now I have more time... that is a very nice proof

Who's is it and from what year?

Joe aka Arnia

### Nit for the day

26199 Posted Apr 23, 2000

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:

0.1113333324598...

0.2384932483423...

0.2304095948332...

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

26199

### Nit for the day

Researcher 114836 Posted May 22, 2000

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).

### Nit for the day

Purple Posted Oct 3, 2000

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'

Key: Complain about this post

### Nit for the day

- 1: Researcher 93445 (Dec 2, 1999)
- 2: Phil (Dec 2, 1999)
- 3: Jan^ (Dec 3, 1999)
- 4: Jan^ (Dec 3, 1999)
- 5: Jan^ (Dec 3, 1999)
- 6: Phil (Dec 3, 1999)
- 7: Phil (Dec 3, 1999)
- 8: Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming) (Mar 11, 2000)
- 9: Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming) (Mar 11, 2000)
- 10: Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming) (Mar 11, 2000)
- 11: Researcher 114836 (Mar 21, 2000)
- 12: Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming) (Mar 22, 2000)
- 13: Researcher 114836 (Mar 23, 2000)
- 14: Researcher 31283 (Apr 18, 2000)
- 15: Researcher 31283 (Apr 18, 2000)
- 16: Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming) (Apr 20, 2000)
- 17: 26199 (Apr 23, 2000)
- 18: Researcher 114836 (May 22, 2000)
- 19: Purple (Oct 3, 2000)

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