Hello everyone,
just a quesiton i have been wondering and no one seems to be able to give me a straight answer.

for example i think that .1 repeating has to be equal to 435784395743 reapeating. Infinite numbers go on for ever and ever. So any infinite numbers have to be equal because they never end. Is this right? Discuss and explain.

"So any infinite numbers have to be equal because they never end."

It just isn't so. I don't know where you got this idea. Moreover, whether or not a number has an infinite representation depends partly on the number system you are using. For example fractions in binary will have infinite expansions unless they are finite sums of powers of 1/2. Similary in base 3 notation, these fractions will have infinite expansions uless powers of 1/3. In decimal notation, finite fractions are only for powers of 1/2 or 1/5 or combinations.

1/3>1/9

Nick,

Do you understand what .11111... means?

It it shorthand for

1/10 + 1/100 + 1/1000 + 1/10000 + ...

This is not the same as, for example .453245324352.... which would be

4/10 + 5/100 + 3/1000 + 2/10000 + 4/100000 + ....

See?

So any infinite numbers have to be equal because they never end

Dear Nick Stenson,

First, this is a good question because it lead us to check again collections of infinitely many things.

Please keep thinking about it.



Any number can be distinguished form another number by at least two basic properties, which are:

a) Structure

b) Quantity

Let me show you a nice thing, for example, let us take number 26.

Now we can show that this quantity has more than one internal structure, which is based on the base value that we are using:




Number 26 represented by base 10:

^0- 0123456789
||||||||||
|_||||||||
|__|||||||
|___||||||
Base 10 = |____|||||
|_____||||
|______|||
|_______||
|________|
^1- |
| 1 0
( 2*10 + 6*10 )
^0- 012345678901234567890123456
|||||||||||||||||||||||||||
|_|||||||||_|||||||||_|||||
|__||||||||__||||||||__||||
|___|||||||___|||||||___|||
|____||||||____||||||____||
|_____|||||_____|||||_____|
|______||||______||||__...
|_______|||_______|||__...
|________||________||__...
^1- | 0 | 1 | 2
|_________| |
|___________________|
|_ ...
|



Number 26 represented by base 3:

^0- 012
Base 3 = |||
|_|
^1- | 3 2 1 0
( 0*3 + 2*3 + 2*3 + 2*3 )
^0- 012012012012012012012012012
|||||||||||||||||||||||||||
|_||_||_||_||_||_||_||_||_|
^1- |0 |1 |2 |0 |1 |2 |0 |1 |2
|__| | |__| | |__| |
|_____| |_____| |_____|
^2- | 0 | 1 | 2
|________| |
|_________________|
^3- | 0
|_ ...
|

So, as you can see, even the same quantity can be distinguished by at least two different internal fractal structures.

From this example we can understand that the concept of a Number is some information form, which is based on not less than Structural_AND_Quantitative properties.