Infinite? How can we have infinite numbers between 0 and 1?
Once in my Math class, a system of equations produced infinite solutions represented by a single line. Is it possible to define infinite by means of a finite measure? :confused:

What exact number comes directly after 0? Is it 0.000001? What about 0.0000001? Or 0.000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 0000000000000000000000000000000000001?
Or mybe it could be represented by 0.0...01? Could we use a symbol to represent the smallest possible number and then use such a symbol to define infinity? Huuuummm.... ;)

There is no smallest possible number. Very small numbers are written using scientific notation.
For example:

Converting 0.000,000,000,000,000,000,000,020 grams per carbon atom into scientific notation involves moving the decimal point to the right 23 times.

0.000,000,000,000,000,000,000,020 = 2.0 x 10-²³

Infinite? How can we have infinite numbers between 0 and 1?
Once in my Math class, a system of equations produced infinite solutions represented by a single line. Is it possible to define infinite by means of a finite measure?

There are more than a finite number, so it must be infinite. A single line has an infinite number of points.

What exact number comes directly after 0?

There is no such number. If you try to have one, you can divide it by 2 and you will have a number closer to 0.

does this apply to space also?

I believe there a a finite amount of numbers, but our minds are to limited to imagine that possibility so we use infinite as an easy excuse... An infinite amount of point on a line, although in theory is common, makes little sense to me. Otherwise the line would not be finite.

sevenblu...

it is really straight forward to show that there is an infinite amount of numbers between 0 and 1.

start with any number between 1 and 0, say 1/2
divide it by 2 , here we get 1/4
check if it's greater than zero, here 1/4 > 0
continue until you find a number that equals zero. you will never find a number that equals to zero and you can continue this and come up with new numbers every time.
hence there is infinately many numbers between 0 and 1.

show me your proof that there is a finite amount of numbers? and remember, this is mathematical space and not physics, so don't you dare use Planck's constant :D

It's called Zeno's Paradox. It leads to the idea of limits which is an introductory concept in calculus. I never made it to calculus in school, but I've been working through an online tutorial. Here's a link to the page about <a href="http://www.karlscalculus.org/calc2.html">Zeno's Paradox</a>. It's not too tough to follow.

It leads to the concepts of delta and epsilon. It can be proven that you can get as close (epsilon) to any number (the limit) as you want given a... (I can't really explain Delta easily like this.) I'm sure some more astute mathmeticians in this forum can explain it far better than I am able to. I could cut and paste the formal contract, but it might be kind of hard to follow.

It leads to the concepts of delta and epsilon. It can be proven that you can get as close (epsilon) to any number (the limit) as you want given a... (I can't really explain Delta easily like this.) I'm sure some more astute mathmeticians in this forum can explain it far better than I am able to. I could cut and paste the formal contract, but it might be kind of hard to follow.Epsilon is the new number you want to attain that is closer to the limit than the old number but still hasn't reached the limit. Delta is the change from the old number to the new one. The statement starts with "For every epsilon there is a delta. . . ." With that start you should be able to fill in the rest.

Thanks Fraggle, I was looking for a more informal declaration though. I could easily copy the formal contract. It doesn't make so much sense unless you (I anyway) have followed a few practical explanations.

I thought Epsilon specifies how close to the limit you want to be. So in other words, rather than the new number, I thought it was the difference between the limit and the number you settle with.

does this apply to space also?

No, the smallest length in space is Plank's length. Its a quantum physics concept, forget exactly how small it is but its very VERY small. Same with time

Bear in mind that space and time are "real", whereas mathematics is abstract.
Planck's length is a quantity, not a number.

No, the smallest length in space is Plank's length. Its a quantum physics concept, forget exactly how small it is but its very VERY small. Same with time

Hrmmm... this is often said but you should say it with care. I am no expert in high energy physics... but.... the Planck length and Planck energy are the points (expressed in length scales or energy scales) at which the four fundamental forces would theoretically unify. I am no expert but I do not immediatelly see how this should form some sort of magic barrier to distinguish between quantification of spacetime or not (I can see how it would be related).

I think it is more correct to say that there are some theories which quantify space (and time) but at the moment there is no experimental evidence that supports this case. So until the Nobel-prize winning experiment that proves the quantification of spacetime is performed, it is safer to assume that spacetime is continuous.

Bye!

Crisp

Once in my Math class, a system of equations produced infinite solutions represented by a single line. Is it possible to define infinite by means of a finite measure? :confused:

Yes, you just did it. You can mathematically prove that there are an infinite number of numbers in the interval [0,1], and even a "special" kind of infinity. You could argue that by defining some function f(x) you could obtain all natural numbers N from the interval [0,1] i.e. that there would be a one-to-one mapping from [0,1] to N (the set of natural numbers).

It turns out that this is not possible, there are "more" numbers in [0,1] than in N, i.e. you cannot "label" all numbers in [0,1] by a number from N.

This "kind of infinity" is called "overcountable" infinity, while N is "countably finite". Don't let the word "countable" misguide you, you can never manually count all numbers in N ofcourse; by counting in mathematics one means the operation of "labeling every element in the set by a natural number". This is hence not possible for [0,1].

Could we use a symbol to represent the smallest possible number and then use such a symbol to define infinity? Huuuummm.... ;)

Wel, the number closest to zero is zero. You can prove that if you keep adding zeroes like 0,001 ... 0,000000001 , .... 0,0000000000000001 ... you eventually get zero:

lim_{n -> oo} 10^-n = 0

So in that sense, there is "no smallest number". You should also take care when trying to define "infinity", because you just saw that there are already two kinds of "infinity".

Bye!

Crisp

lim_{n -> oo} 10^-n = 0
Yes, that's exactly whatr I'm talking about. Couldn't we definea finite number to decribe infinite by using that? I mean.... assigning a variable, with a definition....

In math: infinite.
In reality: none.

I mean, using mathematical symbols to solve equations. :bugeye:

Hrmmm... this is often said but you should say it with care. I am no expert in high energy physics... but.... the Planck length and Planck energy are the points (expressed in length scales or energy scales) at which the four fundamental forces would theoretically unify.

Are u sure that this is all it is? I always thought it was this but also that there can be no differentiation (of anything :eek:) on a smaller scale of space or time than the Planck dimensions.