Exploring Infinity: The Concept of Infinite Numbers between 0-1

[AstrophysicsX]

Is it true, that there is an infinite amount of numbers between 0-1? Think about it, what number comes after 0, and before 1? Whenever I try to think about it, my mind goes blank. Discuss.

 

[CJames]

Mathematically speaking, yes. In the physical world, this may or may not be the case, depending on what theory you're talking about or what you're measuring.

 

[AstrophysicsX]

So in reality, there can't be a number directly after 0?

 

[CJames]

It depends on what you mean by "in reality." Mathematically, no matter how close a number is to 0, there is always another number that is smaller, which means that there really are an infinite amount of numbers between 1 and 0.

But this doesn't necessarily hold true for reality. In quantum mechanics, for example, there is no "spin" between 0 and 1, for example. As far as measurements of space or time go, nobody is entirely sure yet.

 

[xts]

The answer depends on what you mean by "number" ?
If you speak about natural numbers, there are no numbers between 0 and 1.
If you speak about rational numbers or real numbers, there are infinities of them, and there is no direct follower of 0.
If you speak about complex numbers, you should think over what means "in between".
If you speak about multiple possible arithmetics, the answer may differ. E.g. arithmetics of decimal numbers with 3 digits ater period is a perfectly valid arithmetics, equivalent to arithmetics of natural numbers, where you have 999 numbers between 0 and 1, and direct follower of 0 is 0.001.
You may create some artificial arithmetics with infinite number of numbers, where 0 has its follower.

 

[Anonymous217]

In this case, I would just assume he means over the reals. Also, do note that over the reals, there's an infinite amount of #s between 0 and 1, between 0 and 1/2, between 0 and 1/4, ..., but only for a finite sequence 0 and 1/n. If you continue that process indefinitely, you will of course converge to 0. Take calculus for more info :]

 

[SteveL27]

So in reality, there can't be a number directly after 0?

It depends on what you mean by reality.

If you mean in the real number system; which is the mathematical model of the intuitive notion of the straight line of Euclid; then the answer is no, there is no next number after 0. No matter how close a number is to zero, as long as the number is even a little big bigger than 0; then there are infinitely many other real numbers between it and 0.

But if you mean in the physical universe that we live in ... to my knowledge, I believe the jury is still out on that question. It's unknown.

 

[gb7nash]

The smallest floating point in IEEE precision is 2^(-1022). http://www.mathworks.com/help/techdoc/ref/realmin.html

If you're asking about real numbers, it doesn't exist.

 

[1MileCrash]

It's been said, but I'll say it again anyway.

There is no number directly after 0. Think about it, what would it be? 0.00000...1? A one after an infinite amount of zeros?? YES, there is are infinitely many numbers between 0 and 1, the majority are transcendental, and yet at the same time there is an infinite amount of rational, irrational, and transcendental numbers between 0 and 1.

But it UNITS of the physical world, it may or may not be the case.

For example, I believe that 1 Planck length cannot be measurably fractioned, ever. I don't think that is universally accepted though, it is just an opinion.

An electron can't be "between" energy levels.

etc.

In short, numerically, there are an infinite amount of different values between 0 and 1. Or even 0 and 0.000000001, as required by the former to be true.

In the real world, there are some physical units that are the "bare bones" of reality, the indivisible particles and states. There is not infinitely many values for electron spin between 0 and 1. There are none.