Can numbers get so small (or so large) that there is kind of a “planck length” effect where you just can’t get any smaller? Or is it really possible to have 1.000000…(infinite)1
EDIT: I know planck length is not a mathmatical function, I just used it as an anology for “smallest thing technically mesurable,” hence the quotation marks and “kind of.”
In: 87
Answer: Math isn’t real. Not in the sense you are thinking of. It is a language that is a NON-PERFECT description of reality, constantly updated to match our understanding of the universe.
Say for example you found the smallest real life thing possible. Let us pretend that it is somehow a component of what we currently consider the smallest possible particle/object. We could still arbitrarily define an area as being less than the whole of that object. This is why numbers are “infinite” –the arbitrary nature of math.
Please note, I am not saying you can’t count things, or math is a lie, or anything else like that. Just that it is a language we use to describe reality.
Infinity is actually not the biggest number. (Not even *technically* a number, more of a concept really)
There are multiple infinites, and some are bigger than others (by a lot).
And I know what you’re thinking. This is actually true, I promise.
If anyone reading is interested in learning more, lookup “how to count past infinity” by Vsauce on YouTube. It’s buried deep, but I believe the answer to your question OP can be found in this video.
Infinity is a concept, not a number. Infinity
+ 1 = Infinity. Therefore is not always equal to itself. You can have Infinity integers, and Infinite numbers between each set of numbers.
It doesn’t make sense to see Infinity as a quantity of something.
You’re asking three different questions.
1. Yes, real numbers can be arbitrarily small or large, and integers can be arbitrarily large. Given any real number, you can divide it in half to get a smaller number, or double it to get a larger number. So there’s no limit on how small or large numbers can be.
2. The Planck length is a matter of physics, not mathematics. We *use* numbers in physics to describe things because they’re useful for that. But even if there is a limit to the indivisibility of the physical world, the real numbers have no such limit.
3. There is no real number “1.000000…(infinite)1”. That’s not meaningful notation. You can make a real number *arbitrarily* small, but not *infinitely* small. This may seem like a technical nitpick, but it’s very important.
Acutally yes! There are infinite numbers between 2 number! In fact There are more numbers between 2 whole numbers then whole numbers themselves! Even though they are both infinite we have proven there are more!
It is difficult to explain in words but the numbers between 2 numbers are part of a class of numbers called uncountable infinites, while whole numbers are countable infinite.
There is an excellent video by vertasium called. “How an infinite hotel ran out of rooms” it covers the same concept I mentioned here where some infinites are larger!
Numbers can always get smaller or larger.
But if you have 1.000(infinity 0s) you can’t have a 1 at the end. There’s no end to infinity. So there’s nowhere to put the 1. It’s kind of like saying “After forever, you can book this hotel room for 1 week.” It doesn’t make sense to have forever plus one week. When does that week happen if it’s already forever? The hotel room can never be booked.
If you want to specify a whole lot of 0s, that’s fine. Maybe 1.00(1 million 0s) and then a 1. That’s a number. And you could always make it smaller by saying 1.00(2 million 0s) and then a 1. But if you ever have infinite, there can never be anything that comes after it. Because infinity has no end.
There is a 0.99 inbetween 0.9 and 1.
There is a 0.999 inbetween 0.99 and 1.
There is a 0.99999999999999999999 inbetween 0.9999999999999999999 and 1.
You can keep going on as long as you want and adding as many numbers inbetween 0.9 and 1 as you want. There are infinitely many numbers inbetween the two. But! Note that all of those are numbers that end. You can have 0.9 with a billion 9s, and that’s still a specific number that fits somewhere in the middle there.
However, the number 0.9 repeating (or 0.9…) is literally and exactly equal to 1, because if the 9s are infinite then there is nothing inbetween 0.9… and 1, which makes them the same thing.
Math itself is not a fundamental part of the universe. It’s a conceptual tool that we use to describe it. The rules of the tool allow us to take any two numbers and derive a number that is greater than the first, but smaller than the second. So yeah…. math is infinite, assuming the universe is too. Here’s the catch: a number requires some way to be represented, be it on paper or in a mind. That representation requires *something* to exist. If that *something* has a limit, then the number would too.
Here’s a counterexample of why the planck length is huge compared to infinitely small numbers: probability will always demand smaller numbers.
For example one roll of the dice has a 1/6 probably of rolling a 4. Two rolls have a 1/36 probability of rolling two 4s. The probability of rolling six trillion 4s in a row is an extremely small number, and it is massive compared to twenty trillion 4s.
Not only is it infinite, but it’s provable that there are more numbers in between numbers than there are numbers.
To be more precise, the set of real numbers is a larger infinite set than the infinite set of integers.
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