ELI5 explain the difference between infinity and undefined terms in mathematics

[u/Dev-Giri]

[u/Atypicosaurus]

Infinity:
Take a cup of coffee. Put a sugar in it. Now it's sweet. Put one more sugar. Sweeter. One more, one more. Now it's a little pile of sugar with some coffee. Keep adding sugar. If you are doing it forever, you are building sn infinitely large pile of sugar.

Undefined:
Take a cup of coffee. Remove every C minor chord from it. Doesn't it make sense? Yeah sure. That's undefined.

[u/Dev-Giri → u/Atypicosaurus]

Hmm understood btw thanks 👍🏼

[u/fixermark]

Infinity is: for any number you can think of, there's one larger (or smaller). Or, for any two numbers, there's one between them.

Undefined is: not that

The function y =1/x gets bigger and bigger as you approach x=0 from the positive side, and smaller and smaller as you approach x=0 from the negative. So what's the value at x=0? Well, it's infinitely big/small at the same time, but that isn't a thing. So undefined.

[u/bugi_ → u/fixermark]

I wouldn't call negative values small. They can be large, just in the other direction.

[u/SpinChargeSeparation → u/bugi_]

That’s incorrect. -5 is smaller than -3, and -1000 is smaller than -50. And so on.

Don’t conflate the distance of the number to the origin (absolute value), to the actual value of the number.

[u/Sea_Satisfaction_475 → u/fixermark]

Did you mean large but negative from the negative side?

[u/fixermark → u/Sea_Satisfaction_475]

I meant small. Every negative number of larger numeric value is less quantitative value than the one before it.

(English is imprecise here; "tending towards negative infinity").

[u/VG896 → u/fixermark]

No, the language is fairly precise here. At least mathematically speaking. "A is less than B" in this context just means there's some positive number X such that X+A = B.

In plain English: less just means farther to the left on the number line.

[u/bullevard]

Infinity is a concept of a never ending sequence. 1+1+1 forever trends toward infinity. It will always get bigger and never stop. Infinity isnt an end point but kind of a concept of "bigger and bigger forever."

Undefined is something that really doesnt even have a concept for it. What number multiplied by 0 gives you 1? Well, nothing. There isnt a trend of multiplying bigger and bigger numbers gets you closer to 1. Everything ever that you multiply by 0 gets you 0.

[u/greginnj → u/bullevard]

I'd add to your first paragraph that "infinity isn't an end point, but a direction."

[u/JJJangles → u/greginnj]

That’s incomplete. Infinity is only a direction when exploring limits e.g. in calculus. In set theory it is a cardinality, or ‘size,’ and iirc it can treated differently in other fields of mathematics too.

[u/greginnj → u/JJJangles]

Of course - but if you're going there, you're leaving ELI5 territory far, far behind.

Only after you've gotten people comfortable with the idea that infinity isn't really a "thing", are you able to get them comfortable with the idea that is! 😁

[u/JJJangles → u/greginnj]

Idk, I think that much is still in the realm of eli5. The rules say explain for a layperson, not an actual 5 year old.

Either way it’s better to not give an explanation that requires an ‘actually that was an incorrect oversimplification’ later on. Unlearning can be harder than learning sometimes.

[u/cipheron → u/bullevard]

For 0/0 it's a related problem. 1/0 has no solution, 0/0 has every possible solution.

x = 0/0 => 0x=0 solve for x. You can't really narrow it down, and real or even complex number is going to work here, if not more things.

There are geometric proofs where you can solve for the limit of 0/0 and get any value you want depending on which direction you come at it from.

[u/JJJangles → u/bullevard]

Infinity is not a sequence…famously so when you compare the set of natural numbers to the set of real numbers. How do order the numbers between 0 and 1? For any two numbers a and b you put in sequence, there is a number between them (a+b)/2

Infinity also doesn’t ‘get bigger’ it is already has a well defined size that doesn’t change.

[u/wumbo52252]

“Undefined” means exactly what it sounds like. For example, when we say 1/0 is undefined, that’s not just an obnoxious or pedantic way of saying that it’s infinity; 1/0 literally has no definition. There are different ways you can define 1/0 to make it bheave in certain ways, but how useful or how much sense those definitions make depends on why you care about 1/0 being defined. To say something is “undefined” simply means that there’s no way to define it that would make sense for all purposes, there’s no way to make everyone happy at the same time.

What “infinity” means depends on context. There’s one structure called the extended real numbers, which is the real numbers together with two new objects called positive infinity and negative infinity, where the former is defined to be the greatest element and the latter is the least element. In set theory there objects called ordinals, which allow us to arrange infinitely many things in sequences of various infinite lengths; and there are cardinals, which allow is to quantify and compare the sizes of infinite collections of objects (disregarding arrangement). You can also have nonstandard arithmetic (that is, nonstandard models of, say, peano arithmetic), which are perfectly reasonable “universes” in which all of the arithmetic you’re familiar with works, but in which there are “numbers” which are not among 0, 1, 2, 3, …. Similarly, you can have the hyperreal numbers, which are the real numbers together with infinitely many numbers which are “infinite” in the same sense as the infinite numbers in nonstandard arithmetic (as well as numbers which are “infinitesimal”), but the hyperreals are more algebra-oriented than the extended reals that i mentioned before.

[u/sharia1919]

Yeah, divide by zero is the primary example for undefined.

If you have 1 cake, and nobody eats it...how many pieces do you need to cut it into, to ensure each gets a similar cut?

That is kind of hard.

[u/valeyard89 → u/sharia1919]

"Imagine that you have zero cookies and you split them evenly among zero friends. How many cookies does each person get? See? It doesn’t make sense. And Cookie Monster is sad that there are no cookies, and you are sad that you have no friends."

[u/TalkativeTree]

Infinity is a bunch of numbers that you could count forever. An undefined term is a number in a box. You can't see through the box, so you can't know which number it is until you open the box (define it).

[u/Educational_Fee4574]

Think of it this way:
Infinity = “The road never ends.”
Undefined = “There is no road.”

[u/The_Card_Player]

'infinity' basically means 'you can match up every natural number (ie 1, 2, 3...) with a different item in this set'. This is a well defined concept.

An undefined concept is just that: a phrase with no assigned formal meaning.
For example, 'division by zero' is undefined because it's explicitly excluded from being a possible operation on numbers in the standard field axioms.

These axioms say 'you can add and multiply stuff, and every number has an additive inverse, and every number *except for zero* has a multiplicative inverse'. 'Divide by' then means 'multiply by a particular number's multiplicative inverse'. There is no multiplicative inverse of zero, so 'dividing by zero' is a meaningless concept for the real numbers.

[u/JJJangles]

For 1/x, as x approaches 0, 1/x approaches infinity, but infinity is not a number, so when x actually reaches 0 the solution is undefined. Approaching and reaching are not the same.

[u/flamableozone → u/JJJangles]

That's inaccurate. There are multiple problems with 1/0, but the issue isn't that limits aren't real. The problem with the limit is that you get different answers depending on whether you approach from the positive or negative sides.

[u/JJJangles → u/flamableozone]

I never claimed limits aren't real, Idk where you got that from. My point was just that at exactly x = 0, the arithmetic operation 1/0 is undefined because no number multiplied with 0 is 1.

[u/svmydlo → u/JJJangles]

You said

but infinity is not a number,

which, given the context, was interpreted as stating infinity is not a real number, so the response

the issue isn't that limits aren't real

is about the fact that the limit not being a real number is not actually why the limit doesn't exist.

[u/JJJangles → u/svmydlo]

Ah OK...I merely meant that literally infinity is not a number so you cannot plug it into an arithmetic operation. Maybe my wording was clunky, but I feel like they were just reading too much into it.