Following on from my thread where I got wrecked for being brain damaged, and posting about dividing by zero, it is time for round two!
Lightening round: Electric boogaloo!
Episode 3: "Glutton for punishment"

You can read that thread here if you like or skip over.
https://devrant.com/rants/4931841/...

Can we divide by zero? Is there some representation where thats the case? And what are the implications if we can?

In this round Devranters, you will be challenged to determine if OP is 1. insane, 2. a genius, 3. high on mushrooms. One contestant will be eliminated. The winning team will get a bag of rice and sunscreen, while the other team will have to vote to send someone home from the island.

Get ready.

Heres the full rant because DR wouldn't post it for some reason:

https://pastebin.com/qBg80ujN

Too lazy to get a pen. How does your proposed mapping work for something like 2 sin(x) / x, where the limit is 2, rather than one?

Off the top of my head, it kinda works*. If you make the parallel to i, let's use j for inf coz lazy. The above works coz you've got 2j, the other j's cancel. There are different types of infinity (like the number of numbers is a countable infinity, the number of numbers between 0 and 1 is uncountably infinite), so j, 2j, are values on the countably infinite line, j^j represents types of uncountably infinite stuff.

*Problem is stuff kinda tends to break down when infinity is involved. If you can show some sensible results for some of the following:
tan(x) lim x-> 0
integral of delta function being heaviside function
heaviside function of (x-n) having a discontinuity (normally its got a weird value at n).

Then whether it's actually of any use, it's an interesting idea but as I said, stuff breaks when infinity is involved, there are a lot of tricks to avoid it.

And if people wanna give me shit for not knowing anything, I've got a degree in physics.

Division isn't a basic mathematical operation. Basic operations are multiplication (with a scalar) and addition (subtraction is just addition with negative numbers). Division isn't a part of that. It doesn't exist in number theory, it doesn't exist in most fields of mathematics actually.

Division is a theoretical inverse of multiplication, not a mathematical operation, and it's only defined as such, i.e. "Can we find a number such that x*0=5?" "No" is a perfectly valid answer to that question. And that's what division is all about.

@hitko This is even more obvious when it comes to vector spaces (or linear algebra in general): Can we find a number such that x*[1,2,3] = [2,-3,5]? No, because those two vectors don't have the same (or opposite) direction. "No" is a valid answer.

@hitko I mean... By your own logic, the only true base operation is addition.. Subtraction is addition with negative numbers, multiplication is addition applied a given number of times.

The problem is, e, the exponent number, has a nasty habit of showing up in nature, meaning that exponentiation and differentiation are something of a primitive, from which you have e^-1 = 1/e...

@atheist

> there are different types of infinity

I always forget that detail.

If this were in a chemistry lab, you would be like "Uhh, thats not a good idea. I don't recommend it."

And then I'd do it anyway, and BOOM.

eyebrows gone, or worse.

@atheist if you look a bit crooked at uncountable infinities like the numbers between 0 and 1, would it be wrong to say they're really just transformations over powers of 10 from the countable infinities of the integer number line?

@Wisecrack nope, they're fundamentally different, in that, while you can't ever "reach the end" of countably infinite numbers, you can at least write a way to list them. ie 1, 2, 3, etc will "eventually" go through every countable integer number, but you can create a single decimal number that contains every countable integer by just doing 0.<insert above list of numbers here>, so a single number between 0 and one is in itself countably infinite. Whereas "all" the numbers between 0 and 1 are an infinity of infinities, that there's no way to progress through in a manner you'll eventually reach the end.

This does a decent job of explaining types of infinity
https://youtu.be/elvOZm0d4H0

@-red it's just a projection onto a higher dimensional space, encoding information that can't be presented in current number line, same as imaginary numbers. Doesn't need to be started from current axioms. I did complex analysis at uni so I've probably got a bit of an advantage on this sorta stuff, but it's a similar basic idea.

> j^j represents types of uncountably infinite stuff.

Actually, thats clever. Didn't think of that.

Does it make sense to generally think, of 'inf' not as a specific number, but as a set of numbers, or an operation over the set?

So if you did inf/inf, regardless of the set we're talking about, the result would come out to

[1, 1, 1...]?

Actually it occurs to me that a continuous function is a kind of uncountable infinity, because it covers the set of all numbers that it intersects, not just the integers, no?

> tan(x) lim x-> 0

I'm not at all familiar with calculating limits so I couldn't tell you.

Imaginary numbers just say "numbers work the same, but this thing that's currently undefined is defined like this". The parallel between root(-1) and 1/0 is there.

@Wisecrack the difficulty of infinity is you've got an infinity of infinities, j^j, then you can have an infinite infinity of infinities, ie j^j, j times, and an infinity of infinity of infinities, And an infinity of those, so you've got an infinite dimensional space with limited relevance to the real world beyond "types of stuff fall on one of these dimensions"

@Wisecrack yeah, 0 to 1 is just one small continuous function domain.

there's probably also some weirdness, in that j*j=j^j I think

or at least, j*j vs j^j needs to be clearly defined by

@atheist

> The parallel between root(-1) and 1/0 is there.

Well I'm glad to know I'm not entirely insane.

I think my mistake here is treating 'inf' as an arbitrarily large number, or point. Guilty as charged.

But I 'm not familiar with any other way to represent that relationship between some variable and some other variable that is arbitrary preciseness. or arbitrarily large in value relative to the other.

I mean how do you express that right?

Theres A, and then theres B, and however big B is, its just another number on the line of countable integers for example. But suppose you *did* want to say "this number or set of numbers are so large as to be uncountable in theory and/or practice."

More fun with infinity
https://youtu.be/PCu_BNNI5x4

@Wisecrack you kinda get a bit of it on paste bin when you're talking about remainders, there's weirdness there, but if 5/0 is 5 times 1/0, and 1/0 is "one unit of a certain type of infinity", then its another axis in your number system, but orthogonal to the current number line. Like imaginary numbers, (3+4i) where 3 is the real component and 4 is the imaginary component. But mapping back and forth is probably tricky because 1/0 just plain breaks stuff.

@atheist Imagine for a moment you have

n/0.

N is arbitrarily large.

Now replace plain zero, with k like mentioned before.

n is effectively so large, as to reduce k to zero

if we were to take k/n

As an aside if we're talking about an arbitrarily large point on a line--*arbitrarily large relative* to another point, k, then wouldn't j*j and j^j be distinct? We're treating the arbitrary point as arbitrary because its either an infinite set, or its so large as to be impossible to enumerate

in any amount of time.

If its simply a point without any symbolic purpose, then j*j is obviously reduceable to j^2, but if its a symbol, then it doesn't necessarily have to be the case that j*j == j^2, though that would be convenient for not breaking another convention.

@Wisecrack thing is, you're looking at it in terms of our current number line from positive infinity to negative infinity. that breaks down. You can't replace 0 in n/0 with k because eg you get a different meaning if K is positive or negative. Which, if you look at tan (or 1/tan), as x tends to 0, you get a different result depending on if you approach from positive side or negative side.

It's been quite a while since I've done this stuff, so take everything I'm saying with a pinch of salt, but I'm semi coherent

@atheist "Which, if you look at tan (or 1/tan), as x tends to 0, you get a different result depending on if you approach from positive side or negative side."

That behaviors kind of interesting though, makes me wonder what kind of utility it might have, even if its wrong.

What if k is constrained to the same sign as n?

What if k is signless? What are the implications?

> Its been a while since I've done this stuff

Drunk you knows way more than sober me, and I am grateful.

@Wisecrack 😂 Speaking of which, I'm going to bed because it's 2am and the woman I'm messaging as a distraction from maths has gone to sleep.

@atheist oh man I'm sorry to hear it.

"We're up all night to get lucky" as the song goes.

And instead we're doing math lol.

Thanks for it all, and thanks for the links.

Sleep well man. Sleep Well.

5/0 is undefined. 5/0 is never *infinity*.

What kind of crazy hole did the writer crawl out from?

My pick: High on mushrooms/weeds

@daniel-wu I'm not high on mushrooms.

I just choose to live in other dimensions sometimes.

@daniel-wu root(-1) is also undefined. Meet my friend, i.

@atheist Please don't bring that stupid "calculus for kids" oversimplification form into this. How do you express (-3)*(-5) as a series of additions? You repeat addition a negative number of times? How about 0.5*0.3? You do it 0.3 times? And even if you could do that, how would you express sqrt(2)*sqrt(8) (two irrational number which give an integer result)? Basic mathematical operations aren't defined for discrete values, they're defined based on what happens to every value in the (numeric, vector, ...) space. Addition moves every value in the space by a fixed amount, multiplication stretches the distances between the values by a fixed amount. In these terms, division is asking "Can we unscale the resulting space back into the original?"

The same goes for exponentiation; thinking of it as a series of multiplications is just an oversimplification they teach to children. Mathematically, exponentiation "bends / rotates" the space, and logarithm is like asking "Can we transform it back?"

@Wisecrack re different behaviour for +/- infinity, if it were to work it would need to be somewhat reversible, so there'd need to be a way to represent "different types of infinity". So to show some "use", it'd be nice to be able to describe a difference between abs(x) /x and x/x. Not easy, because those kinds of functions are notoriously fiddly.

@hitko All this is what mathematical analysis is about, but I tried to keep it out of this because it's not something most people ever learn about. A fundamental thing to keep in mind is that an operation is only invertible when it preserves the cardinality, e.g. multiplication preserves cardinality unless it scales the entire space into a single value (multiplication by 0).

@hitko this is my point. You introduced the naive oversimplification, but the complexity of exponentiation is very present in nature. e^x + e^-x is the shape a hanging string forms. e^-x = 1/e^x, so we kinda have to accept either division as a primitive, or exponentiation as a primitive, but as the two are coupled, one gives the other.

@atheist They aren't coupled. People who claim they are coupled are people who think that the sum of all positive integers is -1/12, because they never learned mathematical analysis.

@hitko they're "coupled" in the sense that you can define division in terms of multiplication by a negative exponent. If exponents are parts of the axioms, then surely division is too, whether you describe it as a bending of a vector space or a rescaling.

I'm not saying that division by zero is useful, some amazing insight, I do think it's interesting to discuss where it might break down without dismissing it out of hand.

@hitko if you have any other examples of where this breaks down, I'd be interested. My point is I enjoy this kinda conversation. Keeps the old brain gears tuned up.

@atheist It breaks as you get into negative basis and more complex (e.g. decimal, matrix) exponents and spaces, where you can no longer calculate it as multiplication. Say, what's a^x if a is a negative number? For real x, it's an oscillating series with exponentially increasing amplitude in complex space. What if a is a (square) matrix? There's no such thing as division by a matrix, but the result is still well defined as a new matrix of the same shape.

@hitko interesting that, notationally, we use a^-1 to denote matrix inverse, giving a parallel to division in this context. Then there are non-invertible matrices, sorta parallel to 1/0 (although linearly inseparable matrix causing information loss is a "bigger" non-invertible thing than 1/0).

@atheist Well, a^-1*a = 1 still holds for matrices the same it does for numbers, except that when dealing with matrices, "one unit" (1) is an identity matrix. That's why a^-1 = 1/a (division) only "holds" for basic numerical spaces, while a^-1*a = 1 (multiplication, exponentiation) "holds" for all mathematical spaces.

@hitko On that note, a/3 = b is defined for matrices as an "inverse" to the multiplication by a scalar.

@Wisecrack
@atheist
@-red
& @hitko

I am reading this stuff.
Mumble 'oh no'.
Wife goes 'what's wrong? What's happening??' as if something bad occurred.
I tell her.
'There's people who do heavy maths.
They attempt to divide by zero and then say things like axioms and elementary functions.
And there is this thing with two different infinities.'
She calls bullshit.
Proper slates me.
'There's only one infinity.'
I Google the shit.
Britannica.com says
"there's three different types of infinity."
We decide to go to bed and call it a day.
I'm going to sleep well, fully ignoring all the infinite types of infinities.
Thank you lads =D