INVENTING A NUMBER SYSTEM 2 | Follow Up

[@aresharesh8671]

"negative infinity can't be the same as positive infinity" *projective geometry wants to know your location*

[@Yoshimaster96smwc]

*casually divides by zero*

[@livedandletdie]

It isn't though. Not even in projective geometry, the point at which infinity and negative infinity meet up on an arbitrary surface is technically a second 0. From which you could project the same thing and end up at the old 0.

[@ahmedouerfelli4709]

Alexandrov extension wants him as well

[@IONATVS]

“Ω” is the symbol used for the “second zero” beyond the highest degree of both positive and negative infinity. It has the most infinite absolute value of all infinites, but also no polarity like zero.

[@Hwd405]

Regardless of whether or not it being an "extra zero" means it's not a point at infinity, the "abject nonsense" cited in the video - that sequences tending towards infinity and sequences tending towards negative infinity can't possibly tend to the same value - isn't nonsense at all when working in a one-point compactification of the real or complex numbers. Whether or not you call that point infinity is up for interpretation but I don't think it really matters either way since the two sequences still definitely converge to the same point, regardless of whether or not it seems to be "nonsense".

[@sketchesofpayne]

The very fact that you can arrive at equally valid or invalid answers is in itself what makes it "undefined." It's in the name.

[@yglyglya]

No replies for 3 years? Lemme fix ðat

[@Eic17H]

In a way it's the opposite of undefined, it's overdefined

[@louisng114]

Mathematician here. Even if we are to only consider real numbers, 0^0 is still undefined. The problem with the limit approach is that 0^0 is not a limit to begin with. While the limit of x^x as x approaches 0 is indeed 1, the limit of 0^x as x approaches 0 (from the right) is 0. The expression 0^0 is simply not defined to be any value.

[@Pietro-qz5tm]

Explain then Newton's binomial formula for (0+1)^n

[@JustMastermind]

​@@Pietro-qz5tm It was simply defined that 0^0 is 1 for the sake of the formula working. If it would be chosen otherwise you would have the answer undefined as well.

[@t0mstone581]

When you said infix and circumfix, i would have thought of something like "twen-six-ty", "si-twenty-x", etc.

[@veggiet2009]

Which is more purely a linguistic answer than a number system answer. In other words you can't express them with digits, only words. Which could work for your language, but I'd want a number system too... I guess you could do something like 2-6-0 for 26, or create a special suffix character like 206~ that would represent the "ty" in a numeral form

[@walavouchey]

Hmm, you could maybe even just split the character for the number in half. 28 might be something like Ɛ23. (Imagine splitting the 8 in half to create Ɛ and 3)

[@veggiet2009]

@@walavouchey that's a fascinating idea, you'd effectively have 3 symbols for each digit.

[@fuseteam]

@@veggiet2009 that would be bad for a number system how would you express 206? or 260?

[@fuseteam]

@@walavoucheynow that sounds interesting

[@jayasuryangoral-maanyan3901]

numberphile has a good video on how 0^0 being undefined works

[@jayasuryangoral-maanyan3901]

and then I check the description and it's there

[@anotheremptiness4316]

@@jayasuryangoral-maanyan3901 lol well i didn't notice it until you mentioned it so your comment is still useful

[@mskiptr]

Even without complex numbers you have lots of other ways to approach 0^0: Let's look at x^y - x->0 & y->0 When you take (x,y)=(1,1),(½,½),(¼,¼)… you get limit of 1. If you go with (x,y)=(1,0),(½,0),(⅓,0)… the limit will be 1 and if (x,y)=(0,⅓),(0,⅙),(0,⅑)… limit will equal 0 But if your sequence was (x,y)=(½,1),(¼,½),(⅛,⅓)… you will get ½ *(a half!)*

[@jan_Eten]

You just rewrote xˣ, x⁰, 0ˣ, and {2⁻ˣ,¹⁄ₓ}.

[@mskiptr]

@@jan_Eten What I was doing is I took various pairs of sequences (a_n, b_n) that approach 0 and plugged them into a_n^b_n to see what we will get. If 0^0 was a uniquely defined number, we could expect that method to always produce the same result - no matter how 0^0 it is approached. And indeed. These sequences are not random and may have nice formulas. That was on purpose, because special cases are nice for showing counterexamples. Edit: Oh, and you're right that this can also be shown with limits of functions. I just used sequences instead because then I was able to easily write the examples as numbers and not just formulas. I think it's more understandable this way - especially to people not knowing (pre-)calculus and stuff.

[@Jallorn]

I mean, it's not that the calculator says, "I checked and it's one," it's that the calculator has a certain pathway programmed in, and in many cases the pathway for x^0 is simply, "set N=1, print N" and the programmer didnt add the extra lines to say, "if x=0, print undefined, else (the stuff from before)"

[@anotheremptiness4316]

ok but personifying computers is an easy way to explain without hauling out the boring "correct" termonology

[@Jallorn]

@@anotheremptiness4316 There's probably good language that still conveys that the calculator is only outputting what it's programmed to do and not really doing much real math beyond counting.

[@anotheremptiness4316]

@@Jallorn but that'd be unnecessary. this isn't a video about the technical workings of calculators. i really don't get your point.

[@Jallorn]

@@anotheremptiness4316 I don't really see that it is unnecessary, as it would really reinforce that a calculator is not really sufficient evidence in a mathematical definition.

[@anotheremptiness4316]

@@Jallorn ok i guess i can see where your coming from. i just think the personification fit the format of the video better ¯\_ (ツ) _/¯

[@NikolajLepka]

Here's an idea for nummerical grammar: What if the root number stays a noun, like 20, and then the specificity becomes an adjective. So if your language has different grammatical rules for adjectives and nouns, you'd be able to get that sort of distinction, and even free up word-order. So to borrow the endings system from Esperanto where -o denotes noun and -a denotes an adjective, you could have Sesa-vento to be 26 a "sixy twenty", or Venta-sesa-cento to be 126 a "twentyy sixy, hundred"

[@Speireata4]

I really like this idea. As if the smaller numbers were just "coloring" the bigger numbers. Or rather like there is this big bulk of 20-30 and you would just add a descriptive adjective to indicate at what part of that big lump you are looking right now.

[@NikolajLepka]

Talpa 1987 yeah exactly!

[@ThisCanBePronounced]

I like this idea. I think I would have come upon this idea as well since I've made a mental note to explore the parts of speech in numbers as well as their usage. So a similar ideas is 26 is a "sixed twenty," and 126 is a "twenty-sixed hundred." We are sixifying numbers! 101 is a oned hundred.

[@artiomboyko]

Ah yes, sesa vento, sixth wind (I understand the idea though, it's cool)

[@Prof_Granpuff]

Can confirm Wolfram does complex stuff. As a grad student in math, taking complex analysis, I have found Wolfram very useful lol. I felt your pain when people doubted the truth you were spouting

[@EnderLord99]

Not only is it undefined, it's undefined for the same reason that anything else^0=1: x^2=(x^3)/x x^1=(X^2)/x x^0=(x^1)/x=x/x 0^0=0/0

[@Feranogame]

Shi# I never expected your username to tell me that.

[@thegiftedspriter7427]

You could also make a case for 0^0 being equal to any number based on that. n=0/0, 0n=0, so any number n is technically a valid solution to 0^0.

[@aannleax]

x^2 is not (x^3)/x ...

[@Sovairu]

@@aannleax How is it not?

[@andrewlitvinov7266]

Uhm. I remember powers being like this: x^n is 1 multiplied by x n times. So x^2=1*x*x, x^1=1*x and x^0=1. In that case 0^0 is 1 multiplied by 0 zero times, aka 1.

[@ibot02]

7:49 "[…]we're getting into a situation where we say negative infinity is the same as positive infinity, which is just abject nonsense." Well, there is this thing called projective reals…

[@Hwd405]

And the Riemann Sphere / Extended Complex Numbers!

[@livedandletdie]

Second point 0 infinite distance away... in all cardinal directions that if at that point you repeat the projection will return you to 0. It's apparently a massive lack of understanding that is missed... 0 is defined as half distance from -unit to unit. Nothing else. And infinity and - infinity doesn't ever meet. Because it's nonsensical. Sure you can claim that it occurs in hyperboles but in that case there isn't really infinite distance and the euclidean mapping of coordinates fails to apply to the object that is projected onto. After all a hyperbole ia a straight line in curved space.

[@Hwd405]

The Major that doesn't actually contradict what they've claimed though, does it? It might be the case that the "point at infinity" is really a second zero but that doesn't really preclude that we can interpret the point as being a point at infinity. Certainly from an analytic standpoint, the limit of any unbounded sequence whose absolute value strictly increases in a one-point compactification of either the reals (or even the complex numbers) gives the same point - a point that in that analytic context would usually be considered to be an infinite point (particularly given its nonfinite absolute value). It's really a matter of interpretation and we can't a priori discount the possibility of a number system wherein both "positive infinity" and "negative infinity" represent the same point simply because it's "abject nonsense".

[@d.l.7416]

@@livedandletdie Depending on the context, infinity = -infinity works. Topology uses it, because in topology, *it's consistent*. Same with projective geometry. Basically what I'm saying is we can define things however we want as long as they are consistent. Like intuitively non of calculus actually makes logical sense, like how could 0^0=e but also 0^0 = 1 make sense. But we can define limits in a way such that it does.

[@James-ep2bx]

Okay I'll be that guy, there are some valid types of math where you can divide by 0, but they tend to be specialized

[@fuseteam]

wut? seriously?

[@electroflame6188]

@@fuseteam wheel theory

[@fuseteam]

@@electroflame6188 hmmm so when 0x ≠ 0, x-x ≠ 0 and x/x ≠ 0 dang what kind of wonderland is this o.O

[@tech6hutch]

Not really a type of math, but in JavaScript, 0 / 0 is Infinity. That probably comes from the floating-point specification, not JS specifically, tho.

[@fuseteam]

@@tech6hutch that's just how js is build :p

[@markmayonnaise1163]

When you said affix, I immediately thought of Hindustani! In Hindi and Urdu, the numbers from one to ninety-nine have become fusionalized. i.e, there is ONE synthetic word for each number up to one hundred. These fusional forms are subject to patterns, irregularity, all what you'd expect. I'm trying to do something similar in my conlang.

[@ThisCanBePronounced]

yeah, surprised they didn't mention this!

[@SchmulKrieger]

Like two + ten = twenty?

[@HerbertLandei]

I agree that in the general case, including the complex numbers, 0^0 is undefined. However, when dealing just with integers, it makes a lot of sense to define 0^0 as 1: Multiplication over the integers is a monoid, having the neutral element of 1 (which means, if you multiply by 1, you change nothing). For that reason, the value of the empty product is defined as 1, too. And it would break mathematics if it were not the case, e.g. number systems wouldn't work. The most striking example is combinatorics, where all hell breaks lose if you don't have 0! = 1, which is also an application of the empty product. If you get philosophical with combinatorics, then the empty product (0! or 0^0, however you want to write it) denotes "nothing", and if you set this value to 0, you claim that there is no "nothingness" state. But there is exactly one state of "nothingness" (the state where nothing exists), and we need this state, in the same sense that a number system needs a zero or set theory needs an empty set.

[@felixbomm]

I always thought 0! was just reversely defined as 1!/1 by just defining n! = (n-1)!*n recursively and 1! = 1, since most mathematical operators leave 1 as is. Is this also plausibe?

[@titush.3195]

@@felixbomm More precisely it's defined by the Gamma Function, which is derived from the fact that n! = n×(n-1)!, so indirectly yes. I find the argument "define 0^0 as 1 because 0! = 1" is not a good argument. There are limits in the real numbers that lead to 0^0 = 0 or any other number you want (lim means as x approaches 0 from the right): lim x^x = 1 lim (exp(-1/x^2))^x = 0 lim (exp(-1/x^2))^(-x) = ∞ lim (exp(-1/x))^(a×x) = e^(-a)

[@doyouknowkeplertwentytwob4032]

You can’t just choose.

[@zulthyr1852]

The people of Halkiovna named their numbers after important things that are that number so, 4 is "Halverd hand", 5 is "human hand", and 19 is "temple"

[@sehr.geheim]

As a kid, I used to cpunt numbers in german, no problem with the prefix unit-tens system, until I was shown the english language at age 10, and saw the uncomplicated wonders of tens-units. Counting in german became harder and I commenced to just count in english. Until i learned about the way numbers were counted in japanese... Ohh, this beauty of logic(for normal counting at least)

[@doyouknowkeplertwentytwob4032]

ok

[@michaelchurch1324]

The definition of 0^0 depends on your number system. Set-theoretically, 0^0 is defined and equals 1. However, on R or C, it leads to counterintuitive results (as you've shown) and so mathematicians prefer to leave it undefined, it being relatively easy to find f(x) and g(x) such that f(x) and g(x) approach 0, as x approaches some a, but f(x)^g(x) doesn't go to 1 (and may not converge at all). It's not that 0^0 "can't be" defined as 1; it's just not, if you're doing analysis or calculus, useful to do so. If you're working in the integers, with each number N mapped to a set of N things (e.g., {0, 1, ..., N-1}), then X^Y is (by definition) the size of the set of functions from Y to X; for example, there are 3^2 = 9 functions from {a, b} to {x, y, z}. There is exactly one function from {} to {}, the identity, so 0^0 = 1. Likewise, there are no functions from {} to any X != X, so 0 to any positive natural number is zero. In number systems like R or C where continuity is valued, 0^0 isn't valuable and it would require a special case in discussing limits, so it's easier to just say that 0^0 is undefined. To be pedantic, we define exponentiation on C differently from on N, the natural numbers. On N, we define exponentiation set-theoretically, as above (or, equivalently, using recursive functions). When we're working with reals or complex numbers, though, we define a^b = exp(ln(a) * b), where exp and ln are derived using known Taylor series, e.g. exp(x) = 1 + x + x^2/2! + ... . This breaks down when a = 0, because ln(0) doesn't exist (or is negative infinity, if that's your jam). By this definition, 0^x doesn't exist for any x; but it doesn't lead to the counterintuitive limit behavior to define 0^x = 0 when x > 0, so that is usually added on to the definition.

[@livedandletdie]

Null and 0 are not the same... one is nothing and the other is a constant. Null is the lack of anything and isn't a constant. You can't describe null as 1-1 because it isn't null aka the empty set in set theory is not 0.

[@michaelchurch1324]

@@livedandletdie A logician would say that the numbers are an interpretation layer over the "true" formalist mathematics, which is usually taken to be set theory, because it's impossible to go beyond first-order logic without defining set-ness (on which higher-order logics rely). The natural numbers are traditionally constructed as follows: 0 = {} 1 = {0} = {{}} 2 = {0, 1} = {{}, {{}}} 3 = {0, 1, 2} = {{}, {{}}, {{}, {{}}} N+1 = N U {N} = {0, 1, ... , N}. With addition, multiplication, et al, defined using set-theoretic features of ZFC. (You'll need to take undergrad-level logic and possibly some analysis to understand how the real numbers are constructed- and all those theorems of calculus proven.) Is this set-theoretic formulation useful for practical arithmetic? Probably not. Is it the only way to define number? No. But it's what's typically considered conventional.

[@ahmedouerfelli4709]

@@livedandletdie The cardinal of the empty set is zero. And the cardinal power is defined that way i.e when you have a set E of cardinal a and a set F of cardinal b then b to the power a is defined to be the cardinal of the set of functions from the set E to the set F, that is the mathematical definition of the cardinal power function in cardinal arithmetic and you can prove easily that it is well defined i.e whenever you consider a set E' of same cardinal as E and a set F' of the same cardinal as F you find that the set of functions going from E' to F' has the same cardinal as the set of functions from E to F.

[@ahmedouerfelli4709]

You notice also if 0^0 is not considered to be equal to 1, you cannot write the Taylor series of the exponential function like so : exp(x) = Σ x^n/n! you will have a problem for n=0. 1 is a very natural value to assign to 0^0 in many senses unless you are someone who cares about the continuity of arithmetic operations, in that case you better leave it undefined.

[@ahmedouerfelli4709]

Something else I forgot to say: the fact that 0^x=0 for any x>0 is reached by another definition of the real exponentiation. We define first the natural number exponents with non-negative bases with recursion. Then we define the exponents that are inverses of natural numbers as the n-th roots (i.e the inverses of the n-th power functions respectively which are bijective over the positive numbers). Then we define the rational exponents p/q as the p-th power of the q-th root and we prove it is well defined i.e does not depend on the choice of the numerator and denominator of the fraction, only depends on the rational number itself. We then prove that f(r)=a^r is continuous on Q for any a>0. Finally, since Q is dense, we extend the exponentiation functions to the positive reals by continuity for each a>0. And here you have the exponentiation defined for all positive exponents and for positive bases except a little problem; a zeroth power could not be defined for zero. If the integer power is defined at the beginning to be 0^0=1, the function f(r)=0^r would not be continuous at 0, but if you leave it undefined at the beginning as if it were an irrational number, you will get at the end 0^0=0 after extending by continuity. We can add a final touch by defining for all x>0 : x^-y=1/x^y EDIT: As R⨯Q is dense in R⨯R, if we consider instead the continuous extension of the function f(x,y)= x^y defined on R⨯Q, we will get 0^0 could not be defined as it has no limit.

[@alexanderboukal5332]

Actually, it depends whether we are talking about 0^0 in terms of limits or 0^0 in the context of discrete math. 0^0 in the context of limits is undefined, but 0^0 in the context of discrete math is 1. Or to put it another way, lim of f(x) = 0^x as x --> 0 is undefined, while 0^0 = 1 must be true for a_0 to be the value of the polynomial ∑(a_j*x^j), from j = 0 up to n, at x=0.

[@AlphaFX-kv4ud]

That limit you showed approaches 0, not undefined, right? It just becomes undefined when we consider that there are more ways to look at it than just that one

[@tristanholderness4223]

on the Riemann sphere, positive and negative real infinity (and all the complex infinities) are actually the same infinity. It's actually a really cool tool for complex analysis

[@Divinemakyr]

Artifexian, on your wordbuilding series where you make your own planet--the "northern supercontinent" world, are you planning on making some humanoid species, then adding in empires, and cultures, etc? It looks amazing so far.

[@HBMmaster]

now here's a question, what does mean in nullary? is zero times 0^0 zero or is it undefined? (it's undefined)

[@felipevasconcelos6736]

You can only have nullary if it’s bijective, since otherwise you’d have no digits, but in bijective systems you only have one way to represent each number, and both and and literally any string of zeroes equals undefined in bijective nullaru, making it not bijective. Therefore, nullary doesn’t exist.

[@palatasikuntheyoutubecomme2046]

No, the defining property of bijectivity is that there are no digits

[@barakeel]

please tell me all your assumptions down to the fundamental rules of logic and I might be able to answer you.

[@Pietro-qz5tm]

Nullary has no digits. You may want to consider the empty string as a number (valued 0 as all empty sums are). If you force inside a digit X of whatever value n then all your nullary positional number are X, XX, XXX, XXXX, ... all valued n. Force in the digit 0 and you have that 0_0, 00_0, 000_0 and so on, all have value 0*0⁰=0*1=0 Force in the digit 1 and all your numbers will have value 1. But plain nullary has 0 digit so you can not compose numbers (except, maybe, the empty string that should have value 0).

[@IruzeRetlaw]

This was really interesting to listen to and made me want to share something. This is not to argue with your point really, but just bring in the experience I've had while going to university as a maths student. (As a sidenote: I don't claim that what I'm saying is absolute truth, but just that, from what my professors told me, it is common practice amongst mathmaticians.) One of the first things my math professors told us at the beginning of university was: "Dealing with 0^0 is really messy and if you ask different people you will get different answers, and really there's no way to validate one." That is why at my university it was just flat out defined to be a value(in our case 1, but that's besides the point.) In fact, my professors told us that while 1 is generally accepted, so is 0, infinity and just plain old undefined. What's important is that you tell the people who you are communicating with WHICH of them is your definition of what 0^0 is equal to. Said another way: 0^0 is undefined. There are still ways to use it in calculations though(and these are generally used to my knowledge). You just need to add rules to it yourself, as the generally accepted rules don't cover it.(And even those axioms can be argued over, which is a whole other can of worms that I won't even go into here, because its both so complex that I could write about it for hours, and that I am scared to get it completely wrong.) P.S: I really appreciate all the work you put into your videos and they help me a lot while building my world. P.P.S: Your german pronounciation is really good.

[@hetakusoda2977]

I tried 0^0 on my Android and it returned "Undefined, or 1"

[@zidanez21]

That's why android is better

[@livedandletdie]

It's not 1. In the reals it's a number that has all properties of 1 in real space but isn't strictly 1. That holds true for the imaginaries as well as 0i^0i is a number that overlays i but isn't i. Which is why the number is undefined in the first place because that would force i and 1 to be the same in complex math for 0i^0i=1 and i at the same time.

[@gJonii]

@@livedandletdie that seems nonsense. To begin with, 0i^0i = 0(i^0) = 0. If you mean (0i)^(0i) then you get 0^0 = 1

[@anotheremptiness4316]

ok hear me out… we call things undefined when they have more than one possible answer, right? and the phone returned "undefined *or* 1" which is more than one answer, right? so the answer that your phone gave is actually undefined, and therefore technically correct! what a smart lil android :3

[@gJonii]

@@anotheremptiness4316 we don't call things undefined when there are more than 1 answer. We call them undefined when they are, well, undefined. When we expect set of symbols(like 0^0) to correspond with some object(like a number), but we haven't actually defined what this corresponding object is. So in this case, phone offers two separate ways to define that expression. One of which is to leave it undefined. Either one might be appropriate in some situations, although I can't really remember cases where 0^0 = 1 is an appropriate definition.

[@jobda1211]

So xⁿ has few definitions, which are valid, but in edge case of 0^0 they can differ. 0^0 in for example complex analysis is undefined of course. But in combinatorics xⁿ tells „How many n-elemented subsets of x-elemented set I can create” and You can rearrange „elements” of ∅ into ∅ only in one way. So 0^0=1 or 0^0Є∅ it depends on definition You have choosen.

[@livedandletdie]

Well either way 0^0 is undefined. Ø ^ Ø is 1 it's a major difference, null and zero aren't the same thing. If we took sandpile counting null is defined as [323][212][323] for a 3by3 sandpile. 0 is a number that is more than -1 and less than +1. It's also more than -i and less than i. But the same is true for a lot of numbers and 0 is therefore the limit as these numbers on the n axis x y z w ... etc approaches it. No matter the dimensionality 0 is the limit and is the halfway point between -u and u.

[@Lopsidationy]

In combinatorics 0^0 is defined to be 1. We define it that way so that n^m is the number of length-m strings in an n-letter alphabet, and so that theorems like the Binomial Theorem are true. Whether 0^0 is undefined depends on what context you’re working in.

[@drdca8263]

The Major I usually don’t define < as applying to complex numbers. I wouldn’t say 0 < i

[@ahmedouerfelli4709]

@@livedandletdie Oh my God, there is no standard ordering for the imaginary numbers. And nobody said that Ø ^ Ø is 1 . It is the set of functions that are defined from the empty set to itself which is a singleton containing the empty function i.e the identity of the empty set.

[@juanpablograjalescanseco28]

7:48 Acktually, in proyective geometry.... I'm just kidding :P but do check proyective geometry out, it's a system of geometry where positive infinity equals negative infinity in the real number line, making it into a sort of circle. Wild stuff

[@peppidesu]

7:50 "-infinity = infinity is nonsense" time for wheel theory guys!

[@kyle-silver]

4:06 I wonder if you could do something like "two (and six) order ten" where you give the digits up front and then specify the magnitude at the end. Almost like... Scientific notation! 26 = 2.6×10^1 You could express a bigger number like 260 as "two and six order 100", or a decimal as "two and six order tenth"

[@Mr.Nichan]

I can see this happening in a far future language, if a future society of space travelers gets so used to saying things in scientific notation that it becomes their normal way of speaking and maybe undergoes various changes to make it faster.

[@SotraEngine4]

May I borrow that for one of my conlangs? It's a conlang made by the gods and if you're a demigod, you can affect the conworld with that conlang

[@SchmulKrieger]

You actually do this in German six two order ten > 6-2-ten > sechs-und-zwan-zig. That's why the units come before.

[@88michaelandersen]

You say that 0^0 is undefined, but that term is typically used to describe things like 1/0 or log(-1). Usually, 0^0 is called a indeterminate. The difference is slight, but can be thought of like this. An undefined expression can never be given a value, but an indeterminate one sometimes can. Indeterminate expressions can sometimes be treated as a limit, and that limit might exist, and if that is the case, then the indeterminate expression can be given a value. The problem is, the same indeterminate form can arise from different expressions, which can lead to different limits, which can give different values. The etymology for the use of these terms in math is as follows: undefined cannot be given a definition, indeterminate can take many different values, and determining which to use is not possible without context.

[@louisng114]

Indeterminate form are for limits. If an expression is not a limit, such as 0^0, it is not an indeterminate form; it is just undefined.

[@ebentually]

The log(-1) has a clear value. I'll assume that you mean the Logarithm of Base 10: That would be log(-1) = ( i*pi/ln(10) If you meant the Natural Logarithm (Base e): That would simply be ln(-1) = i*pi Of course, those solutions both are non-real but they are still well defined

[@danielrhouck]

In computer science, usually we define 0^0 as 1; I'm pretty sure IEEE 754, which defines the floating point numbers your computer uses when integers aren't good enough, defines 0^0 as -1. That's probably why a lot of calculators do. But it turns out that IEEE 754 floating-point numbers are not the same system as real numbers, so just because 0^0 = -1 for floats doesn't mean it does for reals, or for complex or quaternions or ...

[@qxtr5853]

2:55 Omg a German word not butchered! I congratulate you on your pronunciation ability.

[@cadr003]

I know this isnt what you meant as an infix system of counting, but I can totally imagine a language that has numerical affixes that become infixed due to phonotactical changes.

[@sunriselg]

You don't need complex numbers to see it's undefined. If you plot x^y for real x and y and you approach (0,0) from different directions, you get different answers.

[@f424m0nd]

I said this exact thing in another reply I made to someone. It's even illustrated in the Wikipedia article about "zero to the power of zero".

[@Lumegrin]

1/0 is complex infinity, something which WolframAlpha will tell you If you don't understand complex infinity it's basically another neutral number along with zero that is the slope of a vertical line

[@MrDowntemp0]

How about a mixed radix system that starts at seximal, and after nifty goes to duodecimal. And maybe it starts with or includes another base to make it bijective as well?

[@barakeel]

The problem is not to know what 0^0=1 is but to know what you mean by 0, ^, = and 1. If you can answer that then for well chosen definitions it"s possible to prove that 0^0=1.

[@John_Weiss]

Slight Problem: When discussing 0^0, you're not dealing with the function, x^x. You're working with x^*y*. So you need to take the limit as *both* x and y go to 0 from either direction. When you do this, you get different answers, depending on what order you take the limits, which directions you approach from [positive or negative direction], etc. Ergo, "0^0" is undefined.

[@Exilum]

Pretty much agree with what's said. People just forget that while 1 is the likely outcome, it isn't the only outcome. Depending on what kinds of uses you have and the maths you do, 1 does not always make sense.

[@ColinPaddock]

One way of doing infix: two-six-wenty = 26 One-nine-wenty = 19 Three-seven-wenty = 37 Seven-four-three-wenty-hundred = 743 Two-zero-nine-five-wenty-hundred-thousand = ???? Not sure that adds any clarity or utility to the more compact reading off digits like a phone number. I was going to give examples of nested infixation, but my brain went BSOD.

[@larho9031]

the ???? is 2,095

[@kibo7838]

for utility, you could separate for different applications. Say that we're talking orders of magnitude rather than specific ammount? in english this would be a term like "in the thousands" or something, in this method, you could just use the language equivalent of "thousand" to get the same message across. It also would make sense for something like measuring in engineering, where as important as the actual number, is what precision it is your dealing with. so normally you'd write 2000 +/- .5 (two thousand plus or minus a half, as I was taught in high school physics) to indicate that the measurement of a thousand units is precise to the unit (it doesn't deviate by more than half a unit), whereas you'd say 2000 +/- 100 to say it's precise to the hundreds, so, it's within the 1900 to 2100 range. With a system like proposed, that distinction is done more naturally, so you could say, for the same numbers, two-wenty-hundred-thousand, to mean, "two thousand, precise to the tens" or just two-thousand to say something more like "two thousand, eyeballing it" Of course, some stuff needs to be adjusted, so that two-hundred-thousand is not ambiguous as to wether you mean two "hundred thousands" (2 * 100,000) or two thousand precise to the hundreds (2* 1000 +/- 100)

[@0hate9]

Dead Centrist position #0^^2: zero to the power of zero is 0.5.

[@doublespoonco]

what

[@jbdbibbaerman8071]

And you know why they're called a dead centrist?

[@lizzyb.8009]

this implies that 0.5 is halfway between 1 and "undefined" which i'm not okay with. i submit, therefore, that 0^0 = 0.somewhat-defined which is a bit awkward for most maths, so let's five it a letter name. i propose | since you can't tell if it's a lowercase L or a capital i (and it's actually just a vbar - yes, that already has its own mathematical use and meaning but we're talking about 0^0 here. it's already proven what a horrible little gremlin it is, so using a symbol that will continue to fuck up your equations is only fitting)

[@veggiet2009]

It would be good if our math notation had ways to specify context and scope more generally, like "given the positive real numbers: 0ⁿ"

[@yourcurtainsareugly]

But... we do. Without fancy formatting and characters here you could probably use R+ here and be understood.

[@ahmedouerfelli4709]

Brother, I have some things to tell you. 1- In order to deduce that: [if x^y has no limit at (0,0) then it could have no image there], you must assume that x^y is continuous at (0,0) while it is not necessarily the case. 2- Even if we suppose that x^y is continuous at (0,0) ; the approach that considers the limit when x tends to 0 of x^x is not correct, we must not force equality between the base and the exponent, the correct approach is to use an arbitrary sequence (x[n],y[n]) that tends to (0,0) and see if x[n]^y[n] will always tend to 1 or not. In the case y=(1/ln(x)) (which tends to 0 when x tends to 0 as well) we obtain x^(1/ln(x))=e which tends obviously to e not to 1 and this still applies when we restrict ourselves to the real numbers, we do not need to seek for complex numbers to prove there is no limit. And by the way, when you suppose that (x,y) tends to (0,0) in a particular way that x=y all along i.e the limit of x^x when x tends to 0, you still obtain 1 no matter what sequence you choose, even when you use complex numbers. Another thing is that if you consider that approach to see if you could define convenient value to 0/0 and you just calculate the limit when x tends to 0 of x/x (using complex numbers or just reals doesn't matter) you will find that 0/0 should be equal to 1. Whereas, if you use an arbitrary sequence (x[n],y[n]) that tends to (0,0) you will find that x/y could tend to anything. 3- The square root function is only defined for positive real numbers and zero. It is not defined at -1. i is a square root of -1 in the sense that i² = -1 ; -i is also a square root of -1 in that sense ; and in the same sense 1/2 + i sqrt(3)/2 is a cube root of 1 also. To define the root functions properly and unambiguously, mathematicians chose the positive reals as valid things to be put under a radical and the result is the positive root always. If you work it out a little bit you will find no proper way to extend this to more than the positive real numbers, while still having a neat rule to chose one of the many complex roots as a result of the function. So it is not because (-2)²=4 that radical 4 should equal -2. 4- Believe it or not, theory-wise, Wolfram Alpha is no better than a little calculator in the sense that it cannot teach you mathematics, it is no more than a tool of calculation and graphical representation no matter how sophisticated it is. 5- There is no rule stating that 0 to the power of anything is equal to zero. There is one which states that zero to the power of any positive (non-zero) real number is equal to zero. 6- Not all mathematicians would say that. Those who say 0^0 is undefined are dealing with the power function used in calculus. Actually there are many power functions with different definitions each used in various fields which it happens that they coincide on the majority of numbers. For example, the analytical power is undefined at (0,0) ; the complex power is undefined there as well. In abstract algebra, 0^0 is 1. In ordinal arithmetic it is 1 as well ; in cardinal arithmetic it is also 1. It is like the fact that in measure theory and integration theory infinity times 0 is 0. But in real analysis it is not defined. For example the definition of power in cardinal arithmetic is as so: A^B is defined to be the cardinal of the set of functions going from a set E of cardinal B to a set F of cardinal A. So 0^0 as a cardinal is 1 because the cardinal of the empty set is 0 and there is only one function that is defined from the empty set to itself i.e the empty function. In abstract algebra if an operation has a neuter element then the zeroth power of any element what so ever is defined to be that neuter element. The negative powers are defined to be the inverse of the positive powers whenever an element has an inverse otherwise it is not defined. A third power for example needs the property (x*x)*x=x*(x*x) verified in order to be defined. The first and second powers are always defined no matter what algebraic structure you are dealing with.

[@AnarchoAmericium]

To a Set theorist or Category theorist, 0^0 = 1, since |Y|^|X| is the number of functions from the set X of size |X| to the set Y of size |Y|. To an analyst, 0^0 is undefined because weirdness with limits. So the answer is: depends on the context and how you want to view 0 and exponentiation.

[@ahmedouerfelli4709]

Yes indeed

[@elygolden]

I'd argue that 0^0 = 1 simply because it's defined that way. When defining natural number exponentiation it is much more convenient to say that 0^0 = 1 then leave it undefined, and there are a lot of algebraic identities that depend on 0^0 = 1. That said there are other contexts where it makes less sense to define 0^0, like calculus (and your explanation seems to be much more calculus-y so that makes sense).

[@vytah]

It makes sense to assume zero to the power integer zero is one and zero to the power real zero is completely undefined. Integer-zeroth power is really handy when evaluating polynomials and polynomial-like series and when doing any kind of discrete maths, and usually in those contexts x^0=1 for all x, while real-zeroth powers show up in calculus and analysis, where x^0 requires special handling in every case. Also, the quoted rule "0^anything=0", it's "0^anything positive=0", as 0^(-1) is undefined with no exceptions.

[@alsatusmd1A13]

In complex analysis, 0^0 can still in fact be treated as equal to 1 because when you take the limit of f(z)=z^0 as z>0 from any direction, you also take the limit of f(z)=1 as z>0 from any direction since any finite number not multiplied by itself is 1 and 0 is finite, and since that limit is 1 from every direction by definition, rolling back to the original function yields 1 as the limit of f(z)=z^0 as z>0 from every direction.

[@Double-Negative]

Here's another way to write out the limit and get a different answer. As x->0, ln(x) -> -infinity. As x->0, 1/ln(x)-> 0 so both x and 1/ln(x) are approaching 0. So let's use those lim x->0 (x^(1/ln(x))) = e. so 0^0=e and 0^0=1. It can't be both, so it's undefined. (Side note: In harmonic arithmetic, -inf=inf=1/0, so boo yah.)

[@RenCarl1sle]

I'm currently working on a world for a novel and a philosopher in the world's history tried to introduce a base-11 system as certain primes are considered sacred due to association with certain gods and 11 is the most sacred due to association with the Moon Goddess and her two moons.

[@RenCarl1sle]

@kotonpap There are some hints of the attempt in the language, however decimal remains the dominant base due to the Sun God whose sacred primes are 2 and 5 as the culture observes 5 seasons with each season lasting 2 "months". The 11th month is split into interseasonal weeks.

[@RenCarl1sle]

@kotonpap No worries. I may try and work out how it would have worked but for now I'm focusing on other elements of the world, the languages and the magic system as I continue to write the story.

[@RenCarl1sle]

@kotonpap Thank you

[@Liggliluff]

I wouldn't call 0⁰ to be undefined; to what if seen, it's quite defined. It's 0 and 1, depending on your perspective. Or are there any other equations where it goes towards a different number?

[@James-ep2bx]

On the calculators issue just remember calculator answers should come with a "according to the method used" addendum

[@Sean-of9rs]

0^0 is definitely not defined for the real numbers. That is the correct answer. But for fun, let's nerd about it. 0^0 is occasionally used as 1 for convenience (e.g. in the way Maclaurin series are usually written, they would be undefined at x = 0, but we take 0^0 = 1 so they evaluate to the constant term at 0) or to simplify a definition. There is one case, however, where 0^0 is absolutely unambiguously defined as 1: cardinal and ordinal numbers. If 0 refers to the cardinal zero, then 0^0 is the number of functions from a set with zero elements to a set with zero elements. This is clearly 1. Ordinals work similarly. So the question of what 0^0 is depends on what type of number you're working with.

[@Divinemakyr]

This channel is a good 11/10.

[@tabbytacocat]

my marching band show this year was based off that painting of the girl with the balloon

[@sciverzero8197]

so are we going to get a followup on base-i numbers in language now? I demand this.

[@dlevi67]

i demands this

[@tparadox88]

Infixing units sounded more to me like "twenfourty". And it seems to make a little sense to have a system where your base^N is a prefix that counts how many of the power and a suffix that defines the power, infixed with the base^N-1

[@ElderUnikirin]

Trolls in Discworld seem to have a pure restricted numbering system. In Monstrous Regiment a troll demonstrates their counting skills: "One, two, many, lots".

[@koibubbles3302]

The biggest thing that bugs me is the graph for X^x. I don’t know where you got the graph from because I put it into wolfram alpha and I got a different graph, but the negative x portion is horrifically wrong. For example, from your graph, supposedly x^x has a solution at x = -0.5. Which implies that -0.5^-0.5, or 1/√-0.5, is zero, which is obviously impossible. The real graph converges on zero as x goes to negative infinity. The 1/x explanation is a common one but one that I don’t really like, because it makes it seem as if 1/0 actually has a value, but just that it has too many. This is not true. In reality, 1/0 is undefined due to the definition of division. a/b = c is defined so that b * c = a. So 1/0 is saying that 0 times some number is equal to 1. Since there is no number that this statement is true for, 1/0 is undefined. 0^0 can be considered undefined, but for the purposes of the average person it is typically going to make more sense for it to be considered 1. I personally like this definition because taking a to the power of b multiplies a, b times. 0^0 is multiplying 0, 0 times. So the comment that “when a is zero the answer is always zero” doesn’t even matter because you’re not even multiplying zero in the first place. The reason you get one is because 1 is the multiplicative identity, meaning any number can be regarded as that number times one. So you can think of a^3 being (a*a*a)*1*1*1*1*1*… so when it’s a^0 it turns into ( )*1*1*1*1*1*… or, just 1. I already mentioned that using a graph to determine an answer can be problematic, but the comment about imaginary numbers here is also wrong. I’m not very good with imaginary numbers, but I do know that taking numbers to the power of i creates a multivalued function. That is to say, any number to the power of i has multiple answers. So you could take this complex number approach to literally any function where an exponent is allowed to vary and every number would come out as undefined. This is obviously a terrible way to derive a value as it literally completely erases exponentiation as an operation. Why imaginary functions are allowed to be multivalued but real ones are not, i do not know, and i take issue with the concept myself, but if i had to pick between what we have now and no multivalued functions whatsoever, i would obviously pick the former.

[@merren2306]

11:00 what's great about inputting 0^0 in wolfram alpha is that it shows you why it is indeterminate: it shows different limits at 0, specifically, lim[x->0](x^x) = 1 AND lim[x->0](0^x) = 0

[@y11971alex]

One instance we might count large numbers is discussing money, and the smaller the currency the larger we count. Indonesia’s rupiah is very small, so people there regularly count tens of thousands or even millions. The introduction of national politics also involves larger numbers, such as when we speak of a nation’s statistics, like GDP and population, which are invariably high.

[@mehrheitler]

This is the first time in my life I hear about people who think that 0^0=1 existing. Modern education at its best.

[@Lucaazade]

In the majority of mathematics, it makes sense to interpret 0^0 as 0 not being multiplied any times. This is the same as the product . = 2^0 = 0! = 1. Calling it undefined is, at best, being very pedantic because exactly one area of mathematics doesn't agree to call it 1, and not even mentioning the fact that it is usually given the value of 1 is extremely misleading. Here's some modern education for you: en.m.wikipedia.org/wiki/Zero_to_the_power_of_zero

[@Tobarius]

0^0 _is_ an edge case, but it is not simply an edge case that leads to an undefined answer. The Numberphile video you linked even talks about how it is possible to argue the different perspectives in different circumstances. Like, if you are using complex numbers, by all means, say 0^0 is undefined. Otherwise, 0^0=1 is valid. tl;dr: 0^0 is sometimes 1

[@genybr]

5:00 - can someone write what he said please? I'm not native, so it's uneasy to undertand that word ("balanceturnity"??)

[@minewarz]

"balanced ternary", it's a number system they spoke about in the last video.

[@zerstor3897]

As MineWarz said, "Balanced ternary." It's the number system they mentioned where the only digits you have are -1, 0, and 1.

[@genybr]

Thank you, some native pronounctuation is hard to be recognised by foreigners.

[@thefafala]

To be more precise at 8.50, therefore the right limit of x^x and its left limit is 1, we can define by continuity an OTHER function which is x^x if x =/= 0 and 1 if x = 0. In fact this way of building new function taking in account a function not defined on a connected space and its limits is really a classic stuff in maths (and this procedure can lead to approximation if not well-explained). You have to keep in mind that these two functions are not the same. To be really convinced that 0^0 is not define, you have to understand what "not defined" means. Broadly, "not defined" is when we cant tell what a mathematical sentence means, so "she cant have only one answer". For instance we know 0/0 is not defined, take f(×) = ln (1-x) / x and g(x) = 2*x / x for x > 0. By studying the limit to 0 at rigtht, we have lim f = -1 and lim g = 2. So it seems that "f(0)= 0/0 = -1" and "g(0) = 0/0 = 2". In fact there is a confusion of what is a limit and what is a value well-defined. And 0/0 doesnt have a real answer so its undefined classical mathematics. Same results goes to 0^0. In limits study we also have as well-known undefined sentences : 0*infinity infinity / infinity infinity - infinity

[@MrRyanroberson1]

also you omitted the all-important orange line which denotes the imaginary component, though that also coincidentally approaches 0

[@el.k9776]

3:41 Russian speaker here, this is up to 11 here. Actually, 11 is "odinnadtsat' ", archaic for "odin na desyat' ", or one over ten. Tens are interesting too. 40 is not "chetyrdsat" but "sorok" cuz sacral stuff and 90 is "devyanosto", "nine over a hundred", which drives me nuts о великий и могучий, ты беспощаден

[@yinchenxu5249]

If you apply the same method, you will get 0/0=1, since x/x approaches 1 when x approaches 0.

[@orly4672]

Limits with functions approach to 0/0 are called Removable Singularities and they can be any value that you can set it to be depending on the function. For example the limit of the function (x^2 - 4)/(x + 2) is equal to 2, despite being 0/0.

[@Wesyan1999]

Base factorial is an interesting limited base, the number a_n a_(n-1)... = a_n*n! + a_(n-1)*(n-1)! + ... And 0

[@billionai4871]

if you have a species that's based around robots, and you wish to remove our bias torwards binary (being easier to detect presence/absence than counting different levels), you could use an unary base, since that's how a pure Turing Machine operates on. That could be really interesting

[@unrealization6478]

I sympathise with the situation of releasing any youtube video that mentions 0^0, because several maths youtube channels in my radar have casually mentioned it in passing before and it seems like youtube audiences will never fail to incorrectly call you out on it en masse. Its nice that people want to participate in discussions, but I wish they'd do an elementary fact-check first. Well, so the internet goes.

[@jasoningram3510]

I like explosives ending cause I see it like you. Zero in division and raising is like an acidic bath. I don't want to have one.

[@Nemo_Anom]

I imagine the problem with circumfixing numbers would be when you started to create larger numbers, it would get very confusing. There are so many ways to slice the cat, as long as they are more or less linear, you can group things however you want to. Even something like "two six dozens and six and eight"= 158.

[@ollllj]

infix-prefix,suffix... notations for numbers make more sense in bases, where 1 number has multiple ways to be expressed, as in phi-base

[@sehr.geheim]

When my students argue with me about that, I always ask them why they think that 1 + 1 is 2. „Because it's obvious “ is not an awnser. We defined the „+“ sign as an addition, which is admittedly an obvious function, but powers are much different. It doesn't make immediate sense what 2 ^(1.0148182+π/e) means, but the calculator has an awnser that can be checked. But why would you calculate that? Why would you ask what 0^0 equals? Sometimes you just define 0^0 as 1 because you maybe are working with real numbers, but then you only use that definition because it's the only one beneficial to your problem. Sometimes you interpret x/0 as infinity, but only because you know that you are working with exclusively positive numbers and because limits are benefitial to your equation. I don't care how many apples everybody gets if Jenna divides them upon 0 people, and saying everybody gets infinitely many is obviously far from the truth, because there isn't anybody to be included in „everybody“

[@didles123]

The context of the limit in this case is 0^x, since we were only considering base 0. So in this situation, 0^0 = 0 does make sense.

[@andrasfogarasi5014]

The correct answer is that the value of 0^0 is one of the rare things in mathematics which isn't entirely concrete. If we're only working with natural exponents, it is most common that 0^0 is defined as 1. This makes many formulas a lot simpler, as otherwise they would all need to make exceptions for 0. The binomial theorem for example does not work if one of the sides of the sum you're exponentiating is 0. Unless you define 0^0=1, in which case it gives completely correct results. (1+0)^2 = (1^2 * 0^0) + 2*(1^1 * 0^1) + (1^0 * 0^2) is correct if 0^0=1. But otherwise, the binomial theorem needs an exception to be put into it. In fact, as far as I'm aware, defining 0^0=1 does not create any contradictions within mathematics. So I have no idea why some mathematicians oppose it so much.

[@Alkis05]

That is not so. 0^0 = 1 in the context of things like the binomial expansion, 0^0 is just a notational short cut. It is completely possible to enunciate the the sum in such a way that it doesn't iterate over i,j = 0. Yes, the formula is not so pretty then, but then you don't need to define 0^0 = 1. You can perfectly define a polinomial, for example, as: p(x) = a_0 + sum_{k=1} (a_k*x^k) Of course that is cumbersome, so in the context of these formulas you adopt the notation 0^0 = 1, but not actually contradicting that the expression 0^0 is undefined in general.

[@13ellamy]

when you talked about infixing digits within larger numbers, i imagined a system sort of like, if you wanted to say 26, you would say basically “two, and six, tens” this would probably be a case where there’s a digit for each number 1-10, rather than 0-9, and then when written numerically, to write 26 you would have something like “26X” with X for example being the digit for ten. It could get interesting, requiring an individual symbol for 100 and 1000 and so on. If the symbol for 100 was Y, then you could have 26X for twenty six 36Y for three hundred and six, with no X being specified since there are no tens and then 326XY for three hundred and twenty six, with the spoken wording being something like “three (and two (and six) tens) hundreds”

[@WadelDee]

I once heard that mathematicians knew that 0^anything = 0 but they simply collectively decided to arbitrarily declare 0^0 = 1 because anything^0 = 1, fully knowing that that would mean that 0^anything = 0, except for 1, meaning that there is a bump in the curve 0^x.

[@comteniark9548]

Me at a liquor store and also me using a restrictive system: Give me many alcohol please

[@morpheon_xyz]

To be honest, I've been thinking of developing my own counting system for a while and how it'll work, and instead of writing anything down as numbers, it follows a pattern and then it goes from there, especially for visual purposes, although I may give each numeral a name for when saying it, because it'll be part of a spoken & written conlang I'm trying to create at this time, so still early development lol. My idea uses a hexagonal pattern with a central reference point, then you count around it like a watch face from 1 - 6 represented as dots around the central axis point, then when you reached 6, you keep it stationary and add the 1 to 6, then it's represented as 6&1, then 6&2 etc till you reach reach 6&5, then you keep the 6&5 and add 1, then it becomes 6&5&1, 6&5&2, and once you reached 6&5&4&3&2&1, then you've got a total of "21" in base 10, which fills up the entire hexagonal pattern, but when saying the number, you'll say it as if it's 4&1, you know that it's in fact the same as 6&5&4&1, the 6&5& is just committed for simplicity's sake, so 4&1 = 16 in base 10. It's still a huge work in progress as I'll need to figure ways out to name and represent the levels beyond a full hexagon pattern, plus also naming them like we have, 10, 100, 1000 etc, but just now in the sense of how my numbering system works. It's wonky and weird, and whether I'll keep it is a different story altogether, but once I've played around with it for longer, then only I'll be able to know whether I like it or not 😅 Also, in reference to the video as well, in my home language we say units-tens, and when writing we usually write left to right, which sometimes gets the confusion going as you're writing tens-units, so English definitely has it's advantage in counting and accurate writing methodology there. Definitely prefer the tens-units way of doing things though, and I think that's kinda the only thing English really has going for it on the simplistic and we'll thought out way 😂

[@kiro9291]

wow so many people actually didn't know 0^0 is undefined

[@Mleko11110]

In world of computers, where -0 exist, dividing by 0 is possible.

[@sodiboo]

7:50 well...... that's actually the one that makes the most sense in the complex number sphere where 1/x = infinity and it's at the pole opposite of 0, and inf+0 are the only 2 numbers where the argument does not matter and they equate only by magnitude alone

[@joshuahillerup4290]

0^0 is undefined in complex numbers, and it's 1 in natural numbers. Basically, the answer is more complicated than that Numberphile video.

[@deangeloenriquez1603]

Joshua Hillerup just made the same comment before I saw this one

[@sunriselg]

You don't need complex numbers to see it's undefined. If you plot x^y for real x and y and you approach (0,0) from different directions, you get different answers.

[@joshuahillerup4290]

@@sunriselg I didn't see you need complex numbers. But with natural numbers it's unambiguously 1, at least with a set theoretic definition, as the number of functions from the empty set to the empty set is 1.

[@livedandletdie]

@@joshuahillerup4290 stop using set theory for 0^0 as the empty set and 0 are not the same thing.

[@livedandletdie]

It's never 1. There is only 1 system where 0 the number and constant to the power of zero the number and constant holds true and that is the surreals which also has division by 0 defined. And without dividing by 0 you can't define 0^0. Due to the expanded form of the power rule which is a division.

[@somnvm37]

So, Talking about how we don't use big numbers in real life, convinced me to use 12-based system, without being scared to switch. I'll probably not go above 72 (which is as much as you can go counting by your fingers) And If I need to cound in big numbers, I'd just make a litle simple system for that too.

[@koppadasao]

3:45 Yes. In Danish, and to a lesser degree, Norwegian, you can say, tre, treogtredve, (three, three and thirthy), which can either mean 333 or 3.33, depending on context. You can even double it up saying such as tredve, tre, treogtredve, (thirthy, three, three and thirty) meaning 30,333 Well, perhaps not the same…

[@Mr.Nichan]

No, that's not exactly what he was talking about, because he was imagining both of the 3s to be refering to the same part of the number. In your example each of the 3s is specifying a separate number. Maybe an example this might be: samha = three theph = six num = twenty sul = thirty samnumha = 23 thenump = 26 samjulha = 33 thejulph = 36 It's pretty hard to imagine this evolving naturally, as I think circumfixes usually like idioms, coming from prefixes and suffixes that were originally separate but have now become essentially one combined word due to semantic shift or loss of of other constructions with the same affixes. It might be a little more like if, say, that twenty, thirty, etc. are actually phonologically motivated infixes in the digit word. Otherwise, maybe the two parts separated actually originally were separate morphemes, but ones which now depend on the number. Maybe different numbers have different suffixes because they come from different languages. Maybe numbers have suffixes inflected for intrinsic classes like even-vs-odd, size-class, cultural significance, or perhaps (probabably not in any natural descendant of our human languages) multiple bases(/other number systems) are being encoded in the same word. PS. There is actually a similar thing to your Danish examples in English, when talking about certain things. "Three thirty-three" = ($)333 or $3.33 (funny how we choose to get ambiguous when money is involved) (In other contexts, "three thirty three" usually means 333, and 3.33 is "three point three three") "Ten one ninety-one" = 10,191 AG (date Dune is set, obviously impportant for conversation)

[@koppadasao]

@@Mr.Nichan Noticed it after I wrote the comment, but before I posted it

[@disneian]

I can't believe from a video on conlang we get into the debate on whether 0^0 = 1...

[@t1nytim]

I mean the answer is both/it depends. Rather than it is 1 or it is undefined, which I imagine why the big argument occurs. x^0 creates a empty or nullary product, which equals 1. Then in combinatorics a 0-tuple is also equal to 1. In set theory you also get an empty set, which also means it is equal to 1. Then when it comes many polynomials (and other functions) it is required that 0^0 to equal 1 in order for them to work, such as the binomial theorem. Despite all that, there are many instances where maths breaks if 0^0 is not undefined. Which is where what it equals appears to more contextual, or at least with my understanding of the mathematics.

[@SnipselOtter]

I'm really glad you know your stuff and stick with it!

[@jan_en_ik]

How did you get a real negative side on your x^x curve? A negative base with a fractional power is complex.

[@Painocus]

In Norwegian you can say 22 as: tjueto (tjue-to) - twenty-two toogtyve (to-og-tryve) - two-and-twenty tottito (to-ti-to) - two-ten-two (although mostly some old people use that form) (note that numbers between 30 and 99 follow the last format. 32 is trettito, tre-ti-to, three-ten-two.)

[@MrMichkov]

So if I restrict my numbers to only the real numbers x^x works out to 1? Or do the complex numbers always bend it towards undefined?

[@tenma628]

I put 0 to the power of 0 on my phone, and it said undefined OR 1

[@RecursiveTriforce]

11:14 In the reals there still are other solution: First: 0^0=0^(1-1)=(0^1)/(0^1)=0/0 Second: x = 0/0 | *0 0*x = 0 Congratulations! We only rearrange x=0^0 to 0*x=0 and now you can see, x is whatever you want it to be... Extra: Doing the second step for 1/0, gives: 0*x=1 That shows 1/0 isn't real (or complex). It can be infinite but doesn't have to be depending on context.

[@barakeel]

(I got this from someone else) 0 = 0 ^ (2 - 1) = (0 ^ 2 / 0 ^ 1) = 0 / 0. So 0 is undefined.

[@ThisCanBePronounced]

On the note of restrictive systems (related, but a different matter) is Spanish's ordinal system. We have latin-based numbers to make ordinal numbers (the same stems that make base 12 be called duodecimal), but most of us don't both after 10. It's only used in formal settings - I don't think we have yam counting. :) So we have first through tenth, but then we usually just say "the 11" or "the number 11," kind of like saying "Where's person #15?" rather than "Where's the 15th person?"

[@oinkymomo]

Another reason 0/0 is undefined: 1*0=0, so if you divide both sides by 0, you get 0/0=1. However, you could replace 1 with 2, and 0/0 would be equal to 2. 0^0, which is equivalent to 0/0, is equal to literally every number

[@jbrains]

I'd learned to distinguish "undefined" from "indeterminate" here. For example, 6/0 is undefined (among real numbers) because there is no (real) number x with the property that x * 0 = 6; however, 0/0 is indeterminate because multiple candidate values are consistent with the relevant properties of arithmetic: 3 * 0 = 0, so it could be 3; -762 * 0 = 0, so it could be -762, and 0 * 0 = 0, so it could be 0. It seems that any real number could reasonably be 0/0, but they can't all be at once, so we decide that we can't determine it. Accordingly, 0^0 is indeterminate, because, as n approaches 0, 0^n approaches 0, but n^0 approaches 1. We can define it, but we can't determine it. I wonder whether the mathematics community typically doesn't care about this distinction, and so labels both of these cases "undefined" except in specific contexts where the difference truly matters.

[@brunojl2]

It really does depend on the eye of the beholder. In set theory we regard 0^0 = 1. I think algebraists are happy with that also. The problem are the ANALysts.

[@John_Weiss]

Yeah, but in set theory, you start by defining the operations and identities on a ring, which requires fixed behaviors on all of them. Calculus, IIRC, treats limits and functions as its fundamental building blocks. I can't recall ever hearing about any branch of mathematics where calculus and set theory meet.

[@Pietro-qz5tm]

@@John_Weiss You can build (a model) of (almost) every mathematical theory inside a set theory (ZFC is usually chosen). Even calculus. Functions are subsets of cartesian products (that are sets of ordered pairs, defined as Kuratowski pairs). Continuity can be defined in topological terms or with epsilon-delta formulas, either way specification axiom (schema) allows the definition of continuous functions. Natural numbers are finite Von Neumann ordinals (or can be build directly from infinity axiom, in one of its forms). Some work is needed to define operations. Integers are equivalence classes of pairs of naturals (where (a,b) ~ (a',b') iff a+b'=a'+b, a pair can be interpreted as a difference). Rational numbers are the field of fractions of the integers (again, it is a quotient set: a set of equivalence classes). There are many constructions of real numbers from rational numbers. The most famous is the one with Dedekind cuts but the quotient of the set of rational Cauchy sequences by the equivalence relation of having infinitesimal difference is probably the most natural construction. Now (after working on enough details to fill a semester course) you have all you need for calculus, built in a set theory. You can do calculus and everything is a set.

[@John_Weiss]

@@Pietro-qz5tm Why am I not surprised that some set theoretician worked this out? 😆 BTW: just so you know my background, it's been 23 years since I finished my PhD in physics (1-sentence abstract: “Where ‘Chaos Theory’ and ’El Niño’ intersect.”) and left academia. So any set theory is limited to what I needed for the abstract algebra class I took as a sophomore, even longer ago! I could still follow the majority of your explanation. Though it still seems inefficient, unwieldy, and just plain bugnutz to reformulate calculus in terms of set theory! 😁😛

[@Pietro-qz5tm]

@@John_Weiss the point is not doing calculus in a convoluted way, there is no gain using set. This kind of construction is useful to study consistency and better understand what must be taken as axiom and what can be proven as theorem. For example, what are real numbers? A definition may be: Dedekind complete Archimedean ordered field. Is this enough to prove, for example, the intermediate value theorem? What about decimal expansions and the 0.99... repeated stuff, are those a good model for real numbers? Is there any difference between possible models of real numbers? Spoiler: no they are all isomorphic so we can say THE real numbers instead of "in this model of real numbers". Also there are interesting facts more related to logic than set theory but where set theoretical informations are relevant. For example consider the first order theory of the real numbers: all first order formulas using 0, 1, +, *, -, ≥, =, 'for all', and 'exists', that are true on the real numbers. There are different models of this theory all equivalent but not necessarily isomorphic; in fact Löwenheim-Skolem says that there must be models of each infinite cardinality, also countable ones. This does not contradict the categoricity of the real numbers (the fact that there is only one Dedekind complete Archimedean ordered field) since Dedekind completeness is a second order formula. In fact at first order it is not possible to define the natural numbers or infinite sequences but other stuff, like properties about any given polynomial, can be written. The interesting bit is Tarski theorem that says that such theory is decidable: there is an algorithm that does not go on forever that given any first order formula in the language of real numbers tells whether it is true or false. This is big deal: we can automatically prove the density property or the fact that any given polynomial has a certain number of root and stuff like that.

[@ChBrahm]

Hey can you make a video on how to worldbuild a planet with 2 suns. But the suns go in opossite directions to one another allowing for a sunset and sunrise to happen at the same time. Like that scene in Final Space season 2 episode 10

[@Lantalia]

0^0 is defined as 1 in most contexts that matter to me (Java, Python, C99, IEEE pow)

[@UnderTrack_]

about the whole 0^0 thing, I would stick to it being equal to 1 for most purposes as it is true for real numbers, and outside of some very specific advanced sciences, like the studying of very specific physic phenomena which needs the use of imaginary/complex numbers in which case it is generaly undefined but localy it may be defined (ei there's a lot of approaches but your physic phenomena may constraint it in a way where your different approachs gives you a common limit); so yeah, with a numerical base, I see it as working with reals and I would therefore use =1 since it's perfectly true with reals. For anything using complex numbers I'd go with undefined if unconstrained and look to define it if it's constrained, hopefully via the exponential form which is easier to deal with in limits around 0

[@alexpfw9896]

My preferred solution to 0/0 is that it’s equal to every single number at once. X/Y = XZ/YZ, essentially when you multiply both sides of a division by the same number the result doesn’t change. Therefore X/Y = X*0/Y*0 = 0/0, and if you turn that around you get 0/0 = X/Y, regardless of what numbers X/Y represent, and because all numbers can be represented as a fraction, 0/0 = every number at once

[@John_Weiss]

Well, considering that in calculus, the derivative of a function is effectively 0/0 for any continuous function, and evaluating the derivative of the set of all functions at any value is the set of real numbers … well, there ya go. The same can be said of integration: it's effectively adding up 0 an infinite number of times, i.e. 0*∞, and since the integral of any function (that can be integrated) evaluated at any point can be any real number … again, 0*∞ is effectively any number.