The Mystery Of The 0th Root 

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an iconic trio: division by 0, 0th root, log base 1

Schrödinger’s root

Zeroth also sounds like the final boss of a video game

We could just have the zeroth root be a multifunction, like with other roots or with the complex logarithm. The zeroth root would have two branches if |z| is not 1, evaluating to either zero or infinity on each branch, and infinitely many branches if |z|=1. Granted, the resulting Riemann surface would be highly pathological, but I think that's to be expected in a case like this.

Yes - as others pointed out, maybe a multifunction could come to the rescue. At 1:20, you ask, how can the 0-th root of 1 be 2 and 3 at the same time. That made me immediately think: well, the square-root of 4 is 2 and -2 at the same time - so having multiple solutions is not such a crazy thing actually. The somewhat weird thing is only that now our set of solutions would become infinite - I guess even uncountably so.

The zeroth root is basically the infinite power, or what you'd get by applying compound interest for an infinite amount of time. Either zero, infinity, or one.

There is another problem AFTER negative numbers: complex numbers: ___---___ I clarify Zeroeth root of Negative/Complex Numbers

3:46 nice! I like this way of showing the "process" of taking limits. :D

The zeroth root actually appears on the "generalized mean". The formula is `(sum(x_i^p)/n) root p` where n is the number of data points, and p is a parameter to control what type of mean to compute. p=-1,1,0.5,2 correspond to the harmonic mean, arithmetic mean, RMS, and SMR respectively. The mean should be undefined when p=0 (zeroth root), but if we used the limit, the parameter surprisingly correspond to the geometric mean, whose traditional formula has no additions nor divisions.

I really like your style of doing math. Reasoning about which approaches you could take. Following them through and reflecting the merit of each approach. And one of those is going to be more suitable in the context of the field. This is totally not what you do in schools, because in schools you don't reason or try dead end paths for enquire. In schools everything has a "right" and "wrong" drawer things need to put in as fast as possible. Neither thinking nor science works this way.

Nice topic the zeroth root - I hadn't thought of this before! Interesting that in the first part you point out that ⁰√1 can take any value, then in the second part you say we had better define ⁰√1 to be 1... For me it's like 0/0 - it can take any value, so is undefined.

your videos are a beacon of clarity and inspiration!

The problem with limits is that it describes how the function behaves at it gets closer and closer to 64^(1/0), but NOT at 64^(1/0) itself. For example, you can have a limit x-> 1 equal to 1 but f(1) can be something completely different, like f(1) = 2, even though it should be 1 if we trust what the limit is telling us. So, even with regular numbers, the argument that “for number a, the limit of this function approaches this value so the function must be this value” is already completely invalid, let alone when using it to explain a number like 0.

the zeroth root is not a function, but a relation. zeroth root of 1 is the set of all real numbers, and the zeroth root of anything else is the empty set

Another way to look at it is to examine the limit for y = exp(x) = lim n->infinity (1+x/n)^n You can get a parameterized version of the log function with n: lim n->infinity y^(1/n) = 1 + (1/n)log y changing parameter a = 1/n y^a = 1 + a log y for a->0 so for power a tends to zero, the result is linear with the log function.

The 0th root has been one of my greatest sources of entertainment in mathematics... and this video brilliantly highlights why. I obviously completely agree with the conclusions reached... and with the conclusion of the linked video. 🙂

Glad to see you back buddy

Very good and unconventional explanation of limits. Great job!

Thanks for the info on the 0th root of a number. I've learned something knew.

awesome intuitive approach

The zeroth root of 1 can be literally any number since any number^0 = 1 (Even 0^0=1 according to many sources). What this really means is that 1^infinity can equal anything, which becomes painfully obvious when you first learn limits.

3:10: "So the zeroth root of 64 should be 64^1/0. And herein lies another problem with the zeroth root: we're dividing by zero." Is it really another problem, or is it exactly the same problem stated more clearly?

The reason why it does matter from which side is the limit is because the power is negative so the limit of n to the 0- of 64 is like saying the limit of n to 0+ of 1/64 and the opposite with the 0.5

With division by 0, approaching 1/x from the left and right results in negative infinity and infinity, and it makes sense to define a number system where those are the same. Makes less sense to try to make 0 and infinity equal I guess.

1:30 Square roots in general have two values, e.g. sqrt(4) is +/-2. So having more than one value isn't immediately a problem.

You know stuff gets crazy when even Bri leaves the expression undefined

I like the idea of it being equal to multiple values. It looks similar to superposition. What is if you try defining it with an "any of" prefix?

Very interesting video and It was as interesting as it was underwhelming: a lot !

How about imaginary-th square root?

Amazing!!!

When it comes to mathematical computations and their limits such as a/0, tan(PI/2 + PI*N), vertical slope, etc... I personally rather list them or determine them to be indeterminate as opposed to "undefined". This for me pertains more to the context of the language than anything else. Take vertical slope for example which is basically the same as a/0 where a != 0, and tan(PI/2). They are basically the same thing. Vertical slope approaches either +/- infinity. For me this isn't undefined. It is however indeterminate because this is a many to one solution. Undefined to me means something that doesn't have a definition. And for things such as division by 0, vertical slope, tan(PI/2), etc... they are actually well defined. It's just that their results don't pass the vertical line test as they have more than one output. Take the general equation of a circle: (x-h)^2 + (y-k)^2 = r^2. Where the point (x,y) lies on its circumference, the point (h,k) is its center and r is its radius. This also doesn't pass the vertical line test because there are two outputs for most of its inputs. Yet, this equation or expression although isn't a function is well defined and isn't considered undefined. In fact, the equation of the circle for all tense and purposes is the same thing as the Pythagorean Theorem: A^2 + B^2 = C^2. It's just that one is relative to circles, where the other is relative to right triangles. Also, the slope of a given line from within the slope-intercept form y = mx+b is defined as (y2-y1)/(x2-x1) = dy/dx = sin(t)/cost(t) = tan(t) where t, theta is the angle that is between the line y = mx+b and the +x-axis. So for me I don't care for this idea that we've been taught that division by 0, and other phenomenon within mathematics is undefined. To me that means it doesn't have a definition, that it's not defined. Now, I can completely agree with indeterminate or ambiguous. As there is a many to one solution, or its output jumps all over the place based on specific ranges within its domain. This is just my take though. What if I were to tell you that that tangent function is actually continuous... Consider the fact that tan(t) = sin(t)/cos(t). We know that both sin and cos are continuous for all of its domain and that its domain is at least All Real Values since they are continuous wave functions that are periodic, oscillatory, rotational and transcendental. And the tan function can be composed of their ratios! Thus, for me, the tangent function also shares those same properties. Yeah, many will argue that the limit of the tangent function isn't the same when taking the left hand from the right hand limits... and to this argument I still wouldn't claim that it is undefined. I would claim that it is either indeterminate or ambiguous. The reason for this, is that tan(PI/2) approaches both + and - infinity and from the left and right hand limits it also approaches 0 besides just +/-infinity. This leads to the conclusion that it isn't undefined as this is no different than vertical slope or division by 0, it's just that we can not determine its output based on that single input as it is a many to one solution. This is what makes it indeterminate or ambiguous.

what happens when you put it on a graph, will it be something like a quadratic where its big with in one set of rules and small in another?

The zeroth root is n^1/0, so it would inherently be undefined. I’m interested to see how you made a video out of it though.

*NICE VIDEO, THANKS, LOVE IT*

For some reason, I can’t add this video to my watch later list. And I can’t add it to my playlist. It was hard to physically watch this video. WTF youtube? I don’t think I’ve ever had this problem. I like this video

My understanding of calculus allows me to agree with your take: That a number (other than 0) "a" raised to the (1/0) power (infinity) (from the left or the right) should remain undefined, it is as if one is trying to define a number which has a 0 factor in the denominator, and while limits allow for this, each equation should be treated with the respect it deserves, every problem is different. I'll note that 0^(1/0) or 0^(infinity) is equal to 0 in all cases when using a limit, such that it can and has been defined properly, but any other number cannot be used in such a manner, it breaks.

Finalmente alguém fez um vídeo sobre esse assunto que eu já passei dezenas de horas fazendo inúmeras abordagens usando álgebra, comparação com exponenciação e até limites no expoente e no índice de das raízes.

that's why i love math. Sometimes things so complicated that they make you laugh.

Nice ! Allright ya earned a sub !

then when x < -1 and we’re talking the limit from the right, it explodes into infinity but half of the time it crosses into the complex world depending on if there’s an even-th root

5:11 I was just starting to think about that... But what about where x < 0?

I always immediately think of roots as the reciprocal exponentiation. So square root = x ^ (1/2) Cube root = x ^ (1/3) Zeroth root = x ^ (1/0) Define 1/0 first , _then_ you can calculate the 0th root.

Bonito ejercicio. Gracias.

IEEE-754 numbers can solve this: 1/+0 = +∞, 1/-0 = -∞ . This is because +0 and -0 aren't "exactly 0", *they represent the limit as we approach 0 from the corresponding side.* This explains most of the weird quirks of FP math

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In my opinion, the function at 5:07 is the best way to define 0th root of a real number. The very last argument you made about the limiting values interchanging when taking x

You could have added a graph showing the limit trends (0 or ∞) on either side of -1 and +1.

Great video! But I dont agree with your final conclusion (of it being too hard to explain). Negative roots are just 1/|the root|, so this makes total sense...

Did anyone else start laughing when seeing the thumbnail? This is the funniest thing I've seen all week. The concept is hilarious impossible.

I like this video. Would have liked to see some consideration of complex numbers and also the gamma function. I doubt it matters with something this ill bahaved though.

0:31 the arrow is in the wrong direction. Example: (-1)²=1 => -1=sqrt(1) which is nonsense.

You are absolutetly Blasphemous! I suscribe!

Hey, just wanted to say I only recently found your channel and I love what you are doing here. I do mathematical modeling in biotech R&D for living, and believe it or not, for most of my life i struggled with math, wspwcially as a kid: the way math was taught to me was very dogmatic, with little room for creativity or debates. It was only in uni when I literally started to build my own math out of necessity that I started to ubderstand what math really is. Now I have a newborn and I am often thinking how would I teach math to him: your approach is what I am going to follow with my kid. You are demistifying math as an a-priori god-given universal truth, and showing instead what math really is all about: a set of logically coherent rules which have consequences and limits. These rules can be whatever you want them to be, as long as you build them in a way that makes sense, but of course the limits, consequences, range of applicability and usefulness of your mathematics will very much vary based on which rules you decide to use. Thanks gor putting your content out there and keep u lp the good work!

The cards for you videos dont appear for me I have a lot of youtube addons so maybe that has something to do with it ?

keeping in mind that this is the definitional inverse function of x^0, it would make sense that 1 maps to every positive and negative value simultaneously, and that 0 itself would be as hard to define in this equation as 0^0. When taking the limit of the function from the right we're effectively re-charting x^infinity, and from the left we're taking x^(-infinity) which is to say (1/x)^(infinity). The behavior of the zeroth root really makes sense for all numbers, with some special behavior at 1 and some undefined value at 0 as per the 0^0 reason.

This COULD make an interesting GRAPH. Plot y = "xth root of k" by plotting y = k ^ (1/x), for some selected values of k. • With k=1, the plot is a horizontal line at y=1. • With k=1.25, the plot has y=1.25 at x=+∞, exploding to y=+∞ as x approaches 0 from the right, AND y=0.8 at x=-∞, collapsing to y=0 as x approaches 0 from the left. • With k=0.8, the plot is a mirror image of the previous graph. The plot has y=0.8 at x=+∞, collapsing to y=0 as x approaches 0 from the right, AND y=1.25 at x=-∞, exploding to y=+∞ as x approaches 0 from the left.

How do you do these animations?

Self evident I’m sure, and bit more work, but I’m confident that a simple graph could better illustrate the the limits and the resulting disconnects. Still a concise explanation.

I dont rlly comment on videos but ur videos r very interesting,thank you.

The zeroth power of any number is one for a very good reason. Exponents are subtracted in division, so if the numerator and denominator have the same exponent meaning the numbers are equal the result is 1. n^x/n^x = n^(x-x) = n^0 = 1. It is not defined to be one, it IS one by calculation.

So hilarious.....but awesomely explained

I would just use a different numbering system entirely where 1/0 is equal to the number v and that 2/0 is thus 2v and 0/0 is 0v which reduces into 0. -1/0 is -v. -2/0 is -2v. infinity/0 is infinity(v) (and thus this is now entering the surreal number territory). i/v is iv, which in a way is just 4. I haven't really played with this nor really just checked if anyone else has done it but I think this is just an easy ass way to deal with the undefined 0s without breaking everything. Some calculators can't handle the square roots of negative values, yet some have imaginary numbers built into it. Some calculators can't handle division by 0, but a calculator that implemented v into it would be able to. In this system, things like the 0th root are no longer mysterious because it's like taking the imaginary root of the number. The general case is simply going to be undefined, or infinitely many solutions possibly I've not thought this out too deeply.. The solution is going to be unknowable, until it suddenly is knowable. anything to the 0th power is best defined as 1. This would mean that, in 0^x, all values of x which are either positive is equal to v, if x is 0 then it is equal to 1, and all negative values of x is equal to 0. It's a piecewise that is actually true here to my knowledge but I'm going to guess my 2am math skills are wrong here because of some random ass shit I didn't think about. v^v is the 0th root of v and is thus not possible as stated above. I would like to imagine this number system as being dimensional. The way these dimensions work are as long strings of every value on their respective number line who then intersect at 0 with all other number lines added to this dimensional approach.These dimensions can then be transformed using functions. This is like bending, pulling, compressing, or cutting the string of the dimension This can be done by application of the new variable w, with w effectively forming a z (real) and series of other axes to define the non-real strings's transformations upon the other strings, wherein the functions applied to dimensions are f(w), then by using f(w)=d^w, with d being the dimension transformed, upon two strings in which one is the imaginary number line and the other is the real number line, then the imaginary number line now forms a cylinder in which the radius is defined by f(w)=d^w and it intersects precisely with the z line where the imaginary number line now becomes real. When this concept is applied to v with the same formula you get a case wherein everything 0 and below is exactly within the real number line except for it being 2 gravitational points where all negative values are at 1 and only x=0 makes the z dimension equal to 0, and then everything else conforms to the 0 axis precisely. It's like a big ass horn. Now you can just keep adding dimensions to other kinds of numbers and you'll get this gigantic mess whenever you change the function for f(w) as it conforms it all into their special intersecting cases. My imagination may have gone wild here but this is literally how I visualize differing number systems when transformed by different functions. I don't know why my brain finds it easier to visualize things in higher dimensions than just the number itself.

Try defining the inverse element of zero. That should do the trick.

And Reload. Those are good too. Not many know about them!

What about if you start using not 0th root but go below zero like negative twoth root. What about let's say, negative second root; -n written on the radical sign? Like -2 root of 5?

Is it actually non defined or do not exist? When we say right hand limit and left hand limit is not equal, limit of x as approaches a does not exist. Then the case we disguise in the videos has one property that i mentioned. The left limit and right limits of the integers between 1 and 0 and bigger than one differs. And also 0th root of 1 is not just equal to one but also equal every real number. So, it seems "DNE" fits more to this analysis.

It’s one of those situations where something is undefined, so redefine it (or use category theory to see why we can/ cant)

f(x) = x^(1/n) is a continuous function for all n. We could look at the behavior of this seqence of functions as n goes to infinity?

I mean, it makes sense to leave it undefined. Anything to the 0th power is 1, so the 0th root of one could be anything, but you can't even take the 0th root of any number other than 1 because the 0th power of any real number could never be anything except for 1. It's kind of like the 0th root of 1 is undefined and the 0th root of any number other than 1 is a domain error.

You can rewrite zeroth root of x as (e^(1/0))^lnx. Rewrite the expression , now you take the limit as n approaches 0 on, and you can see that the positive limit expression is undefined (+inf) at 1, and 0 for all the negative values, and the negative limit expression is undefined at 1 (+inf as well) and 0 for all the positive values. That means that the function has no real value anywhere, because the left side contradicts the right side of the limit, and the one point where both limits agree is undefined.

what if we *do* define it though? As in, what if we do a similar thing to what we do to give -1 a square root, in that we take some 0th root and give it a value thats on its own terms (such as one we could call q) and then try to make this work?

- Daddy, why they say it is impossible to divide by zero? - C'mon son. Sit down. We need to talk about USS Yorktown.

You said the answer. x is the set of all nonnegative real numbers. It's just that you can only take the 0th root of the number 1. But x can be anything. Using undefined /0 to get something else - verboten. BTW what is (-1)^0? BTW what is (i)^0?

The only case it where might work is the 0thRoot(1)={x, x∈R/{0}} because we are basically solving for x^0=1 and that’s defined for all real numbers except 0. And for non 1 numbers we have 0thRoot(x≠1)= ∅ basically no solution for anything else. So I would be implying that 1^(1/0)= x which doesn’t make any sense except if you look at it as x^0=1. So to go from that to the other expression I’ll need to take the 0th power then to evaluate 0/0=1!? Anyway, stuff is weird ignore this comment, it just breaks down.

What is the problem with saying it means to the power of infinity according to the limit

On the complex plane, n^(1/z) has ESSENTIAL SINGULARITY at z = 0. So undefined is the only possible answer.

Given that between 0 and 1 it converges, it may be that the function is a representation of a distribution.

In the third grade when we were exposed to multiplying and dividing for the first time, we learned that multiplication was just repeated addition and division was repeated subtraction. Therefore, division by zero means subtracting zero until you can't subtract anymore. Therefore division by zero equals infinity. That has nothing to do with this video, though, even though it was mentioned as being undefined.

I’m a math fan to!

around 0:40 - is this a generally accepted definition? The one I know defines the root as just the positive solution, otherwise the result is not a number, but a set of possible values (or the root loses its functionality).

you could also treat it as n^0=x, which makes n be anything if x=1 and undefined if x≠1

A few questions: 1:07 You said that any positive real number raised to the exponent of zero equals 1, but negative reals raised to the exponent zero also equal 1, so you don’t need to make that distinction: any nonzero real number will work. In fact, more broadly, any nonzero complex number raised to the exponent zero equals 1. As for zero itself, that’s a bit more debatable. 😄 Those immersed in calculus would say zero to the exponent zero is undefined, but if you ask someone who deals mostly with combinatorics, they’d probably say it equals 1. (The Google calculator, for example, says it equals 1.) 5:16 I’m not sure it’s so cut-and-dried to define the zeroth root of 1 to be 1. The “note” says that 1 to any exponent equals 1, which is true, but any nonzero real raised to the exponent of 0 equals one, which is just as consistent of a pattern, so mathematicians could just as easily define it as ♾️, though technically you could define it as anything you want if you don’t want to call it undefined. (I suppose defining it as 1 gives it a nice symmetry.)

x^(-y) is the same as (1/x)^y; this is why scientific notation for small decimals is generally written as x*10^(-y), such as the pressure of space being notated as 1x10^(-17) torr rather than 0.00000000000000001 torr. So in your examples at 5:25, 64^-(1/x) would equal (1/64)^(1/x) which would be between 0 and 1, while 0.5^-(1/x) would be the same as *1/0.5)^(1/x), which would just be 2^(1/x) and thus be greater than 1...thus the values for 64 coalescing to 0 and the values for 0.5 coalescing to infinity holds true to the earlier definition.

How to Scare Your Math Teacher and/or Math Class: 1. Write 0th root of any number 2. Write “Solve this” under it 3. Let chaos ensue

My brain is too smooth for this

I love this sacrilegious math

What happens with the 0th root of i? (or j for our engineering friends), what about other complex numbers, etc 2, 3, 4... terms? What about 0th root of say 0+3i+4j+5k+6l?

// just note that in geometry, there are objects that square to 1, 0, or -1. // these are square roots of 1. directions in space are such a thing. // these are exactly as weird as i*i=-1. u*u=1. v*v=1. // note that when you square a number, there are multiple possible choices // it is not defined which one it is, without other context x^2=4 implies (x=2 or x=-2 or x=2u or x=2v or ...). which can be converted into: x^2=4 implies x=2 or x^2=4 implies x=-2 or x^2=4 implies x=2u or ... We are not bothered by this, where a multiply by zero loses information about a: 0 a = b if you were to divide by 0, then you are saying that a could be anything. b/0 implies (a=0 or a=1 or a=1/2 or a=-2 or a=i or ...) // note that this is not a problem if a=0. // but by convention, we won't write b=0a as b/a=0 when a=0. 0a = b implies (b=0 and b/a=0) There is something we are not doing right in our notation when we track side-conditions.

Would you say that replacing the zero in the maths system with a square and the square works just like 0, however the square represents space as a zero. Would be a easier way to represent zero being added to maths to represent space.

The cool thing about Undefined, is it is “not defined”. So if you’re looking to make use of applications of the concept of 0-rt(x), then you can define what it should do… in a particular Axiomatic system. Ofcourse, the contradictions matter a lot. There is actually a field of math that focuses on having good axioms (avoiding paradoxes and stuff),, which I haven’t studied. Regardless. It’s kinda fair game. Especially if you’re analyzing 0-rt(x) within the context of some sort of other mathematical model: For example, it could be useful for some specific kind of geometry, who knows.. An example of this is “Fractional dimensions”. The idea is: “Fractional dimensions” don’t exist/ they’re made up (probably: this might be a bad example though, because someone probably analyzes fractional-dimension shapes),,, anyhow… Fractals are easy to explain ~as though they are shapes belonging to a fractional dimensional space. So, it is useful to define things in math,,, when the numbers work out okay.

In 1:07 you said that x to the 0th power is one for any positive real number. However, a negative number to the 0th power equals to 1, too.

I don’t get it, if you put a different input then you get a different output right? Correct me if i’m missing something

Pretty close to the infinite root, simplifying as x^1/infinity, 1/infinity is approaching zero if not already. X^0 is 1 and therefore the infinite root of x is 1. But this works for any equation no matter the number, so they’re all 1. 1^infinity is undefined as well this being one of those reasons. This root also states every number is every other number and is therefore undefined. 0 is the opposite of infinity and negative infinity and as it involves 1/0 which approaches those exact values, and numbers approach two seperate values depending on the way we approach zero.

Can we please redefine the symbol to just mean the nth root and put a 2 next to the square root? It would look so much nicer with other roots

Very tricky... Even myself cant explain that....

another way of finding that is it undefinied is for example. x^1/2 = the sqrt of 2 and x^1/3 = sqrt of 3 so if we were to do the zeroth root we will have to have x^1/0 which is underfined therefore the zeroth root is undefined

a deliciously crazy math problem with more than one case..... Considering heading to the driving range to practice.....

I would have liked to have seen these plotted on a graph. Also what happens if you do this analysis with irrational numbers?

I’m an aspiring mathematician and I’m actually working on division by 0, on the exponential part I found that if x is positive then x^1/0 (basically the 0th root)=1/0 If x=1 then x^1/0=1 And if x is negative then x^1/0=+/-1/0, I can’t detail everything but if y’all really want maybe I’ll make a video on the subject