solving x^x=1 but x is not real!
Вставка
- Опубліковано 7 лис 2023
- How to solve x^x=1? We know 1^1=1 and 0^0 has no agreement (some say 1, some say undefined) but you are about to learn that (2.213...+3.113...i)^(2.213...+3.113...i) will also give you 1. Today we will solve a basic tetration equation x^x=1 but we will use the Lambert W function and get the complex solutions! Check out another cool exponential equation e^e^x=1: • how is e^e^x=1 solvable??
🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron: / blackpenredpen
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
----------------------------------------
Thank you all!
how is e^e^x=1 solvable? ua-cam.com/video/ckc9F0VjZ3k/v-deo.htmlsi=m91azalG4twF0nTo
I solved a x^x^x=2
If you want I can give you the answer
Its unsolvable because we can rewrite it as e^(e^x)=e^0 so we can simplify the base and find e^x=0 which doesn't have any solution neither real or complex
@@saliryakouli1260 We can not rewrite as x=e^x
But we can say x=e^ln(x)
@@leonardobarrera2816 no I said that e^x is equal to 0 because we can simplify when it's the same base
@@saliryakouli1260 But the equation is other, it does not contain 0 in it original expretion
ill stick to the 1
What's wrong with 2.213534+3.1139999i? Don't be such a hater against complex numbers.
BODMAS 9. KIDDING 😅
more like the 0
@@Sir_Isaac_Newton_HUH
@@Sir_Isaac_Newton_0⁰=1 ?
I’ve been watching a lot of your Lambert W function videos lately and they’re awesome!
Randomly yesterday I realized that I could use the Lambert W function to solve for the equation of a separatrix of a system of ODEs in my Numerical Analysis class. So cool to see it randomly pop up in my studies and I knew how to solve it because of your videos. Thank you!
You have two different functions that converge to 0. But do they converge to 0 at the same rate? what happens if you use those two functions to work out the value of 0/0?
[Edit] Sorry, I posted this under the wrong video. I blame autoplay! I meant this to be a response to the 0^0=0 video.
That's the problem, doing this method will result in many different answers, one for each fumtion that you choose, that's why it's undefined.
0/0 is not a value but an informal saying for lim f/g where f and g go to zero. Anyways, it can equal anything.
Lim ax / x = a
Lim lnx / x = inf
@@hybmnzz2658yes, you are right, just that many ignorant people cannot get a clue, that limit of the function is not the same as the value of the function, they don't have to coincide, they even don't have to exist simultaneously
You are the reason I love calculus and algebra so much. Thanks for the great videos!
thanks for your hard work sir, really elevated my spirit to study calc 2 even more.
i have an integral question for you, i think this is quite hard, since wolfram alpha can't gave any indefinite form of this integral:
Integral of Sqrt(t^2 + 3Sin[t] + 4) dt
who let him cook this time because this is some good 2 AM maths
Hey man, I really love your channel and your content, I watched your hours long integral and series videos to get myself through calculus 2 and I was wondering if you have or plan to do anything of the same sort for differential equations, just massive videos solving a bunch of different types of ordinary differential equations.
Here's one video: ua-cam.com/video/e-cTygNbEUE/v-deo.html : )
cheers!
ln(x^x) does not necessarily equal to xln(x) when we extend to the complex numbers, more care in handling the left hand side.
Why?
@@TeFurto777 Oh there is one good example in another comment. (assuming we are working under principal branch) on one hand: ln(e^(2pi i))=ln(1)=0, on the other hand 2pi i ln(e)=2 pi i. But clearly 0 does not equal 2 pi i! So this is a valid example showing ln(a^b) does not equal b ln(a). Of course, this isn't really ln(x^x) but it gives you an idea (if I give an example with ln(x^x) there will be a lot of computations, but if you insist, I'll try).
The issue is branch cut and the definition of log. It turns out if we want to generalize log into the complex plane in a nice way (differentiability/analyticity), the most natural definition would leave a ray of undefined points starting at 0. Turns out there are multiple possible way of defining log based on where the ray is pointing at. In fact, near the ray, the function has a gap. Take a look at the right side of the picture.
functions.wolfram.com/ElementaryFunctions/Log/visualizations/5/02/imagetext/0031/text31.gif (the left graph is the real part of the log function, and right graph is the complex part)
This gap causes issues. And (this part i am not so sure), ln(x^x) and xln(x) will differ by a constant multiple of the "gap" size (which turns out to be 2pi i).
@@qwerty_ytrewq4452 That’s why BPRP wrote ln(exp(2nπi)), which equals 0 when n = 0. So he’s not working with the principal branch only.
@@adiaphoros6842 He did do it for the right hand side, log(0)= 2\pi i n for some fixed n. But he used the incorrect identity log(a^b)=blog(a) on the left side. While he is considering some branches, it isn't a rigorous justification, he didn't justify he is considering all branches and all possible answers.
-If Log is defined on the branch (pi/2, 2pi+pi/2], Log(i^{1/4})= (2pi+pi/4) i while 1/2 Log(i)=(pi+pi/4) i. Note that Log(i^(1/2))-1/2Log(i)= pi, and is -*-not in the form of 2pi i n-* .
Edit: the above example was wrong, it turns out to be the case that log(x^x)-xlog(x)=2pi i n.
Without justification, it could be the case that Log(x^x)=xLog(x)+ci where c is some random real constant, and that BPRP is not justifying why he is only considering the case of 2pi i n. and not proving that log(x^x)-xlog(x)=2 pi i n for some integer n.
Moreover, even with that, say log(x^x)=xlog(x)+2pi i m and log(0)=2 pi i n. Then xlog(x)=2pi i(n-m). Since both n and m are dependent on log, one need to justify that all possible n-m over different log is the set of integers.
@@TeFurto777 Log(z^w) = w * Log(z) ONLY IF -pi < Im( w * Log(z) )
I love your videos, pls upload more often!!
The values you've found are unreal!
That was pretty simply explained thanks so much
My last 3 videos have been the Lambert W function and seeing this, I was able to get the solution without even knowing precisely what the Lambert W function is
incredible work
Very tricky - great👍
Can you do some problems on Rolle's theorem and how to decide on the primitive function that should be used?
Btw, e^W(x) is just x/W(x). So if you don’t want to deal with nested powers, you can write it as 2pini / W(2pini)
right cuz Wx.e^Wx = x
genius!
Never heard of this Lambert function, but now I'm curious.
Can you do more differential equations solutions?
So good!
Could somebody make a playlist of all the BPRP videos where he uses the Lambert W function? I want to binge them.
Thanks PROF 👍
Not, gonna lie I wish this video came out before my Methods 34 exam this year as it could of helped with one of the questions.
Worst mistake is taking Ln both sides
xlnx=0
And picking 1 solution as x=0
how is it equal to 0 when it's equal to 2pi*n*i, that's precisely the point of "finding all the solutions"
Could you argue your point ?
Ln(1) = 0 wym
Not quite! Here we use some number theory at the end sort of.
x = e^lambert w function(2 pi i n)
n is any integer!
If n = 0:
x = e^W(2iπ0)
x = e^W(0)
W(0)→ W(x.e^x) = x → 0 = 0.e⁰ → W(0.e⁰) = 0
x = e⁰
x = 1
So n can be any integer
We can also see that x=0 is an extraneous solution by plugging it back into the multiplicative equation. That gives us an ln(0) term, which is undefined.
It's easy to show that for all n exist an m where both are integers
w(m,2nπi)=2nπi
Is it possible to find a relationship between m and n
Have you thought about using triangular notation?
In my opinion::
X^X = 1, then X = 1 or X ~ 0 (nearly to 0) since X^0 = 1.
Example:
For X = 0.00001, then X^X = 0.99988487.
For X = 1e-10, then X^X = 0.99999999769
Smaller X will give X^X close to 1.
Using ghraphic aid, I also get X = -0.999992, with X^X = 0.999998, (but I don't know how to interpret it)
6:19 the normal exponential power rule cannot be used in case of complex number right??
But you can. Perhaps it's mostly circumstantial, but this can be seen when you consider the complex number in euler's form, and rewrite it in polar form. e^theta(i) * e^beta(i) = (cos(theta)+sin(theta)i) * (cos(beta)+sin(beta)i).
This is further simplified to cos(theta)cos(beta) + cos(theta)sin(beta)i + cos(beta)sin(theta)i - sin(theta)sin(beta) = cos(theta+beta) + sin(theta+beta)i.
We can let (theta+beta) = zeta for simplicity and treat it as a parameter. Meaning the above expression may be simplified to e^zeta(i).
Therefore, e^theta(i) * e^beta(i) = e^zeta(i), which means this specific piwer rule is applicable. The same may apply to the other rules I pressume. Please correct me if I am incorrect in this.
Actually I think you can't justify like that. ln(x^x) won't always be xlnx when talking about complex numbers, lneⁿ = n neither. So it should be a different and accurate way to handle with it, granting the right answer as well. Even if just by justifying the use of this rule.
For exemple: let 1 = e^i2pi and ln both sides
it'd be ln1 = ln(e^i2pi)
i2pi = 0
If I plot the function y(x) = Re(x^x), I get an almost continuous graph that seems to intersect the y-axis at y = 1.
This seems to imply in some sense that x^x approaches 1 as x approaches 0.
It does which makes sense.
Eddie Woo's video on 0^0 goes into this in more detail, I recommend it
That shirt would be so fire with the angry fish instead of x
Fun idea try integrating😅
Accidentally proved 0^0 = 1 🤯
I love how easy it is to prove bullshit in math, like it’s so easy to “prove” 1=-1 with more advanced math and yeah there are rules you’re breaking almost all the time but nobody catching that shit.
Not really. Cuz ln0 is undefined. Even if u define it as -infinity. 0*-infinity is also an undefined expression
not really
Actually you can also “prove” it by taylor series but Ofc it is wrong
To be fair, there are many cases where 0⁰ = 1 is acceptable, like discrete math, combinatorics, or Taylor series.
It is also consistent with the empty product definition, and many programming languages calculate 0⁰ as 1.
As a _limit,_ though, it is indeterminate.
Lambert w function is the biggest W
I think 0 should be a solution, because I think 0^0 should be 1. As an example, lets take 2 to several powers. 2^3= 1*(2*2*2). 2^2= 1*(2*2). 2^1=1*(2). 2^0 = 1. You see how we reduce the number of twos every time? Let's do it with zero. 0^3= 1*(0*0*0). 0^2=1*(0*0). 0^1 = 1*(0). 0^0 = 1
Now let's change the base and power at the same time. 3^3 = 1*(3*3*3). 2^2=1*(2*2). 1^1=1*(1). 0^0=1
Now, let's do 0^x. 0^3 = 1*(0*0*0). 0^2 = 1*(0*0). 0^1 = 1*(0). 0^0 = 1 every time!
These Tshirts are awesome
What exactly is the lambert W function?
0^0 = 1*0/0 = undefined (basically 0 itself is undefined)
That exponent subtraction property of exponents doesn't work for 0, since that just results in 0 ÷ 0 every time. The best argument I can come up with for x⁰ = 1 is just that it is an empty product, meaning that it is the product of no elements, which defaults to 1, since that is the multiplicative identity and you would want the product of a 1-element list to evaluate to said element. This extends to 0⁰, which then becomes 1. Going the other direction with 1/0 for the argument of a product, you still haven't multiplied any (1/0) terms to cause the product to evaluate to undefined.
• Most programming languages accept 0⁰ = 1 as well.
• Taylor expansions of functions like cos(x) or eˣ rely on this.
• It is useful in combinatorics.
I want that shirt of Urs with the W f(x)😊,how can I get it?
What is the W function?
Where can I get that poster?
As a 15 yr old highscooler in Britain (we dont get taught calculus until post 16 btw) I can confirm that you my friend are a magician
Second: what branch w lambert we get?
Good question. I think that question would be where the real fun starts. And by "real" I mean complex.
Can you do x=1/Sq root (x)
x = 1/√x
Multiply by √x on both sides
x√x = 1
Square both sides
x².x = 1
x³=1
x = ³√1
x = 1 (real)
x = -√3/2 ± i/2 (complex)
What is the w?
but isn't i to the power of any multiple of 4 1 so shouldn't you add +4n where n is any integer
why are there no fishes on the t-short?
Could you do
x
∫ (ln[t]/[t^x])dt = 0
1+1/x
The answer is nice
Is it (1+i√3)/2 or (1-i√3)/2?
@@anshumanmondal8317It's the golden ratio
@@anshumanmondal8317how did you manage to get complex numbers from a real-valued function integral 🤨
Could you please demonstrate this in a video?
How is it possible that the sum for x=1 to infinity and for x=0 to infinity of e^x are negative numbers?
I put them in the calculator and I really can't understand how these results are right.
They aren’t, it’s BS.
Ramanujan summation is BS, it doesn’t give right results.
Please , geometry session full complete simple to andvanced lebel and olympiad questions
As amc ,india ioqm,rmo,inmo,imo etc.
The Domain of w Lambert is real or complex?
Please, amc for best books
this dude can figure out why the chicken crossed the road
3:38 wait cant you only use the principal root..?
my casio fx-991 cw says the solve for this eqn is- 1e-50
This looks unreal
sqrt(1) as well?
great thumbnail
I’ve been working on a similar problem, but I can’t seem to solve it. It’s x^x^x^x=9
It's 3am why am I watching this?
Can we do?
Xlnx=i^4
You can have i = i^4 but that just for s the subset in 2πni of 8πki with k being integers in that power. You skip 2π, 4π and 6π angles every 8π rotations as a subset of the 0, 2π, 4π, 6π, 8π, etc rotations in powers of four complex numbers rotations of integer multiples.
Why I'm feeling so disturbed with such a deliberate dealing with complex functions, especially logarithms and exponents which don't provide a single value for each value of complex argument. Especially when he puts multiple values in rhs, but simply cancels exp with log on the left side, while there are multiple values there as well. To me all this videos are to be considered as some type of mental exercise, rather than rigorous solution having anything common to mathematics as a science and just an educational subject, aimed to learn people think critically and outside of the box. But when you omit such crucial details in your solutions that questions even this goal, for you can do something like that in your real life, and it might have much more dramatic effect in your life, than just a bad mark on math 😮
a^0 is also one where a belongs to Z
Or Q
Ofc (a/b)^0 is 1 when b≠0
@@lukapaun8497and also a≠0
What W function is? I totally forgot that.
4:10 where did the red "x" go??
it became e^ln(x) which is the same as x
Good morning
I don't understand why this man would keep shoes in camera frame😅
Whenever I see one of these insane problems where the only real solution is 1 or 0, I know W can’t be far behind…
How do I use W() on my Casio?
You don't
@@kodirovsshik Why is it not there?
Because it is a special function. And not just any but a pretty specific one and is rarely used so it's not put on calculators, similar to other special functions. It is available in different computer algebra systems, in Wolfram alpha it is known as "W", "LambertW" and "ProductLog"
Alternatively you can compute it using a few iterations of newton's method by using the fact that W is an inverse of x*exp(x)
@@kodirovsshik Oh, thanks for your exhaustive answer! If I may further ask: There is nothing special about W() that would make it impossible or illogical to put it on a calculator, right? It's just uncommon so they did not implement it.
Yes sir, exactly. That is to say, if some of the special functions were implemented on a calculator, I would guess that one would be much more likely to see something like (poly)gamma, zeta, elliptic integrals, Si, Ei, li, hypergeometric function, and maybe only then the Lambert W. I might be wrong though, I'm just guessing from what I've seen. I myself sometimes lack W in tools I use.
aap kha se ho. m Bharat se hu. ap acha padata hai
Bro wearing the Lambert but still I am dumb enough as a lamb to notice it until he mentioned it.
I don't know why I'm watching this even tho I don't understand a single thing😅.
Can you justify that Natural log of complex numbers. Because 'mindyourdecision' show some problem with that method in his video dedicated to solving i^i.
It's treated as a different function than natural log, but there is a complex log function, that takes the natural log of the magnitude, and then adds on the angle (plus any integer multiple of 2*pi) times the imaginary unit.
Such that:
log(z) = ln|z| + i*(angle(z) + 2*pi*k)
where k is any integer
The way we can derive it, is as follows:
Let z = r*e^(i*t), and let L = a + b*i, where a, b, r, and t are all integers
Define L such that:
e^L = z
Carry out e^L, based on its polar form breakdown:
e^(a + b*i) = e^a * [cos(b) + i*sin(b)]
Since the magnitude of [cos(b) + i*sin(b)] will always equal 1, no matter what be equals, this means that the magnitude of z is equal to e^a.
|z| = e^a
Which means:
a = ln(|z|)
cos(b) = real(z)
sin(b) = imag(z)
This means that be equals any angle that is a coterminal angle to z, since cos(angle(z)) also equals real(z), and likewise for sin(angle(z)) equaling imag(z). Thus:
b = angle(z) + 2*pi*k
where k is any integer.
Since e^L = z, this means if we solve for a and b, the components of L, the log of z, we show that L = ln(|z|) + i*(angle(z) + 2*pi*k).
What about x^x^x=1. ?
x = 1 🤩
Btw Lambert W doesnt work on this kind of equation (x^x^x = y)
Yes i can use it. Solution equation x^x^x =a is recurency x(n+1)=exp(W(W(x(n)*ln(a))) x(0)=1. For a =2pi*i we have 2.34604680777561+0,67808152886982*i
fish function?
he doesnt solve he simplifies
IMO, if 0! is defined as 1, it seems logical that 0^0 could also be defined as 1.
After all, we KNOW that 0 times any number is zero, but 0! isn't 0???
Tbh, I still don't understand why there is "no agreement". When limits are never actually finding the correct answer. They only find the approaching number on a function line and literally anything could be it depending on the function. While a series LITERALLY adds to the number and CAN'T be dependent on a function, because there is only one function there and you just plug a number in and get the correct answer. e^x series clearly proves 0^0 does equal 1. It makes zero logical sense to try and say take the "limit" when trying to get the CORRECT answer to the EXACT number. Is there another series that I don't know where it can give you 0^0 exactly and be a different answer than 1? We also have the definition of anything to power of 0 is equal to 1, why does that stop at 0? Cause you can think its somehow 0/0 or something?
Can you give a more detailed response to why e^x series proves 0^0 equals 1? Sort of confused because series is also defined as a limit. But I would also say the limit does find actual answer...The formal definition of limit: Epsilon-Delta definition proves the uniqueness and existence of a single limit for every converging sequence, so taking limit does give a unique answer (or I may be confused on what you are talking about). I believe there is no rigorous justification of what 0^0 is, it is more of a debate on which definition is nicer. There are many places where 0^0 is *defined* to be 1 since it is more convenient in many cases.
x^0 for any number besides 0 is 1. 0^x for any number besides 0 is 0. The limit isn't consistent depending on how you approach 0,0 for x^y. That alone is enough for me to think 0^0 is ambiguous.
@@qwerty_ytrewq4452 This my logic, a more mathimatical explaination I would watch "The Most Controversial Number In Math" by BriTheMathGuy.
It may be defined with a limit to make it infinite terms, but it its not actually "approaching" anything, it actually adds to or we say converge to, not approaching. It actually adds terms to the other terms continuing and WILL get to such number in the end. Like geometric series are infact just limited amount of area of a shape and WILL have a limited area or one correct area no matter. If a series does infact converge it would be no difference to an limited area of a shape just more abstract compared to a geometric series, which is called geometric for a reason, actual shapes.
If you can rigorously prove "e^x" 's series is correct, then e^0=0^0/0!. And we know for a fact e^0=1. So if e^0 is equal to one, then 0^0 has to equal to one.
@@chitlitlah But it only approaches a number on a function line and is never the exact number itself. Limits can't prove what the exact number is. It can happen to be the same as the exact number, but it doesn't actually prove to be it. Limits SEEM to get that number, not actually get to that number.
My two cents here: I believe saying x=0 is not a solution because x^x has no agreement is absolutely stupid in the context of this particular function since it approaches 1 at x=0 no matter which side in complex plane you approach it from
Source: Wolfram Alpha
Where is the Wilbet function fish 😳😳😳😳
1
Bro is stuck in a time loop making the same videos ten trillion times
(-1)
Uno a la uno es uno..
I don't understand your first solution. From what I understand, with any complex number multiplied by any complex number you multiply the magnitudes and add the angles. If you square any complex number, then the square of the magnitude has to equal 1 in this case. How is the magnitude of 2.213534 + 3.1139999i squared equal to 1? Wait, we're not squaring. It's to the power of itself. Still, I can't visualize it. Are there any videos showing complex numbers powered by complex numbers?
What is this W( ) function? Spell it for me so I can look it up.
ua-cam.com/video/Qb7JITsbyKs/v-deo.html
damn you love that function, don't you haha
Edel
It's 1 🤷
The answer is 1 my guy
I wonder what shows up if one plots all solutions in complex plane. Too lazy right now.
The points lie on 2 curves, one for negative n and one for positive n. The curves are almost straight lines, and they are complex complements from each other.
I'm in highschool pre-calculus. This is all gibberish.
Find all solutions to x^x =x including real and complex
Does the Lambert W function come from hell? :)
Would someone please write out the list of values of e^W(2nπi) for n=1,-1,2,-2,... in so far W(x) is defined for x=2nπi, and to the extend it can be written out in a UA-cam comment, preferably in the form of the complex notation, a+bi? 🙏
Not only that, the product log also has multiple branches...
@@nanamacapagal8342
Then I would like to see the values from as many branches as can be made room for in a UA-cam comment when each branch exhibits the same number of values as there are branches.
So if there are k subbranches with their values written out, then I would like to have each of them contain k values (corresponding to k values of n).
I think it would be pretty to look at. And I'm also curious as to how many of the values would contain an imaginary part with 4 successive 9's after the 3. decimal place 🙌
@@nanamacapagal8342
Thanks in advance 🙏💯
Using the other branches of the W function results in values for x where x^x is not 1, so I'm going to ignore these branches entirely. All values were generated by typing "exp(LambertW(0, n * 2 * pi * i))" into WolframAlpha. Replace n with an integer first. The first value of the W function is the branch, where "0" refers to the principled one. All numbers in the following table are cut off to 10 decimal places. Also, one quickly realizes that values generated from negative and positive n are just complex complements of each other.
n ↦ exp(LambertW(0, n * 2 * pi * i))
0 ↦ 1
1 ↦ 2.2135834259 + 3.1139999484 i
-1 ↦ 2.2135834259 - 3.1139999484 i
2 ↦ 3.0341971853 + 5.2243275754 i
-2 ↦ 3.0341971853 - 5.2243275754 i
3 ↦ 3.7176120975 + 7.1085493517 i
-3 ↦ 3.7176120975 - 7.1085493517 i
4 ↦ 4.3265783964 + 8.8679145247 i
-4 ↦ 4.3265783964 - 8.8679145247 i
5 ↦ 4.8865678373 + 10.5434364291 i
-5 ↦ 4.8865678373 - 10.5434364291 i
6 ↦ 5.4109995743 + 12.1569423578 i
-6 ↦ 5.4109995743 - 12.1569423578 i
7 ↦ 5.9080018511 + 13.7218136986 i
-7 ↦ 5.9080018511 - 13.7218136986 i
8 ↦ 6.3829537059 + 15.2470120150 i
-8 ↦ 6.3829537059 - 15.2470120150 i
9 ↦ 6.8396407853 + 16.7389157853 i
-9 ↦ 6.8396407853 - 16.7389157853 i
10 ↦ 7.2808511214 + 18.2022723323 i
-10 ↦ 7.2808511214 - 18.2022723323 i
I'd like to see the plot of x for many values of n to see if there's a geometric pattern to it. Also, I wonder if x^x has nontrivial solutions in quaternions, or if the properties of quaternions make the equation unsolvable.
I believe saying x=0 is not a solution because x^x has no agreement is absolutely stupid in the context of this particular function since it approaches 1 at x=0 no matter which side in complex plane you approach it from
You made a horrendous mistake, at 1:38, two factors' product equals zero IF AND ONLY IF one of them is equal to zero, while the other is DEFINED! You can't literally multiply zero by smth nonexistent and get smth existent (zero, in this case) . Where I've studied math you would either get zero ( point) for the solution, or get some points subtracted from your score ( depending of the severity of the examiner) . The same is with 0 to the 0th power, it is undefined, and that's the explanation why, there's not "no agreement" but the result of this operation CAN'T be defined, because of fundamental rule not to devide by zero , you literally can't pick ANY number real or complex that soild be a reasonable value of 0^0.
1¹=1 x=1
me : x = i^4