|i Factorial| You Won't Believe The Outcome 

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For those who wants the value of i! it's roughly 0.498 - 0.155i

That was absolute value. But what is the argument of `i!` ?

i really like this style of teaching - showing us the tools we need and then applying them in the problem. This way it doesn't feel too overwhelming, but if we wanted to, we could still go study the proofs of those tools

The fact you don't write it's about 0.5215640468649398 at the end is scandalous.

Amazing video, the relationship between the Gamma function and pi is just incredible.

make a follow up video in which you calculate arg(i!), having only the absolute value looks kinda incomplete

An easy way to represent or imagine what i is, is to consider it to be equivalent to either a 90 degree rotation or a rotation by PI/4 radians. Consider the following: For some number x we can rotate it by i^n where i = sqrt(-1) and n is an integer value > 0. For each iteration of n, x will be rotated by 90 degrees in a counterclockwise rotation. Therefore we can see that the following expression holds true: f(x) = x.rotateBy(i^4) == x. We rotated the value of x by 90 degrees or PI/4 radians for each increment of n. Thus 90 + 90 + 90 + 90 = 360 degrees or PI/4 + PI/4 + PI/4 + PI/4 = PI. The value of i isn't as imaginary or unreal as one would tend to think by its original definition. What's happening here is that i and -i are respectively orthogonal or perpendicular to their unit counterparts of 1 and -1. To further illustrate this we can consider the sine and cosine functions and compare their waveforms to see their similarities and their differences. They both have the same shape, they both have the same domain and range, they both have the same periodicity of 2PI. These are their similarities. Where they differ is their corresponding inputs and outputs as they are 90 degrees, PI/4 radians, or i horizontal translations of each other. They are out of phase by 90 degrees from each other. Where does this phenomenon come from? Let's look at their triangular definitions based on the properties of right triangles. We know that a right triangle as sides A, B, and C where A & B are the lengths or magnitudes of their two sides and C is the Hypotenuse or the side that has the longest magnitude or length which is opposite of the right angle between the other two side lengths. From this we are able to define the sine and cosine functions in this term based on the ratio or proportions of a given angle that is not the right angle with respect to one of those sides and the hypotenuse. Sine = opp/hyp and Cosine = adj/hyp. The common factor of the sine and cosine is the hypotenuse, their differences rely on the orthogonality of the two side lengths of A and B. We also know from Pythagorean's Theorem that A^2 + B^2 = C^2 has a direct relationship to that of the Trigonometric Functions. This is why we have a Pythagorean Identity amongst the trig functions. When we extend our range and domain from the Reals or basic Euclidean Geometry into the Complex Plane or to Polar Coordinates we can easily see some wonderful properties emerge. e^i*pi = -1, or e^i*pi +1 = 0 e^i*pi = i^2 i = +/- sqrt(e^i*pi) e^i*x = cos x + i * sin x This is all possible simply because 1+1 = 2. How and why? The simple expression of 1+1=2 is the unit circle with its center (h,k) located at the point (1,0) in the Cartesian plane. This is also why there is a direct relationship between the properties of vectors and the cosine function which we call the Dot Product. The orthogonality or pendicularness of numbers can be seen within the Cross Product between various vectors having equivalent unit basis components. This is also why other mathematical operations or functions are very efficient or optimized such as Quaternions, Octonions, Fast Fourier Transforms, and more. Sure, I didn't get into the properties or concepts of Factorials, but that's what this video is for! I just wanted to show another way at looking at the value of i and what it is, what it represents. Yes we know it is defined as the sqrt(-1) by trying to solve for the roots of various polynomials, but this can be a non intuitive way of trying to understand it. If we look at i as being a 90 degree or PI/4 rotational transformation of some initial value where the result of applying this transformation has the effect of causing the output of that transformation on the original value to become orthogonal or perpendicular to its initial state is a better way of seeing the relationship that the complex numbers have in comparison to the real numbers. A simple example considering we are working in the complex plane. If we take the value 1 and map it into the complex plane it will be the vector going from the origin (0,0) to the location (1,0). When we take the value 1 and apply the rotation of 90 degrees or PI/4 radians to it, then this point will translate to the point (0,i) in the complex plane. This is equivalent to saying that 1*i = i. When we apply a second translation of this point at (0,i) by another i we end up at the point (-1,0). This then shows that i^2 = 180 degrees or PI/2 radians or -1. This is one of the many reasons why I love math! I just hope that this might bring some insight to others as another way at looking at something. Now, once one is able to understand the connections that I've made above, and understand what factorials are, then some of the things that were mentioned within this video might make more sense as to what is going on within various functions such as the Gamma function. Instead of trying to think of i as a linear value try to think of it as a curved value... Happy problem solving!

I really don't know how this guy doesn't have at least 1m sub for his good explanation. HE MADE LOVE MATH, THX!

I never thought of this question, thanks for you for giving me ideas to share to my viewers. Also, it is a real number with 'pi'es

Beautiful! That was awesome!. Excellent presentation !.

Such a beautiful result

3:14 I guess the scale of the horizontal axis got stretched a bit 😅 |1-√3i| is pointing more to 0.5-√3i

So, but what is the argument of i! ???

please go into crazy depth sometime, i'd love that!

Beautiful! That was awesome!

Excellent presentation !

Well that just totally blew me away. Have really looked at this stuff from my college days and now I remember why. 😃

I understand Euler's identity but this one's alluded me. I'll come back to it thanks.

A bit above my level but got the gist of some of it. Thanks.

|Gamma(n+1+ib)|²= (πb) Π k=1 to n (k²+b²)/sinh(πb) put b=1 and n=1 or you can find other values too.

Awesome explanation , the gamma function is great function

animations are really cool!

Very nice!

Hey i love your content as your content has Blackdrop background which use much less data so i can enjoy your content.. btw i am a maths lover thanks for this ❤

(IMO) your vids are just top quality the video is the most entertaining to watch for a idea, Remember: Its just My own Opinion on the suggestion, Advice; "Try getting used to making a opinion on a topic youre interested in works for *me works for anybody".

Quickly into the weeds. More power to you

You only showed the distance i! lies from the origin, that's not enough to know it's exact value

The factorial function is simpler than the Gamma function. It's the same integral but with z instead of z - 1.

This class is nothing short of breathtaking! Your ability to make complex concepts easily understandable is truly remarkable.

Nice I always thought it the solution would just be passed as no solution, but to see that the answer is real and complex is quite intriguing 😮

Awesome video. Though you do often confuse i! with |i!|. Saying that i! is something when really you mean |i!|

It's interesting to observe that (x i)! has a bell shape... The Gaussian distribution is an omnipresent invariant of mathematics!

RU-vid is getting scary I thought about i factorial in my head this morning and then I get recommended this video

And I was gone for double integrals lol 💀💀. When I saw the title I tried my self and found | i! | ^2 = integral {from 0 to infinity} of {cos(ln(s))/(1+s)^2 ds}. The answer of your video killed me 😂

I tried doing this for hours and got nowhere. That reflection formula seems to do all of the heavy lifting for this theorem, so now I just wanna go prove that haha.

This could relate to Bose-Einstein condensates and to the midpoint in time between the solution to the Basel problem and its reciprocal.

Nice video! I just don't understand how you can explain what is the modulus of i after the three first high-level minutes of the video. You should stand in a certain level of complexity ^^

I am too new to complex numbers to know how we got to hyperbolic space, as in your mention of hyperbolic sin. Help.

These are cleanest animations I've seen on any math channel! Thanks for the quality content

whats subfactorial of i? and also can you do subfactorials of negative numbers or fractions?

In which world is absolute value good enough??????

8:05 I don't understand this step, how did you magically make a 1 appear here: *i (IΓ(i)I)^2 = i (IΓ(1+i)I)^2* ?

Reposting and slight editing of recent mathematical ideas into one post: Split-complex numbers relate to the diagonality (like how it's expressed on Anakin's lightsaber) of ring/cylindrical singularities and to why the 6 corner/cusp singularities in dark matter must alternate. The so-called triplex numbers deal with how energy is transferred between particles and bodies and how an increase in energy also increases the apparent mass. Dual numbers relate to Euler's Identity, where the thin mass is cancelling most of the attractive and repulsive forces. The imaginary number is mass in stable particles of any conformation. In Big Bounce physics, dual numbers relate to how the attractive and repulsive forces work together to turn the matter that we normally think of into dark matter. The natural logarithm of the imaginary number is pi divided by 2 radians times i. This means that, at whatever point of stable matter other than at a singularity, the attractive or repulsive force being emitted is perpendicular to the "plane" of mass. In Big Bounce physics, this corresponds to how particles "crystalize" into stacks where a central particle is greatly pressured to break/degenerate by another particle that is in front, another behind, another to the left, another to the right, another on top, and another below. Dark matter is formed quickly afterwards. Mediants are important to understanding the Big Crunch side of a Big Bounce event. Matter has locked up, with particles surrounding and pressuring each other. The matter gets broken up into fractions of what it was and then gets added together to form the dark matter known from our Inflationary Epoch. Sectrices are inversely related, as they deal with all stable conformations of matter being broken up, not added like the implosive "shrapnel" of mediants. Ford circles relate to mediants. Tangential circles, tethered to a line. Sectrices: the families of curves deal with black holes. (The Fibonacci spiral deals with how dark matter is degenerated/broken up and with supernovae. The Golden spiral deals with how the normal matter, that we usually think of, degenerates, forming black holes.) The Archimedean spiral deals with dark matter spinning too fast and breaking into primordial black holes, smaller dark matter, and regular matter. The Dinostratus quadratrix deals with the laminar flow of dark matter being broken up by lingering black holes. Delanges sectrices (family of curves): dark matter has its "bubbles" force a rapid flaking off - the main driving force of the Big Bang. Ceva sectrices (family of curves): spun up dark matter breaks into primordial black holes and smaller, galactic-sized dark matter and other, typically thought of matter. Maclaurin sectrices (family of curves): older, lingering black holes, late to the party, impact and break up dark matter into galaxies. Dark matter, on the stellar scale, are broken up by supernovae. Our solar system was seeded with the heavier elements from a supernova. I'm happily surprised to figure out sectrices. Trisectrices are another thing. More complex and I don't know if I have all the curves available to use in analyzing them. But, I can see Fibonacci and Golden spirals relating to the trisectrices. The Clausen function of order 2: dark matter flakes off, impacting the Big Bang mass directly and shocking the opposite side, somewhat like concussions happen. While a spin on that central mass is exerted, all the spins from all the flaking dark matter largely cancel out. I suspect that primordial black holes are formed by this, as well. Those black holes and older black holes, that came late to the Big Bounce, work together to break up dark matter. Belows method (similar to Sylvester's Link Fan) relates to dark matter flaking off during a Big Bang event. Repetitious bisection relates to dark matter spinning so violently that it breaks, leaving smaller dark matter, primordial black holes, and other matter. Neusis construction relates to how dark matter is broken up near one of its singularities by an older black hole and to how black holes have their singularites sheared off during a Big Crunch. General relativity: 8 shapes, as dictated by the equation? 4 general shapes, but with a variation of membranous or a filament? Dark matter mostly flat, with its 6 alternating corner/cusp edge singularities. Neutrons like if a balloon had two ends, for blowing it up. Protons with aligned singularities, and electrons with just a lone cylindrical singularity? Prime numbers in polar coordinates: note the missing arms and the missing radials. Matter spiraling in, degenerating? Matter radiating out - the laminar flow of dark matter in an Inflationary Epoch? Connection to Big Bounce theory? "Operation -- Annihilate!", from the first season of the original Star Trek: was that all about dark matter and the cosmic microwave background radiation? Anakin Skywalker connection?

I've been wondering about complex primes lately. Something like 5i can be dived by 5 and 1 and i and leave no remainder which suggests that a redefinition could be used for primes on the imaginary axis and the real axis. Can we define a complex prime that doesn't exist on either axis but still retains the basic property of a prime? Probably somebody has thought of this before.

You don't need the gamma function. There's a PI function for that. It just doesn't have the -1 in the top. Just write N! In integral form. Gamma is just PI, but shifted to make identity relation easier.

you said click the video hard but the video isn't showing can you please add it too the description

The video starts with wondering what value i! has but ends up with calculating |i!| instead

But what does it mean? And how do these complex factorials behave? And what are they useful for?

btw it looks fun to write |i!|

Bro assembled the infinity gauntlet of math concepts

3:29 where did you get absolutes here?

Brilliant.

I understood some of those words. Hyperbolic is where Goku trained to fight Cell.

So it's about 12/23. That's almost Christmas! :D

Next the value of !|i| 😉

Besides real arguments, can the gamma function be calculated for any point in a non-numeric way? The integral gets so much harder.

Why is the absolute value |i!| the same as i! ? I don't get it... the question was for i! in the beginning, so I am missing one step in the end?

gosh now i need a 3b1b video explaining why pi is here and where the circle is

yeah that's just beautiful

Alright, so we have an exact form for the magnitude of i!. Is there an exact form for the argument thereof?

the brilliant ad seared my eyeballs when the screen went white lol

I tried this on my own and took the following approach: |i!| = |i(-1 + i)(-2 + i)...| We know that (a e^ix)(b e^iy) = ab e^i(x + y), i.e., the magnitude of the product of complex numbers is just the product of the magnitudes. This gives us the product of k = 0 to infinity of sqrt(k^2 + 1), which is just infinity. Why doesn't this approach work? Maybe it is the infinity? I guess maybe this whole approach doesn't make sense, since you never reach 1 repeatedly subtracting 1 from 1 from i.

I mean; technically, i is an integer, in the sense that it’s a whole number of units (1), it’s not a fraction, nor an irrational number. It’s just a complex integer. Yeah, you can’t have i things; but, by that definition, negative numbers shouldn’t (pardon the pun) count as integers, either. I mean, you can’t exactly have a negative number of things, either. You can’t have -3 apples, or anything.

Hey @brithemathguy, love your vids, but here it seems you forgot to answer the actual question: what is i!. Sure enough, it’s close to 0.498-0.1549i, but that isn’t obvious from your non-complex sinh result for the abs value.

Brilliant

Its true, I won't believe the outcome! Because the Gamma function is not unique among meromorphic functions in extending factorial from the natural numbers.

I'm not sure who your intended audience is. You are trying to show people how you can derive this cool result, but you depend on all these properties that you tell people "just believe me on this because it's too complicated".

Is there an additiin version of the factorial? I've had a use for it, from time to time, but no name for the process.

I legit spent a couple hours in office hours with my Professor talking about repeated exponentiation. We called it the star operator! And x Star x increases WAY FASTER than x! It’s so much fun to harass your professors =)

You're right. I dont believe it. Factorial of anything other than an integer is meaningless.

3:23 Your wrong because if you do absolute value of one its positive one so you are still adding

That was fast... Like mind blazingly fast lol

|amazing|

Always fun to know that you don't even understand the question. I thought it would be something like (0+1i) * (0+2i) = -2 (-2+0i) * (0+3i) = (0-6i) (0-6i) * (0+4i) = 24 (24+0i) * (0+5i) = (0+120i) etc.

I used factorials today at my job in the military industrial complex. Weird.

You were right. I don't believe the outcome.

Bruh you calculated the magnitude of i!. But it's a complex number, what should it's argument be?

You know what isn't brilliant? Going from a pleasing black background video to a white background sponsor segment.

i ! is a complex number , you give the answer only for absolute value .However, I like your presentation .

i!i!i!

I actually did find it surprising that the answer is a real number.

Why abs(i) is 1? Since by definition i^2 need to be -1.

(Quizzical look) i is an integer.

The commercial gave me an opportunity to take a breath but I still hate it.

I am not convinced. How can you make the jump from n! = gamma(n+1) in the integers to complex numbers?

i like your funny words magic man

(IMO) Imaginary Numbers is 1 of the topics to study for a idea, Note: Its just My own Opinion on the suggestion, Advice; "Feel free to exchange eachothers own Opinion even mine* to eachother".

So then i! by itself would be plus/minus the final result?

Me too 🙋🙋

(IMO) this might be a amazing video to look for a idea, Disclaimer: Its just My own Opinion on the suggestion, Advice; "Be very proud of having a Opinion on something that counts for eachother including *me.”

You are factorial. Believe in yourself!

Guessing the answer would be 1 factorials are designed to yield the number of possible combinations. i tans alone

sorry, i lack understanding, i do not see the sense of the matter, empty formalisms - whatever should be the sense of e powered by pi?

I was lost as soon the video started and now at the end i don't know my name...

“Not to get too technical” 😂

BlackpenRedpen did this 4 years ago

Now I understand why e^iπ + 1 =0

Why does wolfram alpha show i! as a complex number?

In decimal, it's about 0.52