|i Factorial| You Won't Believe The Outcome

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#ComplexNumbers #GammaFunction #brithemathguy

Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information. Viewers should always verify the information provided in this video by consulting other reliable sources.

[BriTheMathGuy]

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[livedandletdie]

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

[benjamin]

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

[danielmilyutin]

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

[anic]

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

[skilz]

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!

[arslenedhahri]

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

[jectlikeslearning]

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

[Peter_]

I love these weird types of functions;
they are always interesting to take a look at.

[JotoCraft]

3:14 I guess the scale of the horizontal axis got stretched a bit 😅

|1-√3i| is pointing more to 0.5-√3i

[seikomega]

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

[SankalpApharande]

So many loopholes in the solution. This video is perfect example of using incomplete solution that luckily leads to the right answer.

[sharpfang]

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

[bigodeDireita]

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

[ConradSlater-di]

Beautiful! That was awesome!. Excellent presentation !.

[danielbraniff]

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

[thebushmaster]

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

[michaelmcgee]

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

[rhm]

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

[LokangoFreewar]

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* ?