To be clear, my lead animator is the math nerd behind all this. And as always, watch DJ and I talk about it: ukposts.info/have/v-deo/nIKbZ45ugHx0y40.html

If you could turn this format into a video game, you'd have an incredibly powerful tool to teach kids math.

0:07 introduction to numbers 0:11 equations 0:20 addition 1:24 subtraction 1:34 negative numbers 1:40 e^i*pi = -1, euler's identity 2:16 two negatives cancellation 2:24 multiplication 2:29 the commutative property 2:29 equivalent multiplications 2:35 division 2:37 second division symbol 2:49 division by zero is indeterminate 3:05 Indices/Powers 3:39 One of the laws of indices. Radicals introcuced. 3:43 Irrational Number 3:50 Imaginary numbers 3:59 i^2 = -1 4:01 1^3 = -i = i * -1 = ie^-i*pi 4:02 one of euler's formulas, it equals -1 5:18 Introduction to the complex plane 5:36 Every point with a distance of one from the origin on the complex plane 5:40 radians, a unit of measurement for angles in the complex plane 6:39 circumference / diameter = pi 6:49 sine wave 6:56 cosine wave 7:02 sin^2(θ) + cos^2(θ) = 1 7:19 again, euler's formula 7:35 another one of euler's identities 8:25 it just simplifies to 1 + 1/i 8:32 sin (θ) / cos (θ) = tan (θ) 9:29 infinity. 9:59 limit as x goes to infinity 10:00 reduced to an integral 11:27 the imaginary world 13:04 Gamma(x) = (x-1)! 13:36 zeta, delta and phi 13:46 aleph

10:45 orange you've been here for 11 minutes and you've already made a all destroying death laser out of math how

I love the surprise Euler identity early on when just playing with simple addition and subtraction, because it’s just like when your playing with a simple concept in math and stumble across something bizzare/that you have no clue how to understand yet.

I love how he goes from learning basic operations to university level maths

*THE MATH LORE* 0:07 The simplest way to start -- 1 is given axiomatically as the first *natural number* (though in some Analysis texts, they state first that 0 is a natural number) 0:13 *Equality* -- First relationship between two objects you learn in a math class. 0:19 *Addition* -- First of the four fundamental arithmetic operations. 0:27 Repeated addition of 1s, which is how we define the rest of the naturals in set theory; also a foreshadowing for multiplication. 0:49 Addition with numbers other than 1, which can be defined using what we know with adding 1s. (proof omitted) 1:23 *Subtraction* -- Second of the four arithmetic operations. 1:34 Our first *negative number!* Which can also be expressed as *e^(i*pi),* a result of extending the domain of the *Taylor series* for e^x (\sum x^n/n!) to the *complex numbers.* 1:49 e^(i*pi) multiplying itself by i, which opens a door to the... imaginary realm? Also alludes to the fact that Orange is actually in the real realm. How can TSC get to the quantity again now? 2:12 Repeated subtraction of 1s, similar to what was done with the naturals. 2:16 Negative times a negative gives positive. 2:24 *Multiplication,* and an interpretation of it by repeated addition or any operation. 2:27 Commutative property of multiplication, and the factors of 12. 2:35 *Division,* the final arithmetic operation; also very nice to show that - and / are as related to each other as + and x! 2:37 Division as counting the number of repeated subtractions to zero. 2:49 Division by zero and why it doesn't make sense. Surprised that TSC didn't create a black hole out of that. 3:04 *Exponentiation* as repeated multiplication. 3:15 How higher exponents corresponds to geometric dimension. 3:29 Anything non-zero to the zeroth power is 1. 3:31 Negative exponents! And how it relates to fractions and division. 3:37 Fractional exponents and *square roots!* We're getting closer now... 3:43 Decimal expansion of *irrational numbers* (like sqrt(2)) is irregular. (I avoid saying "infinite" since technically every real number has an infinite decimal expansion...) 3:49 sqrt(-1) gives the *imaginary number i,* which is first defined by the property i^2 = -1. 3:57 Adding and multiplying complex numbers works according to what we know. 4:00 i^3 is -i, which of course gives us i*e^(i*pi)! 4:14 Refer to 3:49 4:16 *Euler's formula* with x = pi! The formula can be shown by rearranging the Taylor series for e^x. 4:20 Small detail: Getting hit by the negative sign changes TSC's direction, another allusion to the complex plane! 4:22 e^(i*pi) to e^0 corresponds to the motion along the unit circle on the complex plane. 4:44 The +1/-1 "saber" hit each other to give out "0" sparks. 4:49 -4 saber hits +1 saber to change to -3, etc. 4:53 2+2 crossbow fires out 4 arrows. 4:55 4 arrow hits the division sign, aligning with pi to give e^(i*pi/4), propelling it pi/4 radians round the unit circle. 5:06 TSC propelling himself by multiplying i, rotating pi radians around the unit circle. 5:18 TSC's discovery of the *complex plane* (finally!) 5:21 The imaginary axis; 5:28 the real axis. 5:33 The unit circle in its barest form. 5:38 2*pi radians in a circle. 5:46 How the *radian* is defined -- the angle in a unit circle spanning an arc of length 1. 5:58 r*theta -- the formula for the length of an arc with angle theta in a circle with radius r. 6:34 For a unit circle, theta / r is simply the angle. 6:38 Halfway around the circle is exactly pi radians. 6:49 How the *sine and cosine functions* relate to the anticlockwise rotation around the unit circle -- sin(x) equals the y-coordinate, cos(x) equals to the x-coordinate. 7:09 Rotation of sin(x) allows for visualization of the displacement between sin(x) and cos(x). 7:18 Refer to 4:16 7:28 Changing the exponent by multiples of pi to propel itself in various directions. 7:34 A new form!? The Taylor series of e^x with x=i*pi. Now it's got infinite ammo!? Also like that the ammo leaves the decimal expansion of each of the terms as its ballistic markings. 7:49 The volume of a cylinder with area pi r^2 and height 8. 7:53 An exercise for the reader (haha) 8:03 Refer to 4:20 8:25 cos(x) and sin(x) in terms of e^(ix) 8:33 -This part I do not understand, unfortunately...- TSC creating a "function" gun f(x) = 9tan(pi*x), so that shooting at e^(i*pi) results in f(e^(i*pi))= f(-1) = 0. (Thanks to @anerdwithaswitch9686 for the explanation -- it was the only interpretation that made sense to me; still cannot explain the arrow though, but this is probably sufficient enough for this haha) 9:03 Refer to 5:06 9:38 The "function" gun, now "evaluating" at infinity, expands the real space (which is a vector space) by increasing one dimension each time, i.e. the span of the real space expands to R^2, R^3, etc. 9:48 log((1-i)/(1+i)) = -i*pi/2, and multiplying by 2i^2 = -2 gives i*pi again. 9:58 Blocking the "infinity" beam by shortening the intervals and taking the limit, not quite the exact definition of the Riemann integral but close enough for this lol 10:17 Translating the circle by 9i, moving it up the imaginary axis 10:36 The "displacement" beam strikes again! Refer to 7:09 11:26 Now you're in the imaginary realm. 12:16 "How do I get out of here?" 12:28 -Don't quite get this one...- Says "exit" with 't' being just a half-hidden pi (thanks @user-or5yo4gz9r for that) 13:03 n! in the denominator expands to the *gamma function,* a common extension of the factorial function to non-integers. 13:05 Substitution of the iterator from n to 2n, changing the expression of the summands. The summand is the formula for the volume of the *n-dimensional hypersphere* with radius 1. (Thanks @brycethurston3569 for the heads-up; you were close in your description!) 13:32 Zeta (most known as part of the *Zeta function* in Analysis) joins in, along with Phi (the *golden ratio)* and Delta (commonly used to represent a small quantity in Analysis) 13:46 Love it -- Aleph (most known as part of *Aleph-null,* representing the smallest infinity) looming in the background. Welp that's it! In my eyes anyway. Anything I missed? The nth Edit: Thanks to the comment section for your support! It definitely helps being a math major to be able to write this out of passion. Do keep the suggestions coming as I refine the descriptions!

Masterpiece

10:02 WAAAAAAAH!

It speaks to Alan and his team’s talent on a number of levels that they can even make me feel sympathy for Euler’s number.

Utterly delightful!

09:58 I love seeing my logo so powerful!

never in my life would I have ever thought I would see something tactically reload a math formula...

The reason why I love this series so much isn't just because of the animation and choreography, but because rules of how the world works are established and are never broken. Regardless of how absurd fight scenes play out there's a careful balance to ensure that not a single rule is broken.

The fact that Alan and his team are the first to make math look insane in animation speaks cm3 volumes....

Wow. This animation is as cool as reality if I had to rate it it will be 10/10👍

A meganalysis of this: 1: a number representing a single entity Equal: a sign representing the sum of a equation Plus: a positive function symbol representing addition 1+1=2 Double equation: this is when you put two plus or minus. 10: a even number Parentheses: 2=(1+1) 3=(1+2) Minus: a negative function symbol representing a take away in math Negative: a negative function symbol representing the negativity or the lowest of a number e i pi: a mathematical equation equaling 0 Multiplication: 3x3=9 (1+1+1+1+1+1+1+1) 3×(4) = (1+1+1..) Division: 6/2=3 12/6=2 81/9=9 Exponent: 6+2²=64 4+4²=16 Fraction: ½=20% Square root: sqrt(4)=2 sqrt(16)=4 i.: -1=n i=1. Radian: also known as rad rad=4.8 maximal rad = 6.2 Pi: symbol representing 3.41 tau: symbol representing t below pi Summation: a symbol representing Σ a calculus sum. Function: symbol representing f f(x) f(sin) Integral: the symbol representing the integral of a number so far good Ratio: r=100 Idk i quit😊

This is actually insane. Having just graduated as a math major and honestly being burnt out by math in general, being able to follow everything going on in this video and seeing how you turn all the visualizations into something epic really made my day. Can’t help but pause every few minutes. GET THIS MAN A WHOLE ASS STUDIO.

This man just gave the sentence “imagine maths are a videogame” a whole new meaning

Mathematics smells genocide in any era. This is why the Ego always stuck periodically.

why is it that e^i*pi is genuinely adorable lmaooo

If math lessons were like this, math would for sure be everyone’s favorite subject Edit: well, this blew up fast. Thanks!

I like how Alan didn’t go for a “Brains vs. Brawn” approach, and instead just made a fight to the death with math terms

I love how people dont even know what is the imaginary number and still watch this Imaginary Number (written as 'i') is equal to the square root of -1 which we considered the second dimension in our Real Number Line, the Real Number line with the Imaginary Number line is known as Cartesian Plane, which is basically a place used to graph equations. (More nerdy stuff at 25 likes)

12th standard nostalgia in the most epic cinematic way possible! Thank you so much for this experience.

Can we just appreciate how TSC went from basic addition to the far end of Calculus in under twenty minutes. That is a hell of a learning curve.

To the math nerd that did the equation and to the animator, heavily respected

1:02 TSC is just like "Yay!!... What did I accomplish?"

7:53 Euler's Identity really released their Domain Expansion

I think the sound design is quite an underrated highlight of this animation. The bleeping and clicking as everything falls into place is so satisfying to listen to.

I can see math teachers showing us this video in the future. It's entirely possible. For Grapic Design, our teacher showed us the very first Animator vs. Animation video. And wanted us to see if we could make something similar. That was basically our biggest semester project.

bro you just show the beauty of mathematics. I really love this video and physics video was also the most amazing animation video I ever watched.

This was brilliant. I think some short stories for individual mathematical concepts would be highly educational. I thought the battle was a bit prolonged without much new discernable math was presented, but I take this video to depict the authors struggle with natural logarithms and the concept of "e". Everything was clear and intuitive to that point.

As a person who has taken calculus, I can confirm we fight bosses every day in math class.

this sound design was top notch. The music felt so appropriate for this weird dimension, and the sfx for all the math clinking and plopping felt like it was exactly how math should sound. absolutely stunning.

This taught me math better and faster than 12 years of school ever did

it brought me to tears. Especially the f(x)=9tg and my favorite Euler identity

Can't wait for all the math channels to do breakdowns of this video. It's incredible how much is packed in here.

This feels like it should win some kind of award. Not even joking this is gonna blow up in the academic sphere. People are gonna show this to their classes from Elementary all the way through college. I don't know if people realize just how powerful of a video you've created. This is incredible. You've literally collected the infinity stones. This is Art at its absolute peak. Bravo.

With great knowledge comes great power and with great power comes great responsibility => with great knowledge comes great responsibility!

Is interstellar of math ! 😂 Wonderful, thanks for your animations !

An animation masterpiece ✅ A cinematic masterpiece ✅ A mathematical masterpiece ✅ A physics masterpiece ✅ Cinematography ✅ Sound design ✅ Everything is so perfect

The sound design here is simply masterful, and makes the whole thing feel physical and *very* satisfying.

As a general nerd myself, and especially a math nerd: This video was awesome 🤓😄

bro made math epic

As a mathematician AND a fan of Alan's works, I can't describe how happy I am.

As a math nerd, this is like my new favorite thing. I love how you started out with the fundamentals of math, the 1=1 to 1+1=2, and then steadily progressed through different areas until you're dealing with complex functions. There's so much I can say about this, it's so creative. Good job, Alan and the team.

10:34 The creation of the death star

After watching this, now i just want to learn everything and everything about math...this is just mind-blowing ❤

The sound design is a masterclass on its own

I think this just proves TSC is smarter than anyone alive. He just absorbed, learned, and utilized in combat 14 years worth of math learning in just 14 minutes.

Insane video... Describes what maths is doing under the hood.... DESTRUCTION!!!

Bro TSC is literally him. In 10 minutes he basically conquers the math universe.

This needs like 10 games, 3 books, a Netflix series, a movie on Disney+ and Dreamworks, and way more

I've never seen anything so mathematically accurate while also entertaining.

2:48 NEVER divide ANY number by zero.

I like to come back to this video every once in a while when ive learned more math and understood more parts of the video. Currently just understood the unit circle, radians, and sigma notation!

Can we all take a minute and appreciate the sound design here? It makes the action and visuals so much more enjoyable than they already are!

I love this. I can only understand completely a third of the math presented here. But the fact that Alan made entire battles, wars, swords, and weapons out of just numbers and radiuses and equations is insane and SO creative. I cannot stop watching.

10:00 was most epic scene

This was incredible. Such an amazing visual of math that makes it understandable. And a better story plot than most Hollywood movies nowadays. Lol

The graphic design in this episode was nothing short of phenomenal. The way e^iπ and TSC interact with numbers is so smooth and natural, and they use complicated formulas so creatively, too... Too bad it didn't fit in the narrative of AvA's grand story because this was one of the most beautifully animated episodes I've ever seen from your team

I didn't understand a good portion of the math, but this is the exact chaotic feeling I get when confronted by math. Only difference is that this animation outs me in awe of math rather than in fear of it. Truly a masterful piece

Now that’s a boss battle if I’ve ever seen

I never was one to like math, now i adore it

This really feels like Alan's old animations, before they became a series. Something exists, it gets explored, big fight, bigger fight, biggest fight, then end. No context, nothing. Except, now the quality of animation is something exceptional.

This should legitimately be shown in schools, so much unique intuition for basic concepts in math is shown here

I believe this left the realm of maths💀

You got complex math and not exit... Man i love this

The start was intriguing, the middle was intense, and the end was heartwarming. This isn't just an animation, it's a masterpiece and will be remembered for generations to come.

I love it how under 2 minutes this already shows more math than I have every known in my life

I am actually speechless. This was way more mind blowing than i expected it to be.

I can’t wait to see math youtubers react to this and explain it all. Here’s hoping the community gets this in front of those creators as soon as possible.

as an nerd myself, here's the actual math: 0:06 1 as the unit 0:13 equations 0:18 addition, positive integers 0:34 base ten, 0 as a place holder 0:44 substitution 1:09 simplifying equations, combining terms 1:20 subtraction 1:30 0 as the additive identity 1:34 -1, preview of e^(iπ) = -1 2:10 negative integers 2:16 changing signs 2:20 multiplication 2:28 factors 2:33 division 2:48 division by 0 error 3:03 powers 3:23 x^1 = x , x^0 = 1, x^(-1) = 1/x 3:35 fractional exponents = roots 3:42 √2 is irrational 3:48 √(-1) = i 3:54 complex numbers 4:00 e^(iπ) returns, i*i*i = i*(-1) = i*e^(iπ) 4:15 Euler's formula: e^(iθ) = cosθ + i*sinθ 4:54 e^(iθ) rotates an angle of θ 5:12 complex plane 5:33 unit circle 5:38 full circle = 2π radians 5:55 circle radii 6:36 π 6:41 trigonometry 7:17 Euler's formula again 7:33 Taylor series of e^(iπ) 7:44 circle + cylinder 7:51 (-θ) * e^(iπ) = (-θ) * (-1) = θ 8:22 Euler's formula + complex trigonometry 8:29 sinθ/cosθ = tanθ, function f(x) = 9*tan(πx) 9:01 π radians = half turn 9:57 limits, integrals to handle infinity 10:15 translation 13:01 factorial --> gamma function, n-dimensional spheres 13:31 zeta, phi, delta, aleph (comment by MarcusScience23)

Gracias por todo este maravilloso trabajo.🤩👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏

Jackson game yesterday was amazing. His hold up ball control and link up play is just amazing

i like how in the beginning, euler’s identity escaped quickly, indicating a lack of fundamental understanding. But as the second coming learns more and more, he becomes more and more capable of keeping up with euler, eventually figuring out how to defeat its dimensional trickery with i, and making peace with it once he understands the prerequisites. very well done animation !!

Here's my interpretation of each scene as a second-year undergrad: 0:00 Addition 1:23 Subtraction 1:40 Euler's identity (first sighting) 2:25 Multiplication 2:36 Division 2:48 Division by zero 3:05 Positive exponents 3:29 Zero and negative exponents 3:40 Fractional exponents and square roots 3:50 Imaginary unit, square root of negative one 4:00 Euler's identity (second sighting) 4:44 a + -a = 0 5:18 The complex plane 5:34 The unit circle 5:38 Definition of a radian 5:59 Polar coordinates 6:39 Definition of pi 6:51 Trigonometry and relationship with the unit circle 7:12 Phase shift 7:19 Euler's identity (third sighting) 7:35 Taylor series expansion for e^x, x=iπ 7:50 Volume of a cylinder (h = 8) 8:25 Hyperbolic expansion for sine and cosine 8:30 f(x) = tan(x) 9:28 Infinite domain 10:00 Calculus boss fight 11:00 Amplitude = 100 11:30 Imaginary realm? 12:10 TSC befriends Euler's identity (wholesome) 12:38 i^4 = 1 13:05 Taylor series expansion for e^x, x=π 13:06 Gamma function, x! = Γ(x+1) 13:25 Reunion with Zeta function, delta, phi and Aleph Null Definitely my favourite Animator vs. Animation video yet, and I'm not just saying that because I'm a math student. It really says something about Alan's creativity when he can make something like mathematics thrilling and action-packed. Top notch!

I adore you, Alan! You gave me a passion for mathematics and physics!

i understand nothing but the animation is brilliant.

As an engineer this has got to be the coolest animation I've ever seen. Its so fun to watch and 100% acurate all the time

only Alan Becker can turn a math formula into a sub-orbital laser cannon

4:09 hey, ya dropped this!

Your animation is so well made and nice, I showed it to my math teacher and he basically used it to make his lesson. Wish there will be more of Animation vs Math !

I love how TSC is using more and more complicated maths as time goes on. This guy somehow made maths interesting

I’m not even joking… get this into an animated short film festival, because this could win an Oscar.

This was math taught as real cinema . I am blown away 🎉🎉🎉

I don’t understand mathematics at all, but it’s so nice and interesting to watch everything that happens! Especially the sound is very cool, the sound of interaction with objects. And although I understand almost nothing from these formulas, I can confidently say that mathematics (geometry or algebra) is an art!)

This is literally 100/10. The sounds, the effects, the animation, the accurate equations and the story, they all were hella awesome. Thanks Alan.

TSC discovered the entire realm of calculus in under 15 minutes, seriously one of the coolest parts was when the Euler monster derived from e caught the shot infinity in a limit, and using the 0-∞ integral, that seriously was like a woah moment Another thing i dont see anyone pointing out is aleph null as a behemoth due to it being the smallest infinity, i loved every bit of this, its my third time rewatching

It's cool and very interesting. I a little learn math with this video. It's amazing

Something like Animation vs. Music would be good. Various notes and ornaments fighting for their lives. TSC would probably go deaf with that fortississimo.

not only did alan somehow make Euler's identity badass, he also made all of its alternate forms even more badass

So far, this is the best action movie in 2023!

Math teacher: ❌ Alan Becker’s video: ✅

10:00 that "waaaa!" was so cute

I came here thinking this video came out 6 years ago but no it was only 6 hours. I’m sure I could say plenty that others have said but it’s so good to see fun and creative animations like this still existing on UKposts after all these years and all the hassles on UKposts. No Ads, No Sponsors, No Patreon no Merch Plugins, just the art of animation in its purest form. Incredible work, keep it up.

I'm studying at the Faculty of Math in university right now and every month i come back to this masterpiece to see what new did i learn. When this animation came out i didnt understand anything besides the begining, now i almost got everything, and everytime it gets more and more interesting to analyse every small detail i notice Thanks for it, it helps he understand that im getting better, smarter, and my efforts arent worthless

If you look closely, the bow is made up of a 2, x, and 2, and the arrow is 4. Genius

7:01 i dont know if its the autism but this part is incredibly mesmerising