To be clear, my lead animator is the math nerd behind all this. And as always, watch DJ and I talk about it: kzfaq.info/get/bejne/mriaZrtorqynmo0.html

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

Utterly delightful!

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

<a href="#" class="seekto" data-time="645">10:45</a> orange you've been here for 11 minutes and you've already made a all destroying death laser out of math how

*THE MATH LORE* <a href="#" class="seekto" data-time="7">0:07</a> 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) <a href="#" class="seekto" data-time="13">0:13</a> *Equality* -- First relationship between two objects you learn in a math class. <a href="#" class="seekto" data-time="19">0:19</a> *Addition* -- First of the four fundamental arithmetic operations. <a href="#" class="seekto" data-time="27">0:27</a> Repeated addition of 1s, which is how we define the rest of the naturals in set theory; also a foreshadowing for multiplication. <a href="#" class="seekto" data-time="49">0:49</a> Addition with numbers other than 1, which can be defined using what we know with adding 1s. (proof omitted) <a href="#" class="seekto" data-time="83">1:23</a> *Subtraction* -- Second of the four arithmetic operations. <a href="#" class="seekto" data-time="94">1:34</a> 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.* <a href="#" class="seekto" data-time="109">1:49</a> 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? <a href="#" class="seekto" data-time="132">2:12</a> Repeated subtraction of 1s, similar to what was done with the naturals. <a href="#" class="seekto" data-time="136">2:16</a> Negative times a negative gives positive. <a href="#" class="seekto" data-time="144">2:24</a> *Multiplication,* and an interpretation of it by repeated addition or any operation. <a href="#" class="seekto" data-time="147">2:27</a> Commutative property of multiplication, and the factors of 12. <a href="#" class="seekto" data-time="155">2:35</a> *Division,* the final arithmetic operation; also very nice to show that - and / are as related to each other as + and x! <a href="#" class="seekto" data-time="157">2:37</a> Division as counting the number of repeated subtractions to zero. <a href="#" class="seekto" data-time="169">2:49</a> Division by zero and why it doesn't make sense. Surprised that TSC didn't create a black hole out of that. <a href="#" class="seekto" data-time="184">3:04</a> *Exponentiation* as repeated multiplication. <a href="#" class="seekto" data-time="195">3:15</a> How higher exponents corresponds to geometric dimension. <a href="#" class="seekto" data-time="209">3:29</a> Anything non-zero to the zeroth power is 1. <a href="#" class="seekto" data-time="211">3:31</a> Negative exponents! And how it relates to fractions and division. <a href="#" class="seekto" data-time="217">3:37</a> Fractional exponents and *square roots!* We're getting closer now... <a href="#" class="seekto" data-time="223">3:43</a> Decimal expansion of *irrational numbers* (like sqrt(2)) is irregular. (I avoid saying "infinite" since technically every real number has an infinite decimal expansion...) <a href="#" class="seekto" data-time="229">3:49</a> sqrt(-1) gives the *imaginary number i,* which is first defined by the property i^2 = -1. <a href="#" class="seekto" data-time="237">3:57</a> Adding and multiplying complex numbers works according to what we know. <a href="#" class="seekto" data-time="240">4:00</a> i^3 is -i, which of course gives us i*e^(i*pi)! <a href="#" class="seekto" data-time="254">4:14</a> Refer to <a href="#" class="seekto" data-time="229">3:49</a> <a href="#" class="seekto" data-time="256">4:16</a> *Euler's formula* with x = pi! The formula can be shown by rearranging the Taylor series for e^x. <a href="#" class="seekto" data-time="260">4:20</a> Small detail: Getting hit by the negative sign changes TSC's direction, another allusion to the complex plane! <a href="#" class="seekto" data-time="262">4:22</a> e^(i*pi) to e^0 corresponds to the motion along the unit circle on the complex plane. <a href="#" class="seekto" data-time="284">4:44</a> The +1/-1 "saber" hit each other to give out "0" sparks. <a href="#" class="seekto" data-time="289">4:49</a> -4 saber hits +1 saber to change to -3, etc. <a href="#" class="seekto" data-time="293">4:53</a> 2+2 crossbow fires out 4 arrows. <a href="#" class="seekto" data-time="295">4:55</a> 4 arrow hits the division sign, aligning with pi to give e^(i*pi/4), propelling it pi/4 radians round the unit circle. <a href="#" class="seekto" data-time="306">5:06</a> TSC propelling himself by multiplying i, rotating pi radians around the unit circle. <a href="#" class="seekto" data-time="318">5:18</a> TSC's discovery of the *complex plane* (finally!) <a href="#" class="seekto" data-time="321">5:21</a> The imaginary axis; <a href="#" class="seekto" data-time="328">5:28</a> the real axis. <a href="#" class="seekto" data-time="333">5:33</a> The unit circle in its barest form. <a href="#" class="seekto" data-time="338">5:38</a> 2*pi radians in a circle. <a href="#" class="seekto" data-time="346">5:46</a> How the *radian* is defined -- the angle in a unit circle spanning an arc of length 1. <a href="#" class="seekto" data-time="358">5:58</a> r*theta -- the formula for the length of an arc with angle theta in a circle with radius r. <a href="#" class="seekto" data-time="394">6:34</a> For a unit circle, theta / r is simply the angle. <a href="#" class="seekto" data-time="398">6:38</a> Halfway around the circle is exactly pi radians. <a href="#" class="seekto" data-time="409">6:49</a> 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. <a href="#" class="seekto" data-time="429">7:09</a> Rotation of sin(x) allows for visualization of the displacement between sin(x) and cos(x). <a href="#" class="seekto" data-time="438">7:18</a> Refer to <a href="#" class="seekto" data-time="256">4:16</a> <a href="#" class="seekto" data-time="448">7:28</a> Changing the exponent by multiples of pi to propel itself in various directions. <a href="#" class="seekto" data-time="454">7:34</a> 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. <a href="#" class="seekto" data-time="469">7:49</a> The volume of a cylinder with area pi r^2 and height 8. <a href="#" class="seekto" data-time="473">7:53</a> An exercise for the reader (haha) <a href="#" class="seekto" data-time="483">8:03</a> Refer to <a href="#" class="seekto" data-time="260">4:20</a> <a href="#" class="seekto" data-time="505">8:25</a> cos(x) and sin(x) in terms of e^(ix) <a href="#" class="seekto" data-time="513">8:33</a> -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) <a href="#" class="seekto" data-time="543">9:03</a> Refer to <a href="#" class="seekto" data-time="306">5:06</a> <a href="#" class="seekto" data-time="578">9:38</a> 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. <a href="#" class="seekto" data-time="588">9:48</a> log((1-i)/(1+i)) = -i*pi/2, and multiplying by 2i^2 = -2 gives i*pi again. <a href="#" class="seekto" data-time="598">9:58</a> 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 <a href="#" class="seekto" data-time="617">10:17</a> Translating the circle by 9i, moving it up the imaginary axis <a href="#" class="seekto" data-time="636">10:36</a> The "displacement" beam strikes again! Refer to <a href="#" class="seekto" data-time="429">7:09</a> <a href="#" class="seekto" data-time="686">11:26</a> Now you're in the imaginary realm. <a href="#" class="seekto" data-time="736">12:16</a> "How do I get out of here?" <a href="#" class="seekto" data-time="748">12:28</a> -Don't quite get this one...- Says "exit" with 't' being just a half-hidden pi (thanks @user-or5yo4gz9r for that) <a href="#" class="seekto" data-time="783">13:03</a> n! in the denominator expands to the *gamma function,* a common extension of the factorial function to non-integers. <a href="#" class="seekto" data-time="785">13:05</a> 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!) <a href="#" class="seekto" data-time="812">13:32</a> 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) <a href="#" class="seekto" data-time="826">13:46</a> 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!

<a href="#" class="seekto" data-time="7">0:07</a> introduction to numbers <a href="#" class="seekto" data-time="11">0:11</a> equations <a href="#" class="seekto" data-time="20">0:20</a> addition <a href="#" class="seekto" data-time="84">1:24</a> subtraction <a href="#" class="seekto" data-time="94">1:34</a> negative numbers <a href="#" class="seekto" data-time="100">1:40</a> e^i*pi = -1, euler's identity <a href="#" class="seekto" data-time="136">2:16</a> two negatives cancellation <a href="#" class="seekto" data-time="144">2:24</a> multiplication <a href="#" class="seekto" data-time="149">2:29</a> the commutative property <a href="#" class="seekto" data-time="149">2:29</a> equivalent multiplications <a href="#" class="seekto" data-time="155">2:35</a> division <a href="#" class="seekto" data-time="157">2:37</a> second division symbol <a href="#" class="seekto" data-time="169">2:49</a> division by zero is indeterminate <a href="#" class="seekto" data-time="185">3:05</a> Indices/Powers <a href="#" class="seekto" data-time="219">3:39</a> One of the laws of indices. Radicals introcuced. <a href="#" class="seekto" data-time="223">3:43</a> Irrational Number <a href="#" class="seekto" data-time="230">3:50</a> Imaginary numbers <a href="#" class="seekto" data-time="239">3:59</a> i^2 = -1 <a href="#" class="seekto" data-time="241">4:01</a> 1^3 = -i = i * -1 = ie^-i*pi <a href="#" class="seekto" data-time="242">4:02</a> one of euler's formulas, it equals -1 <a href="#" class="seekto" data-time="318">5:18</a> Introduction to the complex plane <a href="#" class="seekto" data-time="336">5:36</a> Every point with a distance of one from the origin on the complex plane <a href="#" class="seekto" data-time="340">5:40</a> radians, a unit of measurement for angles in the complex plane <a href="#" class="seekto" data-time="399">6:39</a> circumference / diameter = pi <a href="#" class="seekto" data-time="409">6:49</a> sine wave <a href="#" class="seekto" data-time="416">6:56</a> cosine wave <a href="#" class="seekto" data-time="422">7:02</a> sin^2(θ) + cos^2(θ) = 1 <a href="#" class="seekto" data-time="439">7:19</a> again, euler's formula <a href="#" class="seekto" data-time="455">7:35</a> another one of euler's identities <a href="#" class="seekto" data-time="505">8:25</a> it just simplifies to 1 + 1/i <a href="#" class="seekto" data-time="512">8:32</a> sin (θ) / cos (θ) = tan (θ) <a href="#" class="seekto" data-time="569">9:29</a> infinity. <a href="#" class="seekto" data-time="599">9:59</a> limit as x goes to infinity <a href="#" class="seekto" data-time="600">10:00</a> reduced to an integral <a href="#" class="seekto" data-time="687">11:27</a> the imaginary world <a href="#" class="seekto" data-time="784">13:04</a> Gamma(x) = (x-1)! <a href="#" class="seekto" data-time="816">13:36</a> zeta, delta and phi <a href="#" class="seekto" data-time="826">13:46</a> aleph

Masterpiece

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

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.

<a href="#" class="seekto" data-time="598">09:58</a> I love seeing my logo so powerful!

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

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.

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

<a href="#" class="seekto" data-time="168">2:48</a> NEVER divide ANY number by zero.

<a href="#" class="seekto" data-time="473">7:53</a> Euler's Identity really released their Domain Expansion

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

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.

<a href="#" class="seekto" data-time="602">10:02</a> WAAAAAAAH!

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. Part 2 - 25 likes: Fun Fact is that |i| is not equal to i, but 1, beacuse if you look at the Cartesian Plane, you see that the distance from 0 to i is 1 unit, so |i|=1, and so |-i| is also equal to 1. The reason of i being 2root(-1) comes with some other weird facts, such as e^(pi×i)=-1, i^5=i^2, etc... About the Cartesian Plane being used to graph equations, the x-axis is the Real Number Line which is the number we insert in an equation, and the y-axis being the Imaginary Number Line where it's the result of the equation respect to the number inserted. You know the Reals and the Imaginaries, but there are also Complexs, writen in form a+b×i where a and b are real numbers, ex: 5+2i, this equation can't be compressed more than it is beacuse of mathematical number relation reasons. I is the second, j is the third and k is the fourth dimension in graphing equations. (More nerdy stuff at 75 likes)

The sound design is a masterclass on its own

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

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

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

as an nerd myself, here's the actual math: <a href="#" class="seekto" data-time="6">0:06</a> 1 as the unit <a href="#" class="seekto" data-time="13">0:13</a> equations <a href="#" class="seekto" data-time="18">0:18</a> addition, positive integers <a href="#" class="seekto" data-time="34">0:34</a> base ten, 0 as a place holder <a href="#" class="seekto" data-time="44">0:44</a> substitution <a href="#" class="seekto" data-time="69">1:09</a> simplifying equations, combining terms <a href="#" class="seekto" data-time="80">1:20</a> subtraction <a href="#" class="seekto" data-time="90">1:30</a> 0 as the additive identity <a href="#" class="seekto" data-time="94">1:34</a> -1, preview of e^(iπ) = -1 <a href="#" class="seekto" data-time="130">2:10</a> negative integers <a href="#" class="seekto" data-time="136">2:16</a> changing signs <a href="#" class="seekto" data-time="140">2:20</a> multiplication <a href="#" class="seekto" data-time="148">2:28</a> factors <a href="#" class="seekto" data-time="153">2:33</a> division <a href="#" class="seekto" data-time="168">2:48</a> division by 0 error <a href="#" class="seekto" data-time="183">3:03</a> powers <a href="#" class="seekto" data-time="203">3:23</a> x^1 = x , x^0 = 1, x^(-1) = 1/x <a href="#" class="seekto" data-time="215">3:35</a> fractional exponents = roots <a href="#" class="seekto" data-time="222">3:42</a> √2 is irrational <a href="#" class="seekto" data-time="228">3:48</a> √(-1) = i <a href="#" class="seekto" data-time="234">3:54</a> complex numbers <a href="#" class="seekto" data-time="240">4:00</a> e^(iπ) returns, i*i*i = i*(-1) = i*e^(iπ) <a href="#" class="seekto" data-time="255">4:15</a> Euler's formula: e^(iθ) = cosθ + i*sinθ <a href="#" class="seekto" data-time="294">4:54</a> e^(iθ) rotates an angle of θ <a href="#" class="seekto" data-time="312">5:12</a> complex plane <a href="#" class="seekto" data-time="333">5:33</a> unit circle <a href="#" class="seekto" data-time="338">5:38</a> full circle = 2π radians <a href="#" class="seekto" data-time="355">5:55</a> circle radii <a href="#" class="seekto" data-time="396">6:36</a> π <a href="#" class="seekto" data-time="401">6:41</a> trigonometry <a href="#" class="seekto" data-time="437">7:17</a> Euler's formula again <a href="#" class="seekto" data-time="453">7:33</a> Taylor series of e^(iπ) <a href="#" class="seekto" data-time="464">7:44</a> circle + cylinder <a href="#" class="seekto" data-time="471">7:51</a> (-θ) * e^(iπ) = (-θ) * (-1) = θ <a href="#" class="seekto" data-time="502">8:22</a> Euler's formula + complex trigonometry <a href="#" class="seekto" data-time="509">8:29</a> sinθ/cosθ = tanθ, function f(x) = 9*tan(πx) <a href="#" class="seekto" data-time="541">9:01</a> π radians = half turn <a href="#" class="seekto" data-time="597">9:57</a> limits, integrals to handle infinity <a href="#" class="seekto" data-time="615">10:15</a> translation <a href="#" class="seekto" data-time="781">13:01</a> factorial --> gamma function, n-dimensional spheres <a href="#" class="seekto" data-time="811">13:31</a> zeta, phi, delta, aleph (comment by MarcusScience23)

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

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

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 !

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.

Only Alan Becker can make a video about maths and we’ll all genuinely be invested in it. Edit: GUYS PLEASE STOP COMMENTING ON HOW THERE’S OTHER CHANNELS THAT CAN MAKE MATHS-BASED VIDEOS THIS WAS COMMENTED TWO MONTHS AGO AND I WAS JUST IMPRESSED AT HOW ALAN AND HIS TEAM WERE ABLE TO EXECUTE IT I DON’T WATCH VSAUCE

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.

My math teacher: This subject will be important for the rest of your life *Thief breaks into my house at 3 am* Me using math to defend my house: <a href="#" class="seekto" data-time="509">8:29</a>

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

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

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

My favorite part is when orange made an orbital death laser to kill a math equation

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

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.

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

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

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

The ending really conveys that maths does not have limits.

<a href="#" class="seekto" data-time="62">1:02</a> TSC is just like "Yay!!... What did I accomplish?"

Perfect, perfect absolutely perfect Brilliantly creative and boy was it fantastic,a fun creatively made math adventure

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

edit: woah woah woah woah too fast y'all too fast y'all this is gaining too much traction p.s. i'm only thirteen! edit two: too much. WAY TOO MUCH. TOO MUCH ATTENTION, THANK YOU ALL SO SO MUCH, and for those who were kind enough to write out things i missed out, i credited them! <a href="#" class="seekto" data-time="7">0:07</a> introduction to numbers <a href="#" class="seekto" data-time="11">0:11</a> equations <a href="#" class="seekto" data-time="20">0:20</a> addition <a href="#" class="seekto" data-time="84">1:24</a> subtraction <a href="#" class="seekto" data-time="94">1:34</a> negative numbers <a href="#" class="seekto" data-time="100">1:40</a> e^i*pi = -1, euler's identity <a href="#" class="seekto" data-time="136">2:16</a> two negatives cancellation <a href="#" class="seekto" data-time="144">2:24</a> multiplication <a href="#" class="seekto" data-time="149">2:29</a> the commutative property <a href="#" class="seekto" data-time="149">2:29</a> equivalent multiplications <a href="#" class="seekto" data-time="155">2:35</a> division <a href="#" class="seekto" data-time="157">2:37</a> second division symbol <a href="#" class="seekto" data-time="169">2:49</a> division by zero is indeterminate <a href="#" class="seekto" data-time="185">3:05</a> Indices/Powers <a href="#" class="seekto" data-time="219">3:39</a> One of the laws of indices. Radicals introcuced. <a href="#" class="seekto" data-time="223">3:43</a> Irrational Number <a href="#" class="seekto" data-time="230">3:50</a> Imaginary numbers <a href="#" class="seekto" data-time="239">3:59</a> i^2 = -1 <a href="#" class="seekto" data-time="241">4:01</a> 1^3 = -i = i * -1 = ie^-i*pi <a href="#" class="seekto" data-time="242">4:02</a> one of euler's formulas, it equals -1 <a href="#" class="seekto" data-time="295">4:55</a> pi/2 is 45 degrees in radians, also 1/8th of the unit circle. (etxrnaleclipse) <a href="#" class="seekto" data-time="318">5:18</a> Introduction to the complex plane <a href="#" class="seekto" data-time="336">5:36</a> Every point with a distance of one from the origin on the complex plane + also the cartesian plane (kinda) (etxrnaleclipse) <a href="#" class="seekto" data-time="340">5:40</a> radians, a unit of measurement for angles in the complex plane <a href="#" class="seekto" data-time="382">6:22</a> Ellipse (Conic sections in pre-calculus) (etxrnaleclipse) <a href="#" class="seekto" data-time="399">6:39</a> circumference / diameter = pi <a href="#" class="seekto" data-time="409">6:49</a> sine wave <a href="#" class="seekto" data-time="416">6:56</a> cosine wave <a href="#" class="seekto" data-time="422">7:02</a> sin^2(θ) + cos^2(θ) = 1 <a href="#" class="seekto" data-time="439">7:19</a> again, euler's formula <a href="#" class="seekto" data-time="455">7:35</a> Taylor's expansion of E, when n = i*pi <a href="#" class="seekto" data-time="465">7:45</a> Area of a Circle = πr^2 (JiReyAnimation) <a href="#" class="seekto" data-time="467">7:47</a> The sigma function (or summation) (etxrnaleclipse) <a href="#" class="seekto" data-time="470">7:50</a> The formula for the volume of a circular prism, (L)pir^2 (etxrnaleclipse) <a href="#" class="seekto" data-time="505">8:25</a> it just simplifies to 1 + 1/i <a href="#" class="seekto" data-time="512">8:32</a> sin (θ) / cos (θ) = tan (θ) <a href="#" class="seekto" data-time="569">9:29</a> Infinity. <a href="#" class="seekto" data-time="579">9:39</a> Infinity represents an idea for the biggest number, and the fancy R represents all the Real numbers (therefore from -infinity to infinity). The span is another way of saying vector. As TSC shoots, he adds another dimension to the vector. <a href="#" class="seekto" data-time="589">9:49</a> e^i pi expands into a more complex form, as e^2i^2 log((1-i)(1+i)), and 1-i/1+i multiplied by it’s conjugate and simplified gets us i, which means it simplified to -2log i, which is beyond my comprehension. <a href="#" class="seekto" data-time="598">9:58</a> Logarithms (etxrnaleclipse) <a href="#" class="seekto" data-time="599">9:59</a> limit as x goes to infinity <a href="#" class="seekto" data-time="600">10:00</a> reduced to an integral <a href="#" class="seekto" data-time="607">10:07</a> Area under a curve or bounded by two or more curves (etxrnaleclipse) <a href="#" class="seekto" data-time="687">11:27</a> the imaginary world <a href="#" class="seekto" data-time="784">13:04</a> Gamma(x) = (x-1)! <a href="#" class="seekto" data-time="816">13:36</a> zeta, delta and phi <a href="#" class="seekto" data-time="826">13:46</a> aleph and that is all the references in the video to the best of my ability!

There are so many clever references to laws, rules, principles. I had to advanced frame by frame scene by scene many times to catch it all. And I still feel like I must've missed things.

My toddler unexpectedly found this video, and now I can’t stop watching it!! The accuracy 🤌🏽 the juxtaposition of simplicity & complexity. Phenomenal! ❤️

Here's my interpretation of each scene as a second-year undergrad: <a href="#" class="seekto" data-time="0">0:00</a> Addition <a href="#" class="seekto" data-time="83">1:23</a> Subtraction <a href="#" class="seekto" data-time="100">1:40</a> Euler's identity (first sighting) <a href="#" class="seekto" data-time="145">2:25</a> Multiplication <a href="#" class="seekto" data-time="156">2:36</a> Division <a href="#" class="seekto" data-time="168">2:48</a> Division by zero <a href="#" class="seekto" data-time="185">3:05</a> Positive exponents <a href="#" class="seekto" data-time="209">3:29</a> Zero and negative exponents <a href="#" class="seekto" data-time="220">3:40</a> Fractional exponents and square roots <a href="#" class="seekto" data-time="230">3:50</a> Imaginary unit, square root of negative one <a href="#" class="seekto" data-time="240">4:00</a> Euler's identity (second sighting) <a href="#" class="seekto" data-time="284">4:44</a> a + -a = 0 <a href="#" class="seekto" data-time="318">5:18</a> The complex plane <a href="#" class="seekto" data-time="334">5:34</a> The unit circle <a href="#" class="seekto" data-time="338">5:38</a> Definition of a radian <a href="#" class="seekto" data-time="359">5:59</a> Polar coordinates <a href="#" class="seekto" data-time="399">6:39</a> Definition of pi <a href="#" class="seekto" data-time="411">6:51</a> Trigonometry and relationship with the unit circle <a href="#" class="seekto" data-time="432">7:12</a> Phase shift <a href="#" class="seekto" data-time="439">7:19</a> Euler's identity (third sighting) <a href="#" class="seekto" data-time="455">7:35</a> Taylor series expansion for e^x, x=iπ <a href="#" class="seekto" data-time="470">7:50</a> Volume of a cylinder (h = 8) <a href="#" class="seekto" data-time="505">8:25</a> Hyperbolic expansion for sine and cosine <a href="#" class="seekto" data-time="510">8:30</a> f(x) = tan(x) <a href="#" class="seekto" data-time="568">9:28</a> Infinite domain <a href="#" class="seekto" data-time="600">10:00</a> Calculus boss fight <a href="#" class="seekto" data-time="660">11:00</a> Amplitude = 100 <a href="#" class="seekto" data-time="690">11:30</a> Imaginary realm? <a href="#" class="seekto" data-time="730">12:10</a> TSC befriends Euler's identity (wholesome) <a href="#" class="seekto" data-time="758">12:38</a> i^4 = 1 <a href="#" class="seekto" data-time="785">13:05</a> Taylor series expansion for e^x, x=π <a href="#" class="seekto" data-time="786">13:06</a> Gamma function, x! = Γ(x+1) <a href="#" class="seekto" data-time="805">13:25</a> 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'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

This Video Is Really Amazing, I'm A Maths Lover🤩 & I Never Seen Such An Amazing Mathematical Animated Fighting Scene With Perfect Mathematical Logics😲❤

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

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

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.

this was really cool I would love to see second coming or even the other stick figures go up against any of the other symbols or whatever they are in another video

Cancelling Infinity using Limits. That is beyond genius!

the actual math: <a href="#" class="seekto" data-time="6">0:06</a> 1 <a href="#" class="seekto" data-time="13">0:13</a> equations <a href="#" class="seekto" data-time="18">0:18</a> addition, positive integers <a href="#" class="seekto" data-time="34">0:34</a> base ten, 0 as a place holder <a href="#" class="seekto" data-time="44">0:44</a> substitution <a href="#" class="seekto" data-time="80">1:20</a> subtraction <a href="#" class="seekto" data-time="91">1:31</a> 0 <a href="#" class="seekto" data-time="94">1:34</a> -1, preview of e^(iπ) = -1 <a href="#" class="seekto" data-time="130">2:10</a> negative integers <a href="#" class="seekto" data-time="136">2:16</a> double negative makes a positive <a href="#" class="seekto" data-time="140">2:20</a> multiplication <a href="#" class="seekto" data-time="148">2:28</a> factors <a href="#" class="seekto" data-time="153">2:33</a> division <a href="#" class="seekto" data-time="168">2:48</a> division by 0 error <a href="#" class="seekto" data-time="183">3:03</a> powers <a href="#" class="seekto" data-time="203">3:23</a> x^1 = x , x^0 = 1 <a href="#" class="seekto" data-time="210">3:30</a> x^(-1) = 1/x <a href="#" class="seekto" data-time="215">3:35</a> fractional exponents = roots <a href="#" class="seekto" data-time="222">3:42</a> √2 is irrational <a href="#" class="seekto" data-time="228">3:48</a> √(-1) = i <a href="#" class="seekto" data-time="234">3:54</a> complex numbers <a href="#" class="seekto" data-time="240">4:00</a> e^(iπ) returns, i*i*i = i*(-1) = i*e^(iπ) <a href="#" class="seekto" data-time="255">4:15</a> Euler's formula: e^(iθ) = cosθ + i*sinθ <a href="#" class="seekto" data-time="294">4:54</a> e^(iθ) rotates an angle of θ <a href="#" class="seekto" data-time="312">5:12</a> complex plane <a href="#" class="seekto" data-time="333">5:33</a> unit circle <a href="#" class="seekto" data-time="338">5:38</a> full circle = 2π radians <a href="#" class="seekto" data-time="355">5:55</a> circle radii <a href="#" class="seekto" data-time="396">6:36</a> π <a href="#" class="seekto" data-time="401">6:41</a> trigonometry <a href="#" class="seekto" data-time="437">7:17</a> Euler's formula again <a href="#" class="seekto" data-time="453">7:33</a> Taylor series of e^(iπ) <a href="#" class="seekto" data-time="464">7:44</a> circle + cylinder <a href="#" class="seekto" data-time="502">8:22</a> Euler's formula + complex trigonometry <a href="#" class="seekto" data-time="509">8:29</a> sinθ/cosθ = tanθ, function f(x) = 9*tan(πx) <a href="#" class="seekto" data-time="597">9:57</a> limits, integrals to handle infinity <a href="#" class="seekto" data-time="781">13:01</a> factorial --> gamma function <a href="#" class="seekto" data-time="784">13:04</a> n-dimensional spheres <a href="#" class="seekto" data-time="811">13:31</a> zeta, phi, delta, aleph

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

Being an engineer and a math enthusiast I love this❤

Fantastic. The part I really liked was the multiple unskippable adverts that completely broke the immersive spell of the video and gave me a chance to think about financial services, feminine hygiene products, and Amazon Prime.

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

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 absoluately love these animations.

Masterpiece, you need to create more animations! Side note: thanks for inspiring me to post animations lol

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

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

Coming back to this video for like the fifth time and just noticed ANOTHER awesome detail: when Orange pulls out a copy of a number, the "stretch" effect is an equals sign - 1=1, the reflexive property!

As a mathematics teacher, I always dream of explaining math concepts in an interesting and amazing way. Let me say, you have done wonderful work in this regard, even though words are not enough to express my feelings. In my review/reaction video (animation vs math in Urdu Hindi), I tried to explain this masterpiece in Urdu/Hindi for roughly 1 billion people in Pakistan and India!

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

This is BRILLIANT. I don't think I've laughed harder at a maths video in a long time. Probably ever.

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

Some of my favourite things from this masterpiece I noticed: <a href="#" class="seekto" data-time="99">1:39</a> e^iπ = -1 <a href="#" class="seekto" data-time="109">1:49</a> Multiplying by i probably can be represented here as moving to another dimention (of complex numbers) as they're located in a real one <a href="#" class="seekto" data-time="157">2:37</a> The division here for a÷b=c is interpreted as "c is how many times you must subtract b from a to get 0" which easily explains later why you can't divide by 0 <a href="#" class="seekto" data-time="188">3:08</a> The squared number is literally interpreted as a square-shaped sum of single units <a href="#" class="seekto" data-time="252">4:12</a> The e^iπ tries to run away to another dimention again by multiplying itself by i but TSC hits it with another i so i×i=-1 returns it back to real numbers <a href="#" class="seekto" data-time="256">4:16</a> The e^iπ extends itself according to Euler's formula <a href="#" class="seekto" data-time="259">4:19</a> TSC gets hit with minus so he flips <a href="#" class="seekto" data-time="262">4:22</a> The reason why e^iπ rides a semicircle comes from visual explaining of e^iπ=-1. e^ix means that you return the value of a particular point in complex plane which you get to through a path of x radians counterclockwise from 1. Therefore e^iπ equals to -1 because π radians is exactly a semicircle. When the e^iπ sets itself to 0 power (e^i0) it returns back to 1 through a semicircle because well 1 is zero radians apart from 1. <a href="#" class="seekto" data-time="286">4:46</a> When "+1" and "-1" swords cross they make a "0" effect <a href="#" class="seekto" data-time="288">4:48</a> The e^iπ makes a "-4" sword which destroys TSC's "+1" sword making it zero, and as a result e^iπ is now holding "-3". Then the same thing repeats with "-3" and "-2". <a href="#" class="seekto" data-time="293">4:53</a> The "2×2=" bow shoots fours <a href="#" class="seekto" data-time="295">4:55</a> As I explained above, e^(iπ/4) means you move exaclty π/4 radians (quarter semicircle) counterclockwise <a href="#" class="seekto" data-time="306">5:06</a> When you multiply a number by i in complex plane you just actually rotate the position vector of this number 90° counterclockwise, that's where a quarter circle came from <a href="#" class="seekto" data-time="339">5:39</a> Each segment here is a radian, a special part of a circle in which the length of the arc coincides with the length of the radius (it's also shown at <a href="#" class="seekto" data-time="346">5:46</a>); the circle has exactly 2π radians which you can visually see is about 6.283 <a href="#" class="seekto" data-time="398">6:38</a> Visual explanation of π radians being a semicircle <a href="#" class="seekto" data-time="408">6:48</a> Geometric interpretation of sinusoid <a href="#" class="seekto" data-time="428">7:08</a> TSC once again multiplies the sine function by i which rotates its graph 90° <a href="#" class="seekto" data-time="456">7:36</a> The sum literally shoots its addends so the value of n increases as the lower ones have just been used; you may also notice that every next addend gets the value of n higher and higher as well as extends to its actual full value when explodes <a href="#" class="seekto" data-time="465">7:45</a> TSC multiplies the circle by π so he gets the area and can use it as shield <a href="#" class="seekto" data-time="484">8:04</a> TSC uses minus on himself so he comes out from another side <a href="#" class="seekto" data-time="497">8:17</a> The sinusoid as a laser beam is just priceless <a href="#" class="seekto" data-time="542">9:02</a> Multiplying the radius by π here is interpreted as rotating it 180° <a href="#" class="seekto" data-time="563">9:23</a> +7i literally means 7 units up in complex plane <a href="#" class="seekto" data-time="578">9:38</a> Here is some kind of math pun. TSC shoots with infinity which creates the set of all real numbers (ℝ). With every other shot he creates another set which represents as ℝ², ℝ³ etc. It also means span (vector) in linear algebra and with every other ℝ this vector receives another dimention (x₁, x₂, x₃ etc.). <a href="#" class="seekto" data-time="598">9:58</a> The sum monster absorbs infinity (shown as limit) and receives an integral from 0 to ∞ <a href="#" class="seekto" data-time="814">13:34</a> The golden ratio (φ) when approaching e^iπ takes smaller and smaller steps which shorten according to the golden ratio (each step is about 1.618 shorter than the previous one) <a href="#" class="seekto" data-time="826">13:46</a> Aleph (ℵ) represents the size of an infinite set so is presented here as enormously sized number

Some of my favourite things from this masterpiece which I understood: <a href="#" class="seekto" data-time="99">1:39</a> e^iπ = -1 <a href="#" class="seekto" data-time="109">1:49</a> Multiplying by i probably can be represented here as moving to another dimention (of complex numbers) as they're located in a real one <a href="#" class="seekto" data-time="157">2:37</a> The division here for a÷b=c is interpreted as "c is how many times you must subtract b from a to get 0" which easily explains later why you can't divide by 0 <a href="#" class="seekto" data-time="188">3:08</a> The squared number is literally interpreted as a square-shaped sum of single units <a href="#" class="seekto" data-time="252">4:12</a> The e^iπ tries to run away to another dimention again by multiplying itself by i but TSC hits it with another i so i×i=-1 returns it back to real numbers <a href="#" class="seekto" data-time="256">4:16</a> The e^iπ extends itself according to Euler's formula <a href="#" class="seekto" data-time="259">4:19</a> TSC gets hit with minus so he flips <a href="#" class="seekto" data-time="262">4:22</a> The reason why e^iπ rides a semicircle comes from visual explaining of e^iπ=-1. e^ix means that you return the value of a particular point in complex plane which you get to through a path of x radians counterclockwise from 1. Therefore e^iπ equals to -1 because π radians is exactly a semicircle. When the e^iπ sets itself to 0 power (e^i0) it returns back to 1 through a semicircle because well 1 is zero radians apart from 1. <a href="#" class="seekto" data-time="293">4:53</a> The "2×2=" bow shoots fours <a href="#" class="seekto" data-time="295">4:55</a> As I explained above, e^(iπ/4) means you move exaclty π/4 radians (quarter semicircle) counterclockwise <a href="#" class="seekto" data-time="306">5:06</a> When you multiply a number by i in complex plane you just actually rotate the position vector of this number 90° counterclockwise, that's where a quarter circle came from <a href="#" class="seekto" data-time="339">5:39</a> Each segment here is a radian, a special part of a circle in which the length of the arc coincides with the length of the radius (it's also shown at <a href="#" class="seekto" data-time="346">5:46</a>); the circle has exactly 2π radians which you can visually see is about 6.283 <a href="#" class="seekto" data-time="398">6:38</a> Visual explanation of π radians being a semicircle <a href="#" class="seekto" data-time="408">6:48</a> Geometric interpretation of sinusoid <a href="#" class="seekto" data-time="428">7:08</a> TSC once again multiplies the sine function by i which rotates its graph 90° <a href="#" class="seekto" data-time="456">7:36</a> The sum literally shoots its addends so the value of n increases as the lower ones have just been used; you may also notice that every next addend gets the value of n higher and higher as well as extends to its actual full value when explodes <a href="#" class="seekto" data-time="465">7:45</a> TSC multiplies the circle by π so he gets the area and can use it as shield <a href="#" class="seekto" data-time="484">8:04</a> TSC uses minus on himself so he comes out from another side <a href="#" class="seekto" data-time="497">8:17</a> The sinusoid as a laser beam is just priceless <a href="#" class="seekto" data-time="542">9:02</a> Multiplying the radius by π here is interpreted as rotating it 180° <a href="#" class="seekto" data-time="563">9:23</a> +7i literally means 7 units up in complex plane <a href="#" class="seekto" data-time="578">9:38</a> Here is some kind of math pun. TSC shoots with infinity which creates the set of all real numbers (ℝ). With every other shot he creates another set which represents as ℝ², ℝ³ etc. It also means span (vector) in linear algebra and with every other ℝ this vector receives another dimention (x₁, x₂, x₃ etc.). <a href="#" class="seekto" data-time="598">9:58</a> The sum monster absorbs infinity (shown as limit) and receives an integral from 0 to ∞ <a href="#" class="seekto" data-time="826">13:46</a> Aleph (ℵ) represents the size of an infinite set so is presented here as enormously sized number P.S. Thanks to the people in replies who taught me the name of the orange character (The Second Coming), before that I just called him "the guy" here. P.P.S. Also thank you all for the feedback, I'm glad you appreciated my half hour work.

<a href="#" class="seekto" data-time="600">10:00</a> was most epic scene

Awesome, but sidnt cover a lot of this in my High School Math experience. Mostly Trig, bit of probability and statistics. Lowest level university qualifying mathematics, at least in my education systems. So much trig...

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.

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

Love how it starts with basic additions and ends with traveling to a new dimension in just 13 minutes

<a href="#" class="seekto" data-time="0">0:00</a> introducing numbers in addition <a href="#" class="seekto" data-time="81">1:21</a> equality <a href="#" class="seekto" data-time="88">1:28</a> subtraction <a href="#" class="seekto" data-time="101">1:41</a> Introducing eiπ <a href="#" class="seekto" data-time="157">2:37</a> multiplying and dividing <a href="#" class="seekto" data-time="172">2:52</a> falling numbers <a href="#" class="seekto" data-time="188">3:08</a> something weird (6+2)²=1 <a href="#" class="seekto" data-time="205">3:25</a> small numbers <a href="#" class="seekto" data-time="220">3:40</a> introducing new symbol √ <a href="#" class="seekto" data-time="233">3:53</a> weird I (is this a number? <a href="#" class="seekto" data-time="242">4:02</a> meeting eiπ again <a href="#" class="seekto" data-time="256">4:16</a> insane fight <a href="#" class="seekto" data-time="322">5:22</a> some other symbols and lining <a href="#" class="seekto" data-time="339">5:39</a> Meeting θ <a href="#" class="seekto" data-time="411">6:51</a>lining and new symbol π <a href="#" class="seekto" data-time="440">7:20</a> meeting eiπ again <a href="#" class="seekto" data-time="456">7:36</a> new symbols Σ 𝑛 ! ∞ _ () ⁿ <a href="#" class="seekto" data-time="514">8:34</a> new symbols 𝑓 • () <a href="#" class="seekto" data-time="531">8:51</a> insane fight again <a href="#" class="seekto" data-time="595">9:55</a> transform (the boss) <a href="#" class="seekto" data-time="791">13:11</a> last goodby (portal) <a href="#" class="seekto" data-time="816">13:36</a> Weird symbols (friend of eiπ) ζφδא <a href="#" class="seekto" data-time="832">13:52</a> Outro (The + End) Btw I liked my own comment :3

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

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.

Words cannot describe how amazingly animated this is.

i have never been this much intrested on a maths video . im suprised that i am thinking of learning advanced maths now

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.

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.

can you make part 2 please I really want to see division, multiplication and any other math thing

This was awesome, it felt like It as watching some kind of math boss fight or anime

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

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.

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

this would be fun if it was a video game and a live virtual simulation to play and you can be the stick character and do real algebra and do whatever and even do work related things and business all while having fun af

Some Small Details <a href="#" class="seekto" data-time="329">5:29</a> this shows The Second Coming is approximately 1.65 units tall. An average adult male is 1.6~1.8 meters tall. It appears the math space is in SI units, m being the SI unit of length. This also shows TSC is about 165cm tall, or 5' 5". <a href="#" class="seekto" data-time="465">7:45</a> a circle is represented as x^2 + y^2 = r^2. Inserting a pi turns it into the area of a circle, pi*r^2. Inserting 8 turns it into the volume of a cylinder, 8*pi*r^2. <a href="#" class="seekto" data-time="541">9:01</a> since f(x) is 9*tan(x) and tangent turns angle into the steepness of a line, it can latch onto the unit circle. <a href="#" class="seekto" data-time="580">9:40</a> f(dot) represents the tangent function at a given point (throughout this video, we can see a dot used as an arbitary number on the number line), and f(inf) represents the tangent function over the entire number line [0, +inf). An entire number line can be seen as a span of an unit vector, thus each shot increases the dimension of the span. This also implies that TSC is a being that is four-dimensional. <a href="#" class="seekto" data-time="597">9:57</a> Sigma + limit = integral. If you try to derive the definite integral using the sum of rectangles method, you will eventually transform lim(sigma(f(...)) into integral(g(...)). <a href="#" class="seekto" data-time="604">10:04</a> Calculating an integral of a function can be seen as getting the total (polar) area between the function and the number line. Thus the Integral Sword attacks with R2. <a href="#" class="seekto" data-time="691">11:31</a> welcome to the imaginary realm. Hope you like it here.

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

-Simple addition -Counting to ten -Adding 2's -Adding 20's -Reaching 100 -Simplifying addiion -Subtracion -Negatives -Euler's identity -Imaginary number usage -Subtracting negatives -2 - make a + -Multiplication -Some numbers have more factors than others -Division -Division by 0 is not possible -Exponents (squared) -Higher exponents -4 to the -x = 1/4 to the x -Square roots (square numbers) -Square roots (non-squares) -Imaginary number (i) -Euler returns -Euler's formula: cos(π)+isin(π) -Euler and TSC fight -Graphs (x axis and y axis) -Circles and the concept of π -Radius, angles, how to calculate π -Cosines and sines come from π -Euler and TSC fight club 2.0 -Concept of sums (Σ) -Functions -Concept of Infinity ♾️ -Spans -Integrals -The imaginary realm -Imaginary numbers -Euler and TSC make a truce The end... of animation vs math.

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

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.

I need more videos like this ❤

The sound effects are on another level

As a math enthusiast I will admit that everything in this video was really fun to watch, and everything demonstrated was done creatively and understandably. (most of the time) The different ways math was used in these animations was very cool and I'd love to see more sometime. Good job Alan and team!