火柴人 VS 数学(Math)

Let me explain all the things you were confused about in the video. I'm doing this from memory so I hope I won't make any stupid mistakes.

I'll be using programming math notation, so here's what it is:

= is the equal sign

+ is addition

- is substraction

* is multiplication

/ is division

^ is power (2^4 is two to the power of four, so 2^4 = 16)

2:50 - Euler's formula. e^(i*pi) = -1. In a more general sense, e^(i*x) = cos(x) + i*sin(x) (in the case of x = pi by definition cos(pi) = -1 and sin(pi) = 0, so e^(i*pi) = -1 + 0*i = -1 + 0 = -1). The constant e is what's called Euler's number, one of the most important mathematical constants. It is equal to 1 + 1/1 + 1/(1*2) + 1/(1*2*3) + 1/(1*2*3*4) + 1/(1*2*3*4*5)... which equals around 2.718. The constant i is what's called an imaginary number. i^2 = -1, which seems impossible as two negative numbers multiplied by each other always equal a positive number. Pi is also a very important constant in math, equal to around 3.142. It is the number you get if you divide the circumference of any circle by its diameter. For now however you only need to know that e^(i*pi) = -1.

3:00 - Euler's number multiplies itself by i, making it an imaginary number and transporting it to the imaginary number space.

3:46 - This is a method of of quickly dividing large numbers. You substract the divisor from the divided number multiple times until you are left with a number smaller than the divisor. The solution is the number of times you substracted plus the number you are left with divided by the same divisor. This breaks with dividing by 0 (which is left undefined because everything suggests that the solution is infinity, but that would make all numbers equal to each other; I won't explain it here, just google why you cannot divide by 0).

4:34 - You seem to be confused about taking something to a power higher than 2. This is actually quite simple. x^5 for example is x*x*x*x*x, so 4^5 = 4*4*4*4*4 = 16*4*4*4 = 64*4*4 = 256*4 = 1024. Taking something to the power of 0 is shown later. Every number except 0 that is taken to the power of 0 equals 1 for reasons I won't get into here. Negative powers are later shown, which are also very simple. x^-y = 1/(x^y). So 4^-2 = 1/(4^2) = 1/16.

4:50 - Square root. A sqaure root is synonymous with taking a number to the power of 1/2. In programming you usually write the square root as sqrt(), but in math it's written like in the video. It's sort of like a reverse to squaring a number. The sqrt(4), the one shown first in the video, is equal to 2 because 2^2 = 4. The sqrt(9) = 3 because 3^2 = 9. The sqrt(1) = 1 because 1^2 = 1. Sqaure roots of numbers that are not perfect squares (0, 1, 4, 9, 16, 25, 36...) will be hard to write with an unending sequence of numbers after the decimal separator (the little period signifying a non-whole number). The sqrt(2) shown in the video is equal to around 1.414, but the actual number is infinitely long if you want to write it. It is however very useful in math. For example the diagonal of a square is always sqrt(2) times the edge of the square.

5:44 - As I previously stated, this is the constant i. i^2 = -1, so sqrt(-1) = i.

6:08 - Orange throws an i at the Euler's number, which is trying to multiply itself by another i to escape to the world of imaginary numbers. This causes them to combine into i*i, which are equal to -1 by definition. This makes the Euler's number equal to -1*-1 = 1, which is not an imaginary number. This is why it gets thrown out back into the world of real numbers.

6:10 - Here Euler's number uses Euler's formula to become cos(pi) + i*sin(pi), which I explained earlier.

6:18 - Euler's number takes out its own pi, which is equal to 180 degrees in a unit of angle called radians (a radian is a unit of angle in which 1 is equal to the legth of the radius of the circle that the angle creates; this means that by definition pi is half of the circle). More of this will be seen later.

6:50 - Here Euler's number has its pi divided by 4, which makes it move by 45 degrees, which is in radians equal to pi/4. Again, more on this later.

7:02 - Orange uses i to rotate him 90 degrees. This is not because of some radian calculation (i cannot even exist as a value in geometry), but because the imaginary and real axis of number are usually shown together on a plain of so-called complex numbers. The axis of the imaginary numbers is vertical and the axis of the real numbers is horizontal, so by multiplying any complex number by i you "turn" it by 90 degrees anticlockwise. This will be shown more clearly in a second.

7:13 - This is when the plain of complex numbers is shown. The first axis drawn by Orange is the imaginary vertical one. The second is the real horizontal one.

7:34 - These are the radians I talked about earlier. The circle can be divided into six segments the size of one radian (one length of the radius) and a small part that makes the total circle slightly larger. This is because the total angle inside the circle, or 360 degrees, is by definition equal to 2*pi, or around 6.283.

7:39 - Orange takes out a single radian, which, as I said, is the size of the radius of the circle.

8:00 - Here the r is the radius of the circle and the weird 0 with a line through it, which is a Greek letter called theta, is the most common symbol used for an angle, especially in physics, but also in math. Orange uses them to get the angle at which the radius is orientated, which is pi at 180 degrees (facing left).

8:38 - I'm not really sure how exactly Orange got the trigonometric functions here, but the sine and cosine functions are highly connected with pi, as the waves of both create sort of "circles" and go through the x axis in intervals of pi.

9:04 - Again we see the sine function multiplied by i to turn it 90 degrees

9:29 - This symbol that Euler's number turns into here is called sigma, and it is again a Greek letter. It is used to not do what I've done while explaining what e is at the beginning and write out a sum with an ellipsis at the end. It is used to write a sum of any large amount (potentially infinite) of numbers if there is a rule that connects them all. For example to get the sum of all the natural numbers up to ten with each number divided by 2 you start with the big E-like thing (sigma), you write the number you start with below the sigma, starting with n= (in this case n=0, which means you start with 0), then you write the number you end at above the sigma (in this case 10). Finally you write the formula to get the specific numbers to the right of the sigma (in this case we want each number to be divided by 2, so we write n/2). I will from now write it as sigma(below: n=0; above: 10; function: n/2). In the case of the video we use the sum from the definition of e, which is e = sigma(below: n=0; above: infinity; function: 1/n!). The ! sign in math is called the factorial and is the multiplication of all numbers up to that number starting from one. So 3! = 1*2*3 = 6, 5! = 1*2*3*4*5 = 120, 1! = 1 and 0! = 1. In the video we see a different sum because the number that is equal to the sum is e^(i*pi), not e. This is why we get sigma(below: n=0; above: infinity; function: (i*pi)^n/n!) in the video. You can also see that the missiles shot by the sigma sum are actually the elements of the sum one after another ((i*pi)^0/0!, (i*pi)^1/1!, (i*pi)^2/2!...).

9:58 - This is actually a thing in math. A negative vector is its positive version rotated in a vector space. The video later confirms that this is a vector space, which I will point out.

10:25 - Orange divided the sine by the cosine of the same angle (pi or 180 degrees), which equals the tangent of the same angle. In mathematical notation: sin(pi)/cos(pi) = tg(pi). I think this is all combined into a tangent function times 9i, but even if it's not, it's definitely some kind of function even though it doesn't have an x parameter. You can actually see the bullets of the weapon leave a trait of tangent functions if you look closely.

10:57 - Here pi is used to rotate the radius by 180 degrees.

11:34 - This confirms that the circle is indeed a vector space.

12:12 - 9i is used to move the circle up by nine (remember the imaginary axis?).

13:21 - Orange and Euler's number end up in the world of imaginary numbers as they have been multiplied by i. They also end up rotated 90 degrees counterclockwise, which is again the result of them being multiplied by i.

14:22 - Orange spells exit here using the Euler's number as e, a multiplication sign as x, the i constant as, well, i, and a half-covered pi as a t.

14:58 - n! is turned into the gamma function, which I will describe as gamma(n - 1). By definition of the gamma function gamma(n) = (n - 1)!. It is not really that important, just explaining what the weird letter in the equasion is.

15:08 - Euler's number adds more and more "volume" to the circle with each use of the function. It's too complicated to explain in a Youtube comment.

15:28 - The different letters here are different constants in math. I am not really sure what the tall bouncing one is, it is the Greek letter zeta but there is only a zeta function, not a zeta constant. The weird o with a line through it one the left is phi, the famous and very useful golden ratio. Phi = a/b = (a+b)/a = around 1.618. The flying letter is delta, the first Feigenbaum constant, which is too complicated to explain here, but it equals around 4.669. The giant letter at the end is aleph, a type of infinity (yes, there are different types and sizes of infinity in math). There is an infinite amount of alephs, the most commonly used one is aleph-zero, the sum of any infinite set of whole numbers

I know this is all very simplified for the math nerds reading this, but this is a very long comment and I just want to finally finish writing this.

英文原稿转自YouTube，由于时间轴不相同，因此我做了翻译，如下（大家有能力可以去看原视频底下评论：此评论出处视频https://www.youtube.com/watch？v=gKJj5JGR7y4

Alan原视频：https://www.youtube.com/watch?v=B1J6Ou4q8vE）

让我解释一下视频中您感到困惑的所有内容。 我是凭记忆做的，所以我希望我不会犯任何愚蠢的错误。

我将使用编程数学符号，所以它是什么：

= 是等号

+ 是加法

- 是减法

* 是乘法

/ 是除法

^ 是幂（2^4 是二的四次方，所以 2^4 = 16）

1:40 - 欧拉公式。 e^(i*pi) = -1。 从更一般的意义上来说，e^(i*x) = cos(x) + i*sin(x)（在 x = pi 的情况下，定义为 cos(pi) = -1 且 sin(pi) = 0， 所以 e^(i*pi) = -1 + 0*i = -1 + 0 = -1)。 常数 e 就是所谓的欧拉数，是最重要的数学常数之一。 等于 1 + 1/1 + 1/(1*2) + 1/(1*2*3) + 1/(1*2*3*4) + 1/(1*2*3*4 *5)...大约等于 2.718。 常数 i 就是所谓的虚数。 i^2 = -1，这似乎不可能，因为两个负数彼此相乘总是等于正数。 Pi 在数学中也是一个非常重要的常数，大约等于 3.142。 将任何圆的周长除以其直径得到的数字。 但现在您只需要知道 e^(i*pi) = -1。

1:50 - 欧拉数自身乘以 i，使其成为虚数并将其传送到虚数空间。

2:36 - 这是一种快速除以大数的方法。 将被除数多次减去除数，直到得到小于除数的数。 解决方案是减去的次数加上除以同一个除数后剩下的数字。 这会打破除以 0 的情况（这是未定义的，因为一切都表明解是无穷大，但这将使所有数字彼此相等；我不会在这里解释它，只需谷歌一下为什么不能除以 0）。

3:20 - 您似乎对计算大于 2 的幂感到困惑。这实际上非常简单。 例如 x^5 是 x*x*x*x*x，因此 4^5 = 4*4*4*4*4 = 16*4*4*4 = 64*4*4 = 256*4 = 1024 . 稍后将显示取 0 次方。 除了 0 之外的每个数字的 0 次方都等于 1，原因我不会在这里详细介绍。 稍后会显示负幂，这也很简单。 x^-y = 1/(x^y)。 所以 4^-2 = 1/(4^2) = 1/16。

3:38 - 平方根。 平方根与取数字的 1/2 次方同义。 在编程中，您通常将平方根写为 sqrt()，但在数学中，它的写法就像视频中一样。 这有点像数字平方的逆过程。 视频中第一个显示的 sqrt(4) 等于 2，因为 2^2 = 4。sqrt(9) = 3，因为 3^2 = 9。sqrt(1) = 1，因为 1^2 = 1. 不是完美平方的数字的平方根（0, 1, 4, 9, 16, 25, 36...）将很难用小数点分隔符后面的无休止的数字序列（小句号表示 非整数）。 视频中显示的 sqrt(2) 大约等于 1.414，但如果要写的话，实际数字是无限长的。 然而，它在数学中非常有用。 例如，正方形的对角线始终是 sqrt(2) 乘以正方形的边。

3:50 - 正如我之前所说，这是常数 i。 i^2 = -1，所以 sqrt(-1) = i。

4:11 - Orange 在欧拉数上扔了一个 i，该数试图将自己乘以另一个 i 以逃逸到虚数的世界。 这导致它们组合成 i*i，根据定义等于 -1。 这使得欧拉数等于-1*-1 = 1，这不是一个虚数。 这就是为什么它被扔回实数世界的原因。

4:15 - 这里欧拉数使用欧拉公式变成 cos(pi) + i*sin(pi)，我之前解释过。

4:22 - 欧拉数取出它自己的 pi，它等于 180 度，单位为弧度（弧度是角度单位，其中 1 等于该角度所在圆的半径的长度） 创建；这意味着根据定义 pi 是圆的一半）。 稍后将会看到更多内容。

4:54 - 这里欧拉数的 pi 除以 4，这使得它移动了 45 度，其弧度等于 pi/4。 同样，稍后会详细介绍这一点。

5:07 - Orange 使用 i 将他旋转 90 度。 这并不是因为某种弧度计算（i 甚至不能作为几何中的值存在），而是因为数字的虚轴和实轴通常一起显示在所谓复数的平面上。 虚数的轴是垂直的，实数的轴是水平的，因此通过将任何复数乘以 i，您可以将其逆时针“旋转”90 度。 稍后将更清楚地显示这一点。

5:18 - 这是显示复数的简单形式的时候。 Orange 绘制的第一条轴是假想的垂直轴。 第二个是真正的水平的。

5:38 - 这些是我之前谈到的弧度。 圆可以分为六个部分，每个部分的大小为一个弧度（半径的一个长度），还有一小部分使整个圆稍大一些。 这是因为圆内的总角度（即 360 度）根据定义等于 2*pi，即 6.283 左右。7:39 - 橙色取出一个弧度，正如我所说，它是圆的半径大小。

6:02 - 这里的 r 是圆的半径，奇怪的 0 有一条线穿过它，它是一个希腊字母，称为 theta，是最常用的角度符号，尤其是在物理学中，但也在数学中。 Orange 使用它们来获取半径方向的角度，即 180 度的 pi（面向左）。

6:43 - 我不太确定 Orange 到底是如何得到三角函数的，但正弦和余弦函数与 pi 高度相关，因为两者的波都会创建某种“圆”并按间隔穿过 x 轴 圆周率。

7:10 - 我们再次看到正弦函数乘以 i 将其旋转 90 度

7:35 - 欧拉数在这里变成的这个符号称为 sigma，它又是一个希腊字母。 它用来不做我所做的事情，同时在开头解释 e 是什么，并在末尾写出一个带有省略号的和。 如果存在将所有数字连接起来的规则，则它用于编写任何大量（可能是无限的）数字的总和。 例如，要获得最大 10 的所有自然数的总和，每个数字除以 2，您从大的 E 状的东西（西格玛）开始，您将开始的数字写在西格玛下方，从 n= 开始（在 在这种情况下 n=0，这意味着您从 0 开始），然后在 sigma 上方写下结束的数字（在本例中为 10）。 最后，编写公式以获取 sigma 右侧的特定数字（在本例中，我们希望每个数字除以 2，因此我们写为 n/2）。 从现在起我将把它写成 sigma（下面：n=0；上面：10；函数：n/2）。 在视频中，我们使用 e 定义中的总和，即 e = sigma（下方：n=0；上方：无穷大；函数：1/n！）。 这 ！ 数学中的符号称为阶乘，是从 1 开始直到该数字的所有数字的乘法。 所以3！ = 1*2*3 = 6, 5！ = 1*2*3*4*5 = 120, 1! = 1 和 0！ = 1。在视频中我们看到了不同的总和，因为等于总和的数字是 e^(i*pi)，而不是 e。 这就是为什么我们在视频中得到 sigma(下面：n=0；上面：无穷大；函数：(i*pi)^n/n!)。 还可以看到，sigma sum 射出的导弹其实就是一个个的 sum 的元素 ((i*pi)^0/0!, (i*pi)^1/1!, (i*pi) ^2/2！...）。

8:05 - 这实际上是数学中的一个问题。 负向量是其在向量空间中旋转的正向量。 视频稍后证实这是一个向量空间，我将指出这一点。

8:30 - Orange 将正弦除以同一角度（pi 或 180 度）的余弦，等于同一角度的正切。 用数学符号表示：sin(pi)/cos(pi) = tg(pi)。 我认为这一切都组合成一个正切函数乘以 9i，但即使不是，它也绝对是某种函数，即使它没有 x 参数。 如果你仔细观察的话，你实际上可以看到武器的子弹留下了切线函数的特征。

9:03- 这里 pi 用于将半径旋转 180 度。

9:38 - 这证实了圆确实是一个向量空间。

10:15 - 9i 用于将圆向上移动 9 个（还记得虚轴吗？）。

11:26 - Orange 和 Euler 的数最终进入虚数世界，因为它们已乘以 i。 它们最终也会逆时针旋转 90 度，这也是它们乘以 i 的结果。

12:26 - 橙色咒语在这里使用欧拉数作为 e，乘号作为 x，i 常数作为，好吧，i，以及半覆盖的 pi 作为 t。

13:03 - 不！ 变成 gamma 函数，我将其描述为 gamma(n - 1)。 根据伽马函数的定义 gamma(n) = (n - 1)!。 其实这并不重要，只是解释一下方程中奇怪的字母是什么。

13:14 - 每次使用该函数时，欧拉数都会给圆增加越来越多的“体积”。 在 YouTube 评论中解释起来太复杂了。

13:35 - 这里不同的字母是数学中不同的常数。 我不太确定那个高弹跳的是什么，它是希腊字母 zeta，但只有 zeta 函数，而不是 zeta 常数。 左边有一条线穿过的奇怪的 o 是 phi，著名且非常有用的黄金比例。 Phi = a/b = (a+b)/a = 约 1.618。 飞翔的字母是 delta，第一个费根鲍姆常数，这里解释起来太复杂，但它大约等于 4.669。 末尾的大字母是 aleph，一种无穷大（是的，数学中的无穷大有不同的类型和大小）。 aleph 的数量是无限的，最常用的是 aleph-zero，即任意无限整数集的和

我知道对于阅读本文的数学迷来说，这一切都非常简单，但这是一个很长的评论，我只想最终完成这篇文章。