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.

英文原稿轉(zhuǎn)自YouTube，由于時(shí)間軸不相同，因此我做了翻譯，如下（大家有能力可以去看原視頻底下評(píng)論：此評(píng)論出處視頻https://www.youtube.com/watch？v=gKJj5JGR7y4

Alan原視頻：https://www.youtube.com/watch?v=B1J6Ou4q8vE）

讓我解釋一下視頻中您感到困惑的所有內(nèi)容。 我是憑記憶做的，所以我希望我不會(huì)犯任何愚蠢的錯(cuò)誤。

我將使用編程數(shù)學(xué)符號(hào)，所以它是什么：

= 是等號(hào)

+ 是加法

- 是減法

* 是乘法

/ 是除法

^ 是冪（2^4 是二的四次方，所以 2^4 = 16）

1:40 - 歐拉公式。 e^(i*pi) = -1。 從更一般的意義上來說，e^(i*x) = cos(x) + i*sin(x)（在 x = pi 的情況下，定義為 cos(pi) = -1 且 sin(pi) = 0， 所以 e^(i*pi) = -1 + 0*i = -1 + 0 = -1)。 常數(shù) e 就是所謂的歐拉數(shù)，是最重要的數(shù)學(xué)常數(shù)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)...大約等于 2.718。 常數(shù) i 就是所謂的虛數(shù)。 i^2 = -1，這似乎不可能，因?yàn)閮蓚€(gè)負(fù)數(shù)彼此相乘總是等于正數(shù)。 Pi 在數(shù)學(xué)中也是一個(gè)非常重要的常數(shù)，大約等于 3.142。 將任何圓的周長(zhǎng)除以其直徑得到的數(shù)字。 但現(xiàn)在您只需要知道 e^(i*pi) = -1。

1:50 - 歐拉數(shù)自身乘以 i，使其成為虛數(shù)并將其傳送到虛數(shù)空間。

2:36 - 這是一種快速除以大數(shù)的方法。 將被除數(shù)多次減去除數(shù)，直到得到小于除數(shù)的數(shù)。 解決方案是減去的次數(shù)加上除以同一個(gè)除數(shù)后剩下的數(shù)字。 這會(huì)打破除以 0 的情況（這是未定義的，因?yàn)橐磺卸急砻鹘馐菬o窮大，但這將使所有數(shù)字彼此相等；我不會(huì)在這里解釋它，只需谷歌一下為什么不能除以 0）。

3:20 - 您似乎對(duì)計(jì)算大于 2 的冪感到困惑。這實(shí)際上非常簡(jiǎn)單。 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 . 稍后將顯示取 0 次方。 除了 0 之外的每個(gè)數(shù)字的 0 次方都等于 1，原因我不會(huì)在這里詳細(xì)介紹。 稍后會(huì)顯示負(fù)冪，這也很簡(jiǎn)單。 x^-y = 1/(x^y)。 所以 4^-2 = 1/(4^2) = 1/16。

3:38 - 平方根。 平方根與取數(shù)字的 1/2 次方同義。 在編程中，您通常將平方根寫為 sqrt()，但在數(shù)學(xué)中，它的寫法就像視頻中一樣。 這有點(diǎn)像數(shù)字平方的逆過程。 視頻中第一個(gè)顯示的 sqrt(4) 等于 2，因?yàn)?2^2 = 4。sqrt(9) = 3，因?yàn)?3^2 = 9。sqrt(1) = 1，因?yàn)?1^2 = 1. 不是完美平方的數(shù)字的平方根（0, 1, 4, 9, 16, 25, 36...）將很難用小數(shù)點(diǎn)分隔符后面的無休止的數(shù)字序列（小句號(hào)表示 非整數(shù)）。 視頻中顯示的 sqrt(2) 大約等于 1.414，但如果要寫的話，實(shí)際數(shù)字是無限長(zhǎng)的。 然而，它在數(shù)學(xué)中非常有用。 例如，正方形的對(duì)角線始終是 sqrt(2) 乘以正方形的邊。

3:50 - 正如我之前所說，這是常數(shù) i。 i^2 = -1，所以 sqrt(-1) = i。

4:11 - Orange 在歐拉數(shù)上扔了一個(gè) i，該數(shù)試圖將自己乘以另一個(gè) i 以逃逸到虛數(shù)的世界。 這導(dǎo)致它們組合成 i*i，根據(jù)定義等于 -1。 這使得歐拉數(shù)等于-1*-1 = 1，這不是一個(gè)虛數(shù)。 這就是為什么它被扔回實(shí)數(shù)世界的原因。

4:15 - 這里歐拉數(shù)使用歐拉公式變成 cos(pi) + i*sin(pi)，我之前解釋過。

4:22 - 歐拉數(shù)取出它自己的 pi，它等于 180 度，單位為弧度（弧度是角度單位，其中 1 等于該角度所在圓的半徑的長(zhǎng)度） 創(chuàng)建；這意味著根據(jù)定義 pi 是圓的一半）。 稍后將會(huì)看到更多內(nèi)容。

4:54 - 這里歐拉數(shù)的 pi 除以 4，這使得它移動(dòng)了 45 度，其弧度等于 pi/4。 同樣，稍后會(huì)詳細(xì)介紹這一點(diǎn)。

5:07 - Orange 使用 i 將他旋轉(zhuǎn) 90 度。 這并不是因?yàn)槟撤N弧度計(jì)算（i 甚至不能作為幾何中的值存在），而是因?yàn)閿?shù)字的虛軸和實(shí)軸通常一起顯示在所謂復(fù)數(shù)的平面上。 虛數(shù)的軸是垂直的，實(shí)數(shù)的軸是水平的，因此通過將任何復(fù)數(shù)乘以 i，您可以將其逆時(shí)針“旋轉(zhuǎn)”90 度。 稍后將更清楚地顯示這一點(diǎn)。

5:18 - 這是顯示復(fù)數(shù)的簡(jiǎn)單形式的時(shí)候。 Orange 繪制的第一條軸是假想的垂直軸。 第二個(gè)是真正的水平的。

5:38 - 這些是我之前談到的弧度。 圓可以分為六個(gè)部分，每個(gè)部分的大小為一個(gè)弧度（半徑的一個(gè)長(zhǎng)度），還有一小部分使整個(gè)圓稍大一些。 這是因?yàn)閳A內(nèi)的總角度（即 360 度）根據(jù)定義等于 2*pi，即 6.283 左右。7:39 - 橙色取出一個(gè)弧度，正如我所說，它是圓的半徑大小。

6:02 - 這里的 r 是圓的半徑，奇怪的 0 有一條線穿過它，它是一個(gè)希臘字母，稱為 theta，是最常用的角度符號(hào)，尤其是在物理學(xué)中，但也在數(shù)學(xué)中。 Orange 使用它們來獲取半徑方向的角度，即 180 度的 pi（面向左）。

6:43 - 我不太確定 Orange 到底是如何得到三角函數(shù)的，但正弦和余弦函數(shù)與 pi 高度相關(guān)，因?yàn)閮烧叩牟ǘ紩?huì)創(chuàng)建某種“圓”并按間隔穿過 x 軸 圓周率。

7:10 - 我們?cè)俅慰吹秸液瘮?shù)乘以 i 將其旋轉(zhuǎn) 90 度

7:35 - 歐拉數(shù)在這里變成的這個(gè)符號(hào)稱為 sigma，它又是一個(gè)希臘字母。 它用來不做我所做的事情，同時(shí)在開頭解釋 e 是什么，并在末尾寫出一個(gè)帶有省略號(hào)的和。 如果存在將所有數(shù)字連接起來的規(guī)則，則它用于編寫任何大量（可能是無限的）數(shù)字的總和。 例如，要獲得最大 10 的所有自然數(shù)的總和，每個(gè)數(shù)字除以 2，您從大的 E 狀的東西（西格瑪）開始，您將開始的數(shù)字寫在西格瑪下方，從 n= 開始（在 在這種情況下 n=0，這意味著您從 0 開始），然后在 sigma 上方寫下結(jié)束的數(shù)字（在本例中為 10）。 最后，編寫公式以獲取 sigma 右側(cè)的特定數(shù)字（在本例中，我們希望每個(gè)數(shù)字除以 2，因此我們寫為 n/2）。 從現(xiàn)在起我將把它寫成 sigma（下面：n=0；上面：10；函數(shù)：n/2）。 在視頻中，我們使用 e 定義中的總和，即 e = sigma（下方：n=0；上方：無窮大；函數(shù)：1/n！）。 這 ！ 數(shù)學(xué)中的符號(hào)稱為階乘，是從 1 開始直到該數(shù)字的所有數(shù)字的乘法。 所以3！ = 1*2*3 = 6, 5！ = 1*2*3*4*5 = 120, 1! = 1 和 0！ = 1。在視頻中我們看到了不同的總和，因?yàn)榈扔诳偤偷臄?shù)字是 e^(i*pi)，而不是 e。 這就是為什么我們?cè)谝曨l中得到 sigma(下面：n=0；上面：無窮大；函數(shù)：(i*pi)^n/n!)。 還可以看到，sigma sum 射出的導(dǎo)彈其實(shí)就是一個(gè)個(gè)的 sum 的元素 ((i*pi)^0/0!, (i*pi)^1/1!, (i*pi) ^2/2！...）。

8:05 - 這實(shí)際上是數(shù)學(xué)中的一個(gè)問題。 負(fù)向量是其在向量空間中旋轉(zhuǎn)的正向量。 視頻稍后證實(shí)這是一個(gè)向量空間，我將指出這一點(diǎn)。

8:30 - Orange 將正弦除以同一角度（pi 或 180 度）的余弦，等于同一角度的正切。 用數(shù)學(xué)符號(hào)表示：sin(pi)/cos(pi) = tg(pi)。 我認(rèn)為這一切都組合成一個(gè)正切函數(shù)乘以 9i，但即使不是，它也絕對(duì)是某種函數(shù)，即使它沒有 x 參數(shù)。 如果你仔細(xì)觀察的話，你實(shí)際上可以看到武器的子彈留下了切線函數(shù)的特征。

9:03- 這里 pi 用于將半徑旋轉(zhuǎn) 180 度。

9:38 - 這證實(shí)了圓確實(shí)是一個(gè)向量空間。

10:15 - 9i 用于將圓向上移動(dòng) 9 個(gè)（還記得虛軸嗎？）。

11:26 - Orange 和 Euler 的數(shù)最終進(jìn)入虛數(shù)世界，因?yàn)樗鼈円殉艘?i。 它們最終也會(huì)逆時(shí)針旋轉(zhuǎn) 90 度，這也是它們乘以 i 的結(jié)果。

12:26 - 橙色咒語在這里使用歐拉數(shù)作為 e，乘號(hào)作為 x，i 常數(shù)作為，好吧，i，以及半覆蓋的 pi 作為 t。

13:03 - 不！ 變成 gamma 函數(shù)，我將其描述為 gamma(n - 1)。 根據(jù)伽馬函數(shù)的定義 gamma(n) = (n - 1)!。 其實(shí)這并不重要，只是解釋一下方程中奇怪的字母是什么。

13:14 - 每次使用該函數(shù)時(shí)，歐拉數(shù)都會(huì)給圓增加越來越多的“體積”。 在 YouTube 評(píng)論中解釋起來太復(fù)雜了。

13:35 - 這里不同的字母是數(shù)學(xué)中不同的常數(shù)。 我不太確定那個(gè)高彈跳的是什么，它是希臘字母 zeta，但只有 zeta 函數(shù)，而不是 zeta 常數(shù)。 左邊有一條線穿過的奇怪的 o 是 phi，著名且非常有用的黃金比例。 Phi = a/b = (a+b)/a = 約 1.618。 飛翔的字母是 delta，第一個(gè)費(fèi)根鮑姆常數(shù)，這里解釋起來太復(fù)雜，但它大約等于 4.669。 末尾的大字母是 aleph，一種無窮大（是的，數(shù)學(xué)中的無窮大有不同的類型和大?。?。 aleph 的數(shù)量是無限的，最常用的是 aleph-zero，即任意無限整數(shù)集的和

我知道對(duì)于閱讀本文的數(shù)學(xué)迷來說，這一切都非常簡(jiǎn)單，但這是一個(gè)很長(zhǎng)的評(píng)論，我只想最終完成這篇文章。