The mathematician invented a complex number system.

那位发明了一套复杂的字系统。

A little messy? The Mandelbrot set of complex numbers is a little messy. This is chaos. Excuse me.

有一点凌乱？曼德尔布罗特集合才叫有一点凌乱，这简直是一团糟。不好意思。

Complex numbers hard to remember, so systems have been created to simplify geocodes.

复杂的字很难记住， 因此已经创建了系统来简化地理编码。

And his next question was, did then have a complex number of arrows?

然后他的下一个问题是，他有没有复个箭头？

And in fact, quantum mechanics is based on complex numbers.

事实上，量子是基于复的。

You have two-dimensional, which is complex numbers.

你有二维，即复。

To do so, what you need to know about complex numbers is one definition and one fact.

要做到这一点，你需要了解关于复的一个定义和一个事实。

So I love complex numbers, and I love arrays, so I looked at Python.

所以我喜欢复，也喜欢组，所以我看了看Python。

You square the norm of this complex number, gives you a probability to go from one to the other.

你对这个复的模取平方，得到从一个状态转移到另一个状态的概率。

That is, add them all together, as complex numbers, and then divide by the number of points that you've sampled.

也就是说， 将它复全部加在一起，然后除以采样的点。

So I was looking for that and it had complex numbers, a lot of programming languages.

所以我就在寻找这样一个东西，它包含了复，很多编程语言都有。

Which is the study of doing calculus with functions whose inputs and outputs are complex numbers.

这是研究对输入和输出都是复的函进行微积分运算的科。

Fortran 90 probably is my favorite Fortran, because it's got complex numbers, got arrays, and it's pretty high level.

Fortran 90 可能是我最喜欢的 Fortran 版本，因它支持复， 有组，并且是相当高级的语言。

Described by pairs of complex numbers, which is weird.

用复对描述，这很奇怪。

But then mathematicians defined the square root of negative one as a new number called i, opening up a whole new mathematical world of complex numbers.

但后来取 -1 的平方根一个叫 i 的新字，在领域里，这复杂字打开了全新的世界。

So, how do things move in this pair of complex numbers?

那么，在这组复中，事物是如何运动的？

And they're described by pairs of complex numbers, by two complex numbers.

它由一对复，即两个复来描述。

I mean, prime numbers seem wholly unrelated to the continuous world of calculus; Much less when complex numbers end up in the mix.

我的意思是， 素似乎与连续的微积分世界完全无关。当复最终混合在一起时， 情况就更糟了。

For the small price of introducing complex numbers into the discussion, the result kind of just pops out from a few lines of algebra.

只要引入复到讨论中，结果就几乎是从几行代中自然而然地显现出来了。

And your problem with two complex numbers is that's four real numbers.

你处理两个复的问题在于那相当于四个实。