The mathematician invented a complex number system.

那位数家发明了套复杂的数字系统。

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

有点凌乱？曼德尔布罗特集合才叫有点凌乱，这简直是团糟。不好意思。

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

复杂的数字很难记住， 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?

然后他的下题是，他们有没有复数箭头？

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

事实上，量子是基于复数的。

You have two-dimensional, which is complex numbers.

你有二维，即复数。

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

要做到这点，你需要了解关于复数的定义和事实。

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

所以我喜欢复数，也喜欢数组，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.

你对这复数的模取平方，得到从状态转移到另状态的概率。

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， 将它们作为复数全部加在起，然后除以采样的点数。

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东西，它包含了复数，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数的函数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，因为它支持复数， 有数组，并且是相当高级的语言。

Described by pairs of complex numbers, which is weird.

用复数对描述，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.

但后来数家们取 -1 的平方根为叫 i 的新数字，在数领域里，这为复杂数字打开了全新的世界。

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

那么，在这组复数中，事物是如何运动的？

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

它们由对复数，即两复数来描述。

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， 素数似乎与连续的微积分世界完全无关。当复数最终混合在起时， 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.

只要引入复数到讨论中，结果就几乎是从几行代数中自然而然地显现出来了。

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

你处理两复数的题在于那相当于四实数。