The mathematician invented a complex number system.

那位数学家发明了一套复杂的数字系统。

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。

You have two-dimensional, which is complex numbers.

你有二维，即复数。

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

事实上，量子力学是基于复数的。

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。

The story here braids together prime numbers, complex numbers and pi in a very pleasing trio.

这里的故事将质数、复数和圆周率编织一个非常令人愉快的三重奏。

It refers to functions that have complex number inputs and complex number  outputs and which you can also take a derivative of.

它指的是具有复数输入和复数输出的函数，您也可以对其求导。

For some viewers, this all might  leave the lingering question of why exactly complex numbers are so  unreasonably useful in this way.

对于一些观众来说，这一切可能会留下一个挥之不去的问题：为什么复数以这种方式如此不合理地有用。

And here we're evaluating just one of those terms, f(zeta), which essentially  comes down to a product of five complex numbers.

在这里，我们只评估其中一项，f(zeta)， 它本质上可以归结为五个复数的乘积。

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.

但后来数学家们取 -1 的平方根为一个叫 i 的新数字，在数学领域里，这为复杂数字打开了全新的界。

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一起，然后除以采样的点数。

Each component in this superposition is given with an amplitude and a phase often encoded as a single complex number.

这种叠加中的每个分量都有一个幅度和一个相位，通常编码为单个复数。

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

这是利用输入和输出为复数的函数行微积分的研究。

Now just multiply this by your point, represented as a complex number.

现在只需将其乘以您的点（表示为复数）。

The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number.

光子仍然被描述为两个振荡分量的叠加，每个振荡分量都有一定的相位和振幅，并且这些分量通常使用单个复数行编码。

For example, complex numbers are two-dimensional, and the tool we develop here is perfect for finding solutions to complex equations.

例如， 复数是二维的，我们在这里开发的工具非常适合寻找复杂方程的解。

But before I do, I want to lead with a spoiler, which is the fact  that the way we're going to solve this involves the use of complex numbers.

但在此之前， 我想先剧透一下，那就是我们解决这个问题的方法涉及使用复数。

We have strings numbers and boolean Numbers can be integers floats or complex numbers Here's the second question. You have this variable fruit set to Apple. What do you think?

我们有字符串数字和布尔数字可以是整数浮点数或复数这是第二个问题。您将这个变量 fruit 设置为 Apple。你怎么认为？

I find it so satisfying to learn something new, especially when it's something that I never could've imagined having a grasp on like calculus or complex numbers or quantum science.

我发现学习新东西非常令人满意，尤其是当我从未想象过能够掌握微积分、复数或量子科学等东西时。

It's certainly not the only time that complex  numbers are unreasonably useful for discrete math to borrow a phrase.

借用一个短语，这当然不是唯一一次复数对于离散数学来说非常有用。