The mathematician invented a complex number system.

那位数学家发明了一套复杂的数字系统。

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，为复杂数字打开了全新的世界。

This letter is of a particular interest, partly because it's in code, a not very complex number for letter substitution, but also because it was written at the turning point of Napoleon's career.

份书信是用密码写的，用的字母替换法也不难理解，而且书写封信时，拿破仑正处在事业的转折点，因此封信很有趣。

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

有一点？曼德尔布罗特集合才叫有一点，简直是一团。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地理编码。

The more math you do the less crazy that seems because complex numbers  have this bizarre relationship with discrete math, but it really is wonderful there's no two  ways about it.

你做的数学越多， 就越觉得不那么疯狂， 因为复数与离散数学之间有着种奇特的关系，但真的非常美妙， 一点毋庸置疑。

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足，特别是当我从未想过自己能掌握些东西时， 比如微积分、复数或量子科学。

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是第二个问题。您将个变量 fruit 设置为 Apple。你怎么认为？

These were people that were on the mailing list, and they were just asking for things like, hey, we really should have complex numbers in this language.

些是邮件列表中的人，他们只是在询问一些事情， 比如，嘿， 我们真的应该在个语言中加入复数。

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

那么，在组复数中，事物是如何运动的？

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

当然不是复数不合理的适用于离散数学的唯一时刻，借用一下句话。

And there were bugs on how the Windows compiler handled complex numbers.

而且在Windows编译器处理复数的方式上存在一些bug。

And the reason for talking in terms of complex numbers, rather than just saying, " It has two coordinates, " is that complex numbers lend themselves to really nice descriptions of things that have to do with winding, and rotation.

而用复数而不是简单地说“它有两个坐标”来描述，是因为复数能够很好地描述与缠绕和旋转相关的事物。

You're really gonna use the fact that these are the complex numbers.

你真的要利用些是复数个事实。

For example, there's an extension of the complex numbers that naturally live up in four dimensions, they're called the quaternions, and they offer a really elegant way to encode and to work with three dimensions.

例如，复数有一个扩展， 自然存在于四维空间中，它们被称为四元数， 并且提供了一种非常优雅的方式来编码和处理三维空间。

This might seem like a strange step if you don't have much of a history with complex numbers, but describing this distance as a product can be unexpectedly useful.

一步可能会显得很奇怪， 特别是如果你对复数了解不多的话，但将个距离描述为一个乘积可能会意外地有用。

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

所以我就在寻找样一个东西，它包含了复数，很多编程语言都有。

Anyway, it's a bit of a long story, but the one problem is that you have to start, it's crucial that you think in terms of complex numbers and not just real numbers.

总之，有点是个长故事， 但问题之一是，你必须从复数的角度开始思考， 而不仅仅是实数。

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，可能会留下一个挥之不去的问题：复数为什么在种情况下如此不合理地有用。

And  the real reason that complex numbers, as opposed to some other structure, proved to  be useful for us is that we could find a value so that successive products have this cycling  behavior.

与其他结构相比，复数被证明对我们有用的真正原因是， 我们可以找到一个值， 以便连续的产品具有种循环为。