That's what I meant by saying this series is as good as this function.

我说这个级数与这个函数异曲同工就在于此
It will obey all the identities known.

它符合所有的已知特性
Okay. Now, for the punch line.

现在来说一下关键点
I think everyone--Some of you may know what the punch line is,

我想大家...有些人可能知道关键是什么
the punch line is the following.

下面就是关键点
Let us introduce, without any preamble right now, the number i,

虽然没有铺垫但还是让我来介绍一下数字 i
which is the √-1.

也就是 √-1
All I expect of i is that when I squared it, I get minus 1.

我对 i 的想法就是当我平方它时得到 -1
If I cube it, i^3=(i^2)·i, so that's -i,

如果做三次方 i^3=(i^2)·i 也就是 -i
and i^4 is back to +1

而 i^4 又回到 +1
because i^4 is (i^2)·(i^2), and i^2 is -1.

因为 i^4 就是 (i^2)·(i^2) 而 i^2 是 -1
So, if you take powers of i,

因此如果你取 i 的幂