You know, real number that you can represent in the memory.

你知道的，那些能在内存中表示的实数。

You could choose any real number in the interval [0,4].

您可以选择区间[0,4]中的任何实数。

There's an infinite number of floating point real numbers in the world.

世界上有无数的浮点实数。

There was float, which is a real number with a decimal point.

有一浮点数，是一带小数点的实数。

The fact that there are no real number solutions indicates that there are no eigen vectors.

没有实数解的事实表明没有特征向量。

Similarly, there's a sensible notion for scaling a function by a real number.

类似地 有一用实数缩放函数的合理概念。

Or if I say, what's bigger, the number of real numbers or rational numbers?

或者，如我问，实数的数量与有理数的数量相比，哪更多呢？

0 minus 2 is minus 2, 3, minus 1 is 2, for t is a member of the reals.

0减2等于负2,3,减1等于2,因为t是实数的一员。

Now, Cantor says, start writing down a new real number.

康托说，现在开始写下一新的实数。

For example, one part in the final expression looks like some certain real number all times R, the reference wave.

例如， 最终表达式中的一部分看起来始终特定的实数R，即参考波。

And by the end of this process, you'll have a real number between zero and one.

在这过程结束时，你会得到一介于 0 和 1 之间的实数。

So we want to think about all possible pairs of real  numbers and the XY plane comes to mind.

所以我们想要考虑所有可能的实数对，这时就会想到XY平面。

It shows there must be more real numbers between zero and one than there are natural numbers extending out to infinity.

它表明，零和一之间的实数必定比延伸到无穷大的自然数更多。

So you each of them, x1 is a member of the reals, and x2 is a member of the reals.

所以你每人，x1 是实数的成员，x2 是实数的成员。

So if I say to you, what's bigger, the number of water molecules in this thing, or the number of real numbers?

所以， 如我问你， 是这东西里水分子的数量多，还是实数的数量多？

Our first character is Linus the Linelander, whose mind can only grasp the one-dimensional geometry of lines and the algebra of real numbers.

我们的第一角色是莱纳斯线国居民，他的思维只能理解直线的一维几何和实数的代数。

You start thinking about Penrose and Game of Life stuff about the generativity of math itself, like the hyperreal numbers, things like that.

你开始思考彭罗斯和生命游戏等关于数学自身生成性的东西，比如超实数之类的事物。

So an integer or a real number you'd expect to square it and it to be a positive value or perhaps zero right?

所以对于一整数或实数，它的平方你会认为结是一非负数，对吗？

And now all of these points for any arbitrary-- t is a member of the real numbers, will lie on this green line.

现在，对于任何任意的 - t 是实数的成员，所有这些点都将位于这条绿线上。

We have a complete infinite list with each integer acting like an index number, a unique identifier for each real number on the list.

我们有一完整的无限列表，其中每整数都充当索引号，是列表上每实数的唯一标识符。