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

您可选择区间[0,4]中的任何实数。

So mapping a certain token for text or image or actions, each of these tokens gets its own little vector of real numbers that represents this.

因此， 于文本、图像或动作中的某个特定标记进行映射时，每个标记都会获得个属于自己的小实数向量， 用表示该标记。

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

我们的第个角色是莱纳斯线国居民，他的思只能理解直线的何和实数的代数。

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

有个浮数，是个带小数的实数。

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

所于个整数或实数，它的平方你会认为结果是个非负数，吗？

It could be P2 minus P1-- because this can take on any positive or negative value-- where t is a member of the real numbers.

它可是 P2 减去 P1 - 因为它可取任何正值或负值 - 其中 t 是实数的成员。

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

在这个过程结束时，你会得到个介于 0 和 1 之间的实数。

To motivate the case with functions that have 2d inputs and 2d outputs, let's start off simpler, with functions that just take in a real number and spit out a real number.

为了说明具有二输入和二输出的函数的情况，我们先从简单的开始， 即那些仅接收实数并输出实数的函数。

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

所， 如果我问你， 是这东西里水分子的数量多，还是实数的数量多？

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

它表明，零和之间的实数必定比延伸到无穷大的自然数更多。

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。

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

世界上有无数的浮实数。

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，于任何任意的 - t 是实数的成员，所有这些都将位于这条绿线上。

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

或者，如果我问，实数的数量与有理数的数量相比，哪个更多呢？

If we can do that with none left over, well then we know that the set of natural numbers and the set of real numbers between 0 and 1 are the same size.

如果我们能做到这，那么我们就知道自然数的集合和 0 到 1 之间的实数集合是相同的大小。

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

所我们想要考虑所有可能的实数，这时就会想到XY平面。

And since we're talking about real values, it's going to be ordered list of real numbers, and a 2-tuple just says it's an ordered list of 2 numbers.

既然我们谈论的是实数值，它就会是实数的有序列表，而个 2 元组只是说它是个 2 个数字的有序列表。

But to check this logic, Cantor imagined writing down an infinite list, matching up each natural number on one side with a real number between 0 and 1 on the other.

但为了检验这个逻辑， 康托尔想象写下个无限的列表，边的每个自然数与另边的 0 到 1 之间的个实数相匹配。

As a next step, let's turn our attention  towards continuous distributions where your random variable can take on values anywhere  in an infinite continuum, like all possible real numbers.

接下来， 让我们关注连续分布， 其中随机变量可在无限连续区间内取任意值，比如所有可能的实数。

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。