They're just adding integers on each downbeat.

只是在每个强拍上加一个整数。

He labeled these hypothetical integers p and q.

他將這兩個假設的整數 命名為 p 及 q 。

You go ahead, ask! Bosons have integer spin, fermions have half-integer spin!

来啊，你问吧！玻色子自旋整数，费米子自旋半整数！

Thus, p could be expressed as 2a, where a is an integer.

因此，p 可以 2a ，a 是一個整數。

You might say, imagine somebody who spends his eternity counting the integers--1, 2, 3, 4, 5, 6.

比方说 有人用他无限的生命数整数 一二三五六。

And any positive integer, up to a certain limit, can be expressed as a sum of these numbers.

在一定范围内的任何正整数，都可以这些数字之和。

Programming languages have some undefined behaviors, such as what to do with misaligned pointers or when encountering signed integer overflows.

编程语言有一些未定义行，例如如何处理偏离的指针以及遇到带符号整数溢出时该怎么办。

The first two aren't very interesting: all ones, and then the positive integers, also known as natural numbers.

头两条并不是很有趣，全都是 1。接下来是正整数，也被称自然数。

But with integers or int you know the computer representation of integers it's not so clear

但是对于整数或者 int 类型（计算机中的整数类型）就不是很清楚了

But it still is obviously not the integer product of those four numbers

但它显然不是这个数的整数乘积

So that's a an example of where your normal expectations about integer arithmetic may or may not hold

这是个例子说明你对整数运算的正常期望可能成立，也可能不成立

So what you can say from this is integer arithmetic is commutative and associative

所以你可以说，整数运算符合交换律和结合律

It's not a normal sort of mathematical integer arithmetic

这不是一种正常的整数运算

And what the compromise is you can overflow an integer and run out of room

折衷的办法是你可以溢出一个整数然后耗尽空间

Interestingly, even though we can't express irrational numbers as ratios of integers, it is possible to precisely plot some of them on the number line.

有趣的是，即便我们无法将无理数称整数的比例，我们却可以将它准确现在图形之中。

And as you probably might have figured both in the integer arithmetic, even though as this potential for overflow

你可能在整数运算中计算过了，尽管（乘法和加法）两种运算都有溢出风险

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

所以对于一个整数或实数，它的平方你会认结果是一个非负数，对吗？

On the other hand one thing you'll find is even though this arithmetic is not a normal sort of integer arithmetic

另一方面，你会发现尽管这个运算不是一种合理的整数运算

So for example if I define a struct that contains a two integer values A in an array

因此，例如，如果我定义一个结构，其中 a 数组中包含两个整数值

According to Pythagoras Theorem, the diagonal length would be square root of two, but try as he might, Hippasus could not express this as a ratio of two integers.

根据毕达哥拉斯的理论，这个正方形的对角线长度应该根号二，但是无论希帕索斯如何尝试 都不能将根号二变两个整数的比例形式。