chapter 6 Generalizations of number

GEBERALIZATIONS OF NUMBER

ONE GREAT peculiarity of mathematics is the set of allied ideas which have been invented in connexion with the integral numbers from which we started. These ideas may be called extensions or generalizations of number. In the first place there is the idea of fractions. The earliest treatise on arithmetic which we possess was written by an Egyptian priest, named Ahmes, between 1700 B.C. and 1000 B.C. , and it is probably a copy of a much older work. It deals largely with the properties of fractions. It appears, therefore, that this concept was developed very early in the history of mathematics. Indeed the subject is a very obvious one. To divide a field into three equal parts, and to take two of the parts, must be a type of operation which had often occurred. Accordingly, we need not be surprised that the men of remote civilizations were familiar with the idea of two-thirds, and with allied notions. Thus as the first generalization of number we place the concept of fractions. The Greek thought of this subject rather in the form of ratio, so that a Greek would naturally say that a line of two feet in length bears to line of three feet in length the ratio of 2 to 3. Under the influence of our algebraic notation we would more often say that one line was two-thirds of the other in length, and would think of two-thirds as a numerical multiplier. 数学的一个显著特征是与我们开始时的整数相关联的一系列理念。 这些理念可以称为数字的扩展或概括。 首先是分数的概念。 我们所拥有的最早的算术著作是由一位名叫阿赫梅斯的埃及祭司在公元前1700年到公元前1000年之间写成的，这很可能是一本更古老作品的抄本。 这本书主要讲述了分数的性质。因此，这个概念显然是在数学史上很早就发展起来的。 的确，这个主题非常明显。 将一个领域划分为三个相等的部分，并取走其中的两个部分，必定是一种经常发生的操作。 因此，我们不必惊讶于远古文明的人们熟悉二分之三的概念以及相关的观念。 因此，作为数字的第一次概括，我们将分数的概念置于首位。 古希腊人更倾向于以比率的形式来思考这个主题，因此古希腊人自然会说，长度为两英尺的线段与长度为三英尺的线段之间的比率是2比3。在我们的代数记号的影响下，我们更常会说一条线的长度是另一条的二分之三，并将二分之三视为一个数值乘数。

In connexion with the theory of ratio, or fractions, the Greeks made a great discovery, which has been the occasion of a large amount of philosophical as well as mathematical thought. They found out the existence of 'incommensurable' ratios. They proved, in fact, during the course of their geometrical investigations that, starting with a line of any length, other lines must exist whose lengths do not bear to the original length the ratio of any pair of integers—or, in other words, that lengths exist which are not any exact fraction of the original length.

与比率或分数理论相关，古希腊人做出了一个重大发现，这引发了大量的哲学和数学思考。他们发现了“不可通约”比率的存在。实际上，他们在几何研究的过程中证明，从任何长度的线段出发，必定存在其他线段，其长度与原始长度之间的比率不等于任何一对整数的比率——换句话说，存在的长度不是原始长度的任何精确分数。

For example, the diagonal of a square cannot be expressed as any fraction of the side of the same square; in our modern notation the length of the diagonal is tomes the length of the side. But there is no fraction which exactly represents . We can approximate to as closely as we like, but we never exactly reach its value. For example, is just less than 2, and is greater than 2, so that lies between and . But the best systematic way of approximating to in obtaining a series of decimal fractions, each bigger than the last, is by the ordinary method of extracting the square rot ; thus the series is , and so on.

例如，一个正方形的对角线无法用该正方形的边长的任何分数表示；在我们现代的符号中，对角线的长度是边长的 倍。但没有任何分数能准确地表示 。我们可以尽可能接近地逼近 ，但永远无法准确达到它的值。例如， 略小于 2，而 大于 2，因此 位于 和 之间。但获得一系列逐渐增大的十进制分数，最好的系统方法是通过普通的提取平方根的方法来逼近 ；因此，这个序列是 ，依此类推。

Ratios of this sort are called by Greeks incommensurable. They have excited from the time of the Greeks onwards a great deal of philosophic discussion, and the difficulties connected with them have only recently been cleared up.

这种比率在希腊人看来是不可度量的。从古希腊时期以来，它们引发了大量的哲学讨论，与之相关的难题直到最近才得到解决。

We will put the incommensurable ratios with the fractions, and consider the whole set of integral numbers, fractional numbers, and incommensurable numbers as forming one class of numbers which we will call 'real numbers'. We always think of the real numbers as arranged in order of magnitude, starting from zero and going upwards, and becoming indefinitely larger and larger as we proceed. The real numbers are conveniently represented by points on a line. Let OX be any line bounded at O and stretching away indefinitely in the direction OX.

我们将不可度量的比率与分数结合起来，并将整个整数、分数和不可度量数的集合视为一个数类，我们称之为“实数”。我们总是将实数视为按大小顺序排列，从零开始，向上增长，并在我们继续的过程中变得越来越大。实数可以方便地用线上的点表示。设 OX 为任何一条在 O 点有界，并且在 OX 方向无限延伸的直线。

Take any convenient point , A , on it , so that OA represents the unit length; and divide off lengths AB, BC, CD, and so on, each equal to OA . Then the point O represents the number 0, A the number 1, B the number 2, and so on. In fact the number represented by any point is the measure of its distance from O, in terms of the unit length OA. The point between O and A represent the proper fractions and the incommensurable numbers less than 1; the middle point of OA represents , that of AB represents , that of BC represents , and so on. In this way every point on OX represents some one real number, and every real number is represented by some one point on OX.

取任意一个方便的点 A，使得 OA 代表单位长度；并划分出长度 AB、BC、CD 等，每段长度均等于 OA。那么点 O 代表数字 0，点 A 代表数字 1，点 B 代表数字 2，等等。实际上，任何点所代表的数字是它与 O 之间距离的度量，以单位长度 OA 为单位。O 和 A 之间的点表示小于 1 的真分数和不可度量数；OA 的中点代表 ，AB 的中点代表 ，BC 的中点代表 ，依此类推。通过这种方式，OX 上的每个点都代表一个实数，而每个实数也由 OX 上的某个点表示。

The series (or row) of points along OX , starting from O and moving regularly in the direction from O to X, represents the real numbers as arranged in an ascending order of size, starting from zero and continually increasing as we go on.

沿着 OX 的一系列（或行）点，从 O 开始，朝着从 O 到 X 的方向均匀移动，表示按大小升序排列的实数，从零开始，随着我们的继续而不断增加。

All this seems simple enough, but even at this stage there are some interesting ideas to be got at by dwelling on these obvious facts. Consider the series of points which represent the integral numbers only, namely, the points, O,A,B,C,D, &c. Here there is first point O, a definite next point, A, and each point, such as A or B, has one definite immediate predecessor and one definite immediate successor, with the exception of O, which has no predecessor; also the series goes on indefinitely without end. This sort of order is called the type of order of the integers; its essence is the possession of next-door neighbours on either side with the exception of No.1 in the row. Again consider the integers and fractions together, omitting the points which correspond to the incommensurable ratios. The sort of serial order which we now obtain is quite different. There is a first term O; but no term has any immediate predecessor or immediate successor. This is easily seen to be the case, for between any two fractions we can always find another fraction intermediate in value. One very simple way of doing this is to add the fractions together and to halve the result. For example, between and , the fraction , that is , lies; and between and the fractions , that is , lies; and so on indefinitely; Because of this property the series is said to be 'dense'. There is no end point to the series, which increases indefinitely without limit as we go along the line OX. It would seem at first sight as though the type of series got in this way from the fractions, always including the integers, would be the same as that got from all the real numbers , integers, fractions, and incommensurables taken together, that is from all the points on the line OX. All that we have hitherto said about the series of fractions applies equally well to the series of all real numbers. But there are important differences which we now proceed to develop. The absence of the incommensurables from the series of fractions leaves an absence of end points to certain classes. Thus, consider the incommensurable . In the series of real numbers this stands between all the numbers whose squares are less that 2, and all the numbers whose squares are greater that 2. But keeping to the series of fractions alone and not thinking of the incommensurables, so that we cannot bring in , there is no fractions which has the property of dividing off the series into two parts in this way, i.e. so that all the members on one side have their squares less that 2, and on the other side greater that 2. Hence in the series of fractions there is a quasi-gaps where ought to come. This presence of quasi-gaps in the series of fractions may seen a small matter; but any mathematician, who happens to read this , knows that the possible absence of limits or maxima to a class of numbers, which yet does not spread over the whole series of numbers, is no small evil. It is to avoid this difficulty that recourse is had to the incommensurables, so as to obtain a complete series with no gaps.

这一切看起来相当简单，但即使在这个阶段，通过深入思考这些显而易见的事实，仍然可以得出一些有趣的观点。考虑仅表示整数的点的系列，即点 O、A、B、C、D 等等。在这里，首先是点 O，接下来是明确的点 A，每个点，例如 A 或 B，都有一个明确的直接前驱和一个明确的直接后继，除了 O，它没有前驱；而且该系列无穷无尽地延续下去。这种顺序称为整数的顺序类型；其本质是在两侧各有相邻的邻居，唯一的例外是序列中的数字 1。再考虑整数和分数一起，省略对应于不可分比的点。我们现在得到的这种序列顺序是完全不同的。首先有一个项 O；但没有任何项有直接的前驱或直接的后继。这一点很容易看出，因为在任何两个分数之间，我们总能找到另一个在数值上处于中间的分数。实现这一点的一个非常简单的方法是将两个分数相加并将结果对半。比如，在 和 之间，分数 ，即 ，位于其中；在 和 之间，分数 ，即 ，也位于其中；如此继续下去；由于这一特性，该系列被称为“密集”。这个系列没有终点，随着我们沿着 OX 线向前走而无限增加。乍一看，似乎从分数中获得的这种类型的序列（始终包括整数）与从所有实数、整数、分数和不可分比综合而成的序列是相同的，也就是来自 OX 线上的所有点。我们迄今所说的关于分数系列的所有内容同样适用于所有实数系列。但现在我们要展开讨论的重要差异。不可分数在分数系列中的缺失导致某些类别缺乏终点。因此，考虑不可分数 。在实数系列中，它介于所有平方小于 2 的数字和所有平方大于 2 的数字之间。但仅仅考虑分数系列，而不考虑不可分数，这样我们无法引入 ，就没有任何分数具备将系列分成两部分的属性，即使一侧所有成员的平方小于 2，另一侧的平方大于 2。因此，在分数系列中存在一个准缺口， 应该出现在此。分数系列中准缺口的存在似乎是个小问题；但任何偶然阅读此文的数学家都知道，某类数字可能缺乏极限或极大值，而该类数字又不覆盖整个数字系列，这并不是个小恶。正是为了避免这一困难，才诉诸于不可分数，以便获得一个没有缺口的完整系列。

There is another even more fundamental difference between the two series. We can rearrange the fractions in a series like that of the integers, that is , with a first term, and such that each term has an immediate successor and (except the first term) an immediate predecessor. We can show how this can be done. Let every term in the series of fractions and integers be written in the fractional form by writing for 1, for 2, and so on for all the integers, excluding 0. Also, for the moment we will reckon fractions which are equal in value but not reduced to their lowest terms as distinct; so that , for example, until further notice, , , , , &c., are all reckoned as distinct. Now group the fractions into classes by adding together the numerator and denominator of each term. For the sake of brevity call this sum of the numerator and denominator of a fraction its index. Thus 7 is the index of , and also of , and of . Let the fractions in each class be all fractions which have some specified index, which may therefore also be called the class index. Now arrange these classes in the order of magnitude of their indices. The first class has the index 2 , and its only member is ; the second class has the index 3, and its members are and ; the third class has the index 4 , and its members are , , ; the fourth class has the index 5. and its members are , , , ; and so on. It is easy to see that the number of members (still including fractions not in their lowest terms) belonging to any class is one less than its index. Also the members of any one class can be arranged in order by taking the first member to be the fraction with numerator 1. the second member to have the numerator 2, and so on, up to (n-1) where n is the index. Thus for the class of index n, the members appear in the order

翻译：在这两个序列之间还有一个更根本的区别。我们可以像整数那样重新排列分数序列，也就是说，每个序列都有一个首项，并且每个项都有一个直接后继（除了首项外，还有一个直接前驱）。我们可以展示这如何实现。让分数和整数序列中的每一项都以分数形式书写，将1写作 ，将2写作 ，依此类推，所有整数都如此，0 除外。此外，暂时我们将值相等但未化简为最简形式的分数视为不同的；例如，直到另行通知，、、、 等都被视为不同的。现在通过将每个项的分子和分母相加，将这些分数分组为不同的类别。为了简洁起见，将分数的分子和分母之和称为其指标。因此， 的指标是7， 的指标也是7， 的指标也是7。让每个类别中的分数都是具有某个特定指标的分数，因此也可以称其为类别指标。现在按其指标的大小顺序排列这些类别。第一个类别的指标为2，其唯一成员是 ；第二个类别的指标为3，其成员是 和 ；第三个类别的指标为4，其成员是 、、；第四个类别的指标为5，其成员是 、、、；依此类推。很容易看出，属于任何类别的成员数量（仍包括未化简的分数）比其指标少1。此外，任何一个类别的成员可以按顺序排列，第一项是分子为1的分数，第二项的分子为2，以此类推，直到（n-1），其中n是指标。因此在n的那类中，其成员表现的顺序为

The members of the first four classes have in fact been mentioned in this order. Thus the whole set of fractions have now been arranged in an order like that of the integers. It runs thus

前四个类别的成员实际上已经按此顺序提到。因此，整个分数集现在已经按照类似于整数的顺序排列。它的顺序如下：

and so on.

等等

Now we can get rid of all repetitions of fractions of the same value by simply striking them out whenever they appear after their first occurrence. In the few initial terms written down above, which is enclosed above in square brackets is the only fraction not in its lowest terms. It has occurred before as . Thus this must be struck out. But series is still left with the same properties, namely, (a) there is a first term, (b) each term has next-door neighbours, (c) the series goes on without end.

现在我们可以通过在每次出现后简单地划掉相同值的分数来消除所有重复的分数。在上面写下的几个初始项中，（上面用方括号括起来）是唯一一个未化简为最简形式的分数。它之前出现过，表示为 。因此，这个分数必须被划掉。但是，序列仍然保留着相同的性质，即：（a）存在一个首项，（b）每个项都有邻近的项，（c）序列是无止境的。

It can be proved that it is not possible to arrange the whole series of real numbers in this way. This curious fact was discovered by Georg Cantor, a German mathematician still living; it is of the utmost importance in the philosophy of mathematical ideas. We are here in fact touching on the fringe of the great problem of the meaning of continuity and of infinity.

可以证明，无法以这种方式排列整个实数序列。这个奇妙的事实是由德国数学家乔治·康托尔发现的(使用对角线论证)，他仍然健在；这一发现对数学思想的哲学有着极其重要的意义。实际上，我们在这里触及了关于连续性和无限性这一伟大问题的边缘。

Another extension of number comes from the introduction of the idea of what has been variously named an peration or a step, names which are respectively appropriate from slightly different points of view. We will start with a particular case. Consider the statement 2+3=5 . we add 3 to 2 and obtain 5. Think of the peration of adding 3: let this be denoted by +3. Again 4-3=1. Think of the operation of subtracting 3: let this be denoted by -3. Thus instead of considering the real numbers in themselves, we consider the perations of adding or subtracting themL instead of , we consider nad , namely the perations of adding and of subtracting . Then we can add these operations , of course in a different sense of addition to that in which we add numbers. The sum of two operations id the single operation which has the same effect as the two operations applied successively. In what prder are two operations to be applied? Then answer is that it is indifferent, since for example

数字的另一个扩展源于引入一个被称为操作或步骤的概念，这两个名称在稍微不同的角度上是各自合适的。我们将从一个特定的案例开始。考虑这个陈述 2 + 3 = 5。我们将 3 加到 2 上，得到 5。想想加 3 的操作：我们用 +3 来表示它。再来看 4 - 3 = 1。想想减去 3 的操作：我们用 -3 来表示它。因此，我们不再单纯考虑实数本身，而是考虑加法或减法的操作：我们不再仅仅考虑 ，而是考虑 和 ，也就是加上 和减去 的操作。那么我们可以将这些操作相加，当然是在一种不同于数字相加的意义上。两个操作的和是单个操作，其效果与这两个操作依次应用的效果相同。那么这两个操作的应用顺序是什么？答案是无所谓，因为例如 so that the addition of the steps +3 and +1 is commutative.

因此，步骤 +3 和 +1 的加法是交换的。

Mathematicians have a habit, which is puzzling to those engaged in tracing out meanings , but is very convenient in practice, of using the same symbol in different though allied senses, The one essential requisite for a symbol in their eyes is that , whatever its possible varieties of meaning, the formal laws for its use shall always be the same. In accordance with this habit the addition of operations id denoted by + as well as the addition of numbers. Accordingly we can write

数学家们有一种习惯，这让那些致力于追寻意义的人感到困惑，但在实践中却非常方便，那就是在不同但相关的意义上使用相同的符号。他们眼中对一个符号的唯一基本要求是，无论其可能的意义有多种，其使用的形式法则始终保持一致。根据这种习惯，操作的加法用 + 表示，数字的加法也是如此。因此，我们可以写 where the middle + on the left-hand side denotes the addition of the operations +3 and +1. But, furthermore, we need not be so very pedantic in our symbolism, except in the rare instances when we are directly tracing meanings; this we always drop the first + of a line and the brackets, and never write two + signs running. So the above equation becomes

其中，中间的 + 表示操作 +3 和 +1 的加法。此外，除了在我们直接追寻意义的罕见情况下，我们不必在符号上如此拘泥；因此，我们总是省略一行中的第一个 + 和括号，从不连续写两个 + 符号。因此，上面的方程变为 which we interpret as simple numerical addition, or as the more elaborate addition of operations which is fully expressed in the previous way of writing the equation, or lastly as expressing the result of applying the operation +1 to the number 3 and obtaining the number 4. Any interpretation which is possible is always correct. But the only interpretation which is always possible, under certain conditions , is that of operations. The other interpretations often give nonsensical results.

我们将其解读为简单的数字加法，或者作为更复杂的操作加法，这在之前的方程书写方式中得到了充分表达，最后，作为将操作 +1 应用到数字 3 上并得到数字 4 的结果。任何可能的解释都是正确的。但在某些条件下，唯一总是可能的解释是操作的解释。其他的解释常常会给出荒谬的结果。(1+1=2,一个东西加另一个东西等于两个东西，但是在，一杯大豆和一杯核桃混合的条件下，却不能得到两杯，混合物。所以，我们聚焦于操作本身，即1+1=2才是唯一正确的解释)。

This leads us at once to a question, which must have been rising insistently in the reader's mind: What is the use of all this elaboration? At this point our friend, the practical man , will surely step in and insist on sweeping away all these silly cobwebs of the brain. The answer is that what the mathematician is seeking is Generality. This is an idea worthy to be placed beside the notions of the Variable and of Form so far as concerns its importance in governing mathematical procedure. Any limitation whatsoever upon the generality of theorems, or of proofs, or of interpretation is abhorrent to the mathematical instinct. These three notions , of the variable, of form, and of generality , compose a sort of mathematical trinity which preside over the whole subject. They all really spring from the same root, namely from the abstract nature of the science.

这立刻引出了一个问题，这个问题在读者的脑海中必定已不断升起：这一切细致的阐述有什么用？在这个时候，我们的朋友——实用主义者，肯定会插手，并坚持要扫除这些愚蠢的脑中蛛网。答案是，数学家所追求的是普遍性。这是一个值得与变量和形式的概念并列的重要思想，因为它在管理数学过程中的重要性。对定理、证明或解释的任何限制都是令人厌恶的，违背了数学的本能。这三种概念——变量、形式和普遍性，构成了一种数学三位一体，主导着整个学科。它们实际上都源自于这门科学的抽象本质。

Let us see how generality is gained by the introduction of this idea of operations. Take the equation x+1=3; the solution is x=2. Here we can interpret our symbols as mere numbers, and the recourse to 'operations' is entirely unnecessary. But, if x is a mere number, the equation x+3=1 is nonsense. For x should be the number of things which remain when you have taken 3 things away from 1; and no such procedure is possible. At this point our idea of algebraic form steps in, itself only generalization under another aspect. We consider, therefore, the general equation of the same form as x+1=3. This equation is x+a=b, and its solution is x=b-a. Here our difficulties become acute; for this form can only be used for the numerical interpretation so long as b is greater that a , and we cannot say without qualification the a and b may be any constants. In other words we have introduced a limitation on the variability of the 'constants' a and b, which we must drag like a chain throughout all our reasoning. Really prolonged mathematical investigations would be impossible under such conditions. Every equation would at last be buried under a pile of limitations. But if we now interpret our symbols as 'operations', all limitation vanished like magic. The equation x+1=3 gives x=+2, the equation x+3=1 gives x=-2, the equation x+=b gives x=b-a which is an operation of addition or of subtraction, for the rules of procedure with the symbols are the same in either case.

让我们看看通过引入运算这一概念如何获得一般性。考虑方程 x+1=3；解为x=2。在这里，我们可以将符号解释为简单的数字，而对“运算”的引用完全是多余的。但是，如果 x 只是一个数字，方程x+3=1 就毫无意义。因为 x 应该是从1中取出3后剩下的物体数量，而这样的过程是不可能的。在这一点上，我们对代数形式的理解介入了，而代数形式本身只是一种在另一种形式下的概括。因此，我们考虑与 x+1=3 形式相同的一般方程。这个方程是 x+a=b，它的解是 x=b−a。在这里，我们的困难变得尖锐；因为这个形式只能在 b 大于 a 时用于数值解释，而我们不能无条件地说 a 和 b 可以是任何常数。换句话说，我们对“常数” a 和 b 的可变性引入了限制，这个限制我们必须像链条一样拖曳在所有的推理中。在这样的条件下，真正深入的数学研究是不可能的。每个方程最终都会被一堆限制埋没。但是，如果我们现在将符号解释为“运算”，所有的限制就像魔法一样消失了。方程 x+1=3 给出 x=+2，方程 x+3=1 给出x=−2，方程x+=b 给出 x=b−a，这是一种加法或减法运算，因为在这两种情况下，符号的操作规则是相同的。

It does not fall within the plan of this work to write a detailed chapter of elementary algebra. Our object is merely to make plain the fundamental ideas which guide the formation of the science. Accordingly, we do not further explain the detailed rules by which the 'positive and negative numbers' are multiplied and otherwise combined. We have explained above that positive and negative numbers are operations. They have also been called 'steps'. Thus +3 is the step by which we go from 2 to 5, and -3 is the step by which we go from 5 to 2. Consider the line OX divided in the way explained in the earlier part of the chapter , so that its points represent numbers. The +2 is the step from O to B, or from A to C, or (if the divisions are taken backwards along OX') from C' to A', or from D' to B', and so on. Similarly -2 is the step from O to B', or from B' to D', or from B to O, or from C to A.

本书的计划并不包括写一章详细的初等代数。我们的目标仅仅是阐明指导这一科学形成的基本思想。因此，我们不再进一步解释“正数和负数”相乘及其他组合的详细规则。我们在上文中已解释，正数和负数是运算。它们也被称为“步长”。因此，+3 是从 2 到 5 的步长，而 -3 是从 5 到 2 的步长。考虑线 OX，按照本章早些时候解释的方式进行划分，使其点代表数字。+2 是从 O 到 B 的步长，或从 A 到 C 的步长，或（如果沿着 OX' 反向划分）从 C' 到 A' 的步长，或从 D' 到 B' 的步长，等等。类似地，-2 是从 O 到 B' 的步长，或从 B' 到 D' 的步长，或从 B 到 O 的步长，或从 C 到 A 的步长。

We may consider the point which is reached by a step from O, as representative of that step. Thus A represents +1, B represents +2, A' represents -1, B' represents -2, and so on. It will be noted that, whereas previously with the mere 'unsigned' real numbers the points on one side of O only, namely along OX , were representative of numbers, now with steps every point on the whole line stretching on both sides of O is representative of a step. This is a pictorial representation of the superior generality introduced by the positive and negative numbers, namely, the operations or steps. These 'signed' numbers are also particle as carried from O to A , or from A to B.

我们可以将从 O 处一步到达的点视为该步骤的代表。因此，A 代表 +1，B 代表 +2，A' 代表 -1，B' 代表 -2，依此类推。需要注意的是，以前仅有“无符号”实数时，只有 O 的一侧的点，即 OX 上的点，代表数字；而现在，随着步骤的引入，整个线段两侧的每个点都代表一个步骤。这是正负数引入的优越一般性的图示表示，即操作或步骤。这些“有符号”的数字也可以视为从 O 到 A，或从 A 到 B 之间的粒子。

In suggesting a few pages ago that the practical man would object to the subtlety involved by the introduction of the positive and negative numbers, we were libelling that excellent individual. For in truth we are on the scene of one of his greatest triumphs. If the truth must be confessed, it was the practical man himself who first employed the actual symbols + and -. Their origin is not very certain, But it seems most probable that they arose from the marks chalked on chests of goods in German warehouses, to denote excess or defect from some standard weight. The earliest notice of them occurs in a book published at Leipzig, in A.D.(Anno Domini) 1489. They seem first to have been employed in mathematics by a German mathematician, Stifel, in a book published at Nuremburg in A.D. 1544. But then it is only recently that the Germans have come to be liked on as emphatically a practical nation. There is an old epigram which assigns the empire of the sea to the English, of the land to the French , and of the clouds to the Germans, Surely it was from the clouds that the Germans fetched + and -; the ideas which these symbols have generated are much too important for the welfare of humanity to have come from the sea or from the land.

几页前我们提出，实用主义者会反对引入正负数所涉及的微妙之处，这实际上是在诋毁那位优秀的个体。因为事实上，我们正处于他伟大成就之一的现场。如果必须坦白，正是那位实用主义者自己首次使用了实际的符号 + 和 -。它们的起源并不十分确定，但最可能的是，它们源于德国仓库中用粉笔标记在货物箱上的符号，以表示与某个标准重量的超额或缺陷。关于它们的最早记录出现在公元1489年出版的一本书中。看起来，它们最早是在公元1544年由德国数学家斯蒂费尔在纽伦堡出版的一本书中用于数学。然而，德国人最近才被坚决视为一个实用主义国家。有一句古老的格言将海洋的统治权归于英国，土地的统治权归于法国，而云层的统治权归于德国人。显然，德国人正是从云层中获取了 + 和 -；这些符号所产生的思想对人类的福祉而言，太重要了，无法仅仅来自海洋或土地。

The possibilities of application of the positive and negative numbers are very obvious. If lengths in one direction are represented by positive numbers, those in the opposite direction are represented by negative numbers. If a velocity in one direction to the hands of a clock (anti-clock-wise) is positive, that in the clockwise direction is negative. If a balance at the bank is positive, an overdraft is negative . If vitreous electrification is positive, resinous electrification is negative. Indeed, in this latter case, the terms positive electrification and negative electrification, considered as mere names, have practically driven out the other terms. An endless series of examples could be given. The idea of positive and negative numbers has been practically the most successful of mathematical subtleties.

正负数的应用可能性非常明显。如果某个方向的长度用正数表示，那么相反方向的长度则用负数表示。如果顺时针方向的速度是正数，那么逆时针方向的速度就是负数。如果银行的余额是正数，透支就是负数。如果玻璃电气化是正的，那么树脂电气化就是负的。实际上，在后者的情况下，‘正电气化’和‘负电气化’这两个术语，作为单纯的名称，几乎取代了其他术语。可以举出无数个例子。正负数的概念实际上是数学微妙之处中最成功的一个。