IF THE mathematical ideas dealt with in the last chapter have been a
popular success, those of the present chapter have excited almost as
much general attention. But their success has been of a different
character, it has been what the French term a succes de
scandale. Not only the practical man, but also men of letters and
philosophers have expressed their bewilderment at the devotion of
mathematicians to mysterious entities which by their very name are
confessed to be imaginary. At this point it may be useful to observe
that a certain type of intellect is always worrying itself and others by
discussion as to the applicability of technical terms. Are the
incommensurable numbers properly called numbers? are the positive and
negative numbers really numbers? are the imaginary numbers imaginary,
and are they numbers?— are types of such futile questions. Now it cannot
be too clearly understood that, in science, technical terms are names
arbitrarily assigned, like Christian names to children. There can be no
question of the names being right or wrong. They may be judicious or
injudicious; for they can sometimes be so arranged as to be easy to
remember, or so as to suggest relevant and important ideas. But the
essential principle involved was quite clearly enunciated in Wonderland
to Alice by Humpty Dumpty, when he told her, apropos of his use of
words, 'I pay them extra and make them mean what I like.' So we will not
bother as to whether imaginary numbers are imaginary, or as to whether
they are numbers, but will take the phrase as the arbitrary name of a
certain mathematical idea, which we will now endeavour to make plain.
如果上一章讨论的数学思想受到了广泛欢迎，那么本章的思想几乎同样引起了广泛的关注。但它们的成功具有不同的性质，正如法语所称的“a
succes de
scandale”。不仅实践者，连文人和哲学家们也对数学家们对神秘实体的专注感到困惑，而这些实体从其名字上看就被承认是虚幻的。在这一点上，值得注意的是，某种类型的智慧总是通过讨论技术术语的适用性而使自己和他人感到烦恼。不可计量的数字是否可以称为数字？正数和负数是否真的是数字？虚数是否是虚幻的，它们是否是数字？——这些都是此类无意义问题的例子。现在必须明确理解，在科学中，技术术语是随意赋予的名称，就像给孩子取的基督教名字一样。没有关于名称是否正确或错误的问题。它们可能是明智的或不明智的；因为有时可以安排得使其容易记住，或者以便于引出相关和重要的想法。但涉及的基本原则在《爱丽丝梦游仙境》中由亨提·邓普提向爱丽丝清楚地阐明，当他说到他使用语言时：“我给他们额外的报酬，让他们按我喜欢的方式来表达。”因此，我们不再纠结于虚数是否真的“虚幻”，或者它们是否算作数字，而是将这个短语视为某个数学概念的任意名称，我们现在将努力将其阐明。
The origin of the conception is in every way similar to that of the
positive and negative numbers. In exactly the same way it is due to the
three great mathematical ideas of the variable, of algebraic form, and
of generalization. The positive and negative numbers arose from the
consideration of equations like x+1=3, x+3=1, and the general form
x+a=b. Similarly the origin of imaginary numbers is due to equation
, and . Exactly the same process is
gone through. The equation
becomes , and this has two
solutions, either , or
. The statement that
here are these alternative solutions is usually written . So far all is plain
sailing, as it was in the previous case. But now an analogous difficulty
arises. For the equation
gives and there is no
positive or negative number which, when multiplied by itself, will give
a negative square. Hence, of our symbols are to mean the ordinary
positive or negative numbers, there is no solution to , and the equation is in fact
nonsense. Thus, finally taking the general form , we find the pair of solutions
, when, and only
when, b is not less that a. Accordingly we cannot say unrestrictedly
that the 'constants' a and b may be any members, that is , the
'constants' a and b are not, as they out to be, independent unrestricted
'variables' ; and so again a host of limitations and restrictions will
accumulate round our work as we proceed.
The the same task as before therefore awaits us: we must give a new
interpretation to our symbols, so that the solutions for the equations
always have meaning. In
other words, we require an interpretation of the symbols so that always has meaning whether a be
positive or negative. Of course, the interpretation must be such that
all ordinary formal laws for addition, subtraction, multiplication, and
division hold good; and also it must not interfere with generality which
we have attained by the use of the positive and negative numbers. In
fact, it must in a sense include them as special cases. When a is
negative we may write for it,
so that is positive. Then
因此，和之前一样，我们面临着同样的任务：我们必须对我们的符号给出新的解释，以便方程
的解
始终具有意义。换句话说，我们需要对这些符号进行解释，以使得 始终具有意义，无论
是正数还是负数。当然，这种解释必须确保所有普通的加法、减法、乘法和除法的正式法则仍然适用；同时，它也不能干扰我们通过使用正数和负数所获得的一般性。事实上，在某种意义上，它必须将它们作为特例包括在内。当
为负数时，我们可以用 来表示它，使得 是正的。然后我们有： Hence, if we can so interpret out symbols that has a meaning, we have
attained our object. Thus has come to be looked on as
the head and forefront of all the imaginary quantities.
This business of finding an interpretation for is a much tougher job that
the analogous one of interpreting -1. In fact, while the easier problem
was solved almost instinctively as soon as it arose, it at first hardly
red, even to the greatest mathematicians, that here a problem existed
which was perhaps capable of solution. Equations like , when they arose, were simply
ruled aside as nonsense.
However, it came to be gradually perceived during the eighteenth
century, and even earlier, how very convenient it would be if an
interpretation could be assigned to these nonsensical symbols. Formal
reasoning with these symbols was gone through, merely assuming that they
obeyed the ordinary algebraic laws of transformation; and it was seen
that a whole world of interesting results could be attained, if only
these symbols might legitimately be used. Many mathematicians were not
then very clear as to the logic of their procedure, and an idea gained
ground that, in some mysterious way, symbols which mean nothing can by
appropriate manipulation yield valid proofs of propositions. Nothing can
be more mistaken. A symbol which has not been properly defined is not a
symbol at all. It is merely a blot of ink on paper which has an easily
recognized shape. Nothing can be proved by a succession of blots, except
the existence of a bad pen or a careless writer. It was during this
epoch that the epithet 'imaginary' came to be applied to . What these mathematicians
had really succeeded in proving were a series of hypothetical
propositions, of which this is the blank form: If interpretations exist
for and for the
addition, subtraction, multiplication, and division of which make the ordinary
algebraic rules (e.g. ,
&c.) to be satisfied, then such and such results follows. It was
natural that the mathematicians should not always appreciate the big
'If' , which ought to have preceded the statements of their results.
As may be expected the interpretation, when found, was a much more
elaborate affair than that of the negative numbers and the reader's
attention must be asked for some careful preliminary explanation. We
have already come across the representation of a point by two numbers.
By the aid of positive and negative numbers we can now represent the
position of any point in a plane by a pair of such numbers. Thus we take
the pair of straight lines XOX' and YOY', at the right angles, as the
'axes' from which we start all our measurements. Lengths measured along
OX and OY are positive, and measure backwards along OX' and OY' are
negative. Suppose that a pair of numbers, written in order, e.g(+3,+1),
so that there is a first number (+3 in the above example), and a second
number (+1 in the above example), represents measurements form O along
XOX' for the first number, and along YOY‘ for the second number. Thus
(cf. Fig. 9) in (+3,+1) a length of 3 units is be measured along XOX' in
the positive direction, that is from O towards X, and a length +1
measured along YOY' in the positive direction, that is from O towards Y.
Similarly in (-3,+1) the length of 3 units is to be measured from O
towards X', and of 1 unit from O towards Y. Also in (-3,-1) the two
length are to be measured along OX' and OY' respectively, and in (+3,-1)
along OX and OY' respectively. Let us for the moment call such a pair of
numbers an 'ordered couple'. Then, from the two numbers 1 and 3, eight
ordered couples can be generated, namely
正如预期的那样，当解释被发现时，它比负数的解释要复杂得多，读者的注意力必须集中在一些细致的初步解释上。我们已经遇到过用两个数字表示一个点。借助正数和负数，我们现在可以通过一对这样的数字来表示平面上任意一点的位置。因此，我们取两条相互垂直的直线
XOX' 和 YOY'，作为我们进行所有测量的‘坐标轴’。沿 OX 和 OY
测量的长度为正，沿 OX' 和 OY'
反向测量的长度为负。假设一对数字按顺序书写，例如
(+3,+1)，其中第一个数字是 (+3)，第二个数字是 (+1)，它们分别表示沿 XOX'
从 O 开始的第一个数字和沿 YOY' 的第二个数字的测量。因此（参见图 9），在
(+3,+1) 中，沿 XOX' 的正方向测量 3 个单位的长度，也就是从 O 向 X
方向延伸，而沿 YOY' 的正方向测量 1 个单位的长度，也就是从 O 向 Y
方向延伸。同样，在 (-3,+1) 中，长度 3 个单位的测量是从 O 向 X'
方向延伸，1 个单位是从 O 向 Y 方向延伸。在 (-3,-1) 中，这两个长度分别沿
OX' 和 OY' 测量，而在 (+3,-1) 中，分别沿 OX 和 OY'
测量。暂时将这样的一对数字称为‘有序对’。然后，从数字 1 和
3，可以生成八个有序对，即 Each of these eight 'ordered couples' directs a process of
measurement along XOX' and YOY' which is different form that directed by
any of the others.
The processes of measurement represented by the last four ordered
couples, mentioned above, are given pictorially in the figure. The
lengths and together correspond to (+3,+1), OM'
and together correspond to
(-3,+1), and together to (-3,-1), and and together to (+3,-1). But by
completing the various rectangles, it is easy to see that the point
completely determines and is
determined by the ordered couple (+3,+1), the point by (-3,+1), the pint by (-3,-1), and the point
by (+3,-1). More
generally in the previous figure (8), the point corresponds to the ordered couple , where x and y in the figure are
both assumed to be positive, the point corresponds to , where x' in the figure is
assumed to be negative,
to , and to . Thus an ordered couple , where x and y are any positive or
negative numbers, and the corresponding point reciprocally determine
each other. It is convenient to introduce some names at this juncture.
In the ordered couple the
first number x is called the 'abscissa' of the corresponding point, and
the second number y is called the 'ordinate' of the point, and the two
numbers together are called the 'co-ordinates' of the point. The idea of
determining the position of a point by its 'co-ordinates' was by no
means new when the theory of 'imaginaries' was being formed. It was due
to Descartes, the great French mathematician and philosopher, and
appears in his Discours published at Leyden in A.D. 1673. The
idea of the ordered couple as a thing on its own account is of later
growth and is the outcome of the efforts to interpret imaginaries in the
most abstract way possible.
It may be noticed as a further illustration of this idea of the
ordered couple , that the point
in fig.9 is the couple (+3,0), the point is the couple (0,+1), the point the couple (-3,0), the point the couple (0,-1), the point the couple (0,0).
Another way of representing the ordered couple is to think of it as representing
the dotted line (cf.Fig.8),
rather than the point . Thus the
ordered couple represents a line drawn from an 'origin', , of a certain length and in a certain
direction. The line may be
called the vector line form to
, or the step from to . We see, therefore, that we have in
this chapter only extended the interpretation which we gave formerly of
the positive and negative numbers. This method of representation by
vectors is very useful when we consider the meaning to be assigned to
the operations of the addition and multiplication of ordered
couples.
We will now go on to this question, and ask what meaning we shall
find it convenient to assign to the addition of the two ordered couples
and . The interpretation
must, (a) make the result of addition to be another ordered couple, (b)
make the operation commutative so that , (c) make the operation associative so
that
make the result of subtraction unique, so that when we seek to
determine the unknown ordered couple so as to satisfy the
equation
使减法的结果唯一，以便当我们试图确定未知的有序对 时，使其满足方程。 there is one and only one answer which we can represent
by
只有一个且唯一的答案，我们可以用…来表示。 All these requisites are satisfied by taking to mean the
ordered couple
. Accordingly, by definition we put
通过将
解释为有序对 ，所有这些要求都得到了满足。因此，根据定义，我们定义为：
Notice that here we have adopted the mathematical habit of
using the same symbol + in different senses. The + on the left-hand side
of the equation has the new meaning of + which we are just defining;
while the two +'s on the right-hand side have the meaning of the
addition of positive and negative numbers (operations) which was defined
in the last chapter. No practical confusion arises from this double
use.
作为加法的例子，我们有： The meaning of subtraction is now settled for us . We find
that
减法的意义现在已经为我们确定。我们发现, Thus and and $$ (-1,-2)-(+2,+3)=(-3,-5)
$$ It is easy to see that
很容易看出， Also Hence (0,0) is to be looked on as the zero ordered couple. For
example
因此，(0,0) 应被视为零有序对。例如， The pictorial representation of the addition of ordered
couples is surprisingly easy.
有序对加法的图示表示出奇地简单。
image-20241105222403104
Let represent so that and ; let represent so that and . Complete the parallelogram
by the dotted lines and , the diagonal is the ordered couple . For draw parallel to ; then evidently the triangles and are in all respects equal. Hence
, and ; and therefore
设 代表有序对 ，使得 且 ；设 代表有序对 ，使得 且 。通过虚线 和 完成平行四边形 ，对角线 是有序对 。然后画出与 平行的 ；显然，三角形 和 在所有方面是相等的。因此，，且 ；因此， Thus represents the
ordered couple as required. This figure can also be drawn with and in other quadrants.
因此，
代表了按要求的有序对。这个图形也可以在其他象限中绘制，其中 和 位于不同的象限。
It is at once obvious that we have here come back to the
parallelogram law, which was mentioned in Chapter 6, in the laws of
motion, as applying to velocities and forces. It will be remembered
that, if and represent two velocities, a particle
is said to be moving with a velocity equal to the two velocities added
together if it be moving with the velocity . In other words is said to be the resultant of the two
velocities and . Again forces acting at a point of a
body can be represented by lines just as velocities can be; and the same
parallelogram law holds, namely , that the resultant of the two forces
and is the force represented by the
diagonal . It follows that we can
look on an ordered couple as representing a velocity or a force, and the
rule which we have just given for the addition of ordered couples then
represents the fundamental laws of mechanics for the addition of forces
and velocities. One of the most fascinating characteristics of
mathematics is the surprising way in which the ideas and results of
different parts of the subject dovetail into each other. During the
discussions of this and the previous chapter we have been guided merely
by the most abstract of pure mathematical considerations; and yet at the
end of them we have been led back to the most fundamental of all the
laws of nature, laws which have to be in the mind of every engineer as
he designs an engine and of every naval architect as he calculates the
stability of a ship. It is no paradox to say that in our most
theoretical moods we may be nearest to our most practical
applications.
