THE DEFINITION of the multiplication of ordered couples is guided by
exactly the same considerations as is that of their addition. The
interpretation of multiplication must be such that
有序对的乘法定义完全遵循与其加法相同的原则。乘法的解释必须满足以下条件：
() the result is another ordered couple,
() the operation is commutative, so that () the operation is associative, so that () must make the result of division unique [ with an
exception for the case of the zero couple (0,0)], so that when we seek
to determine the unknown couple (x,y) so as to satisfy the equation
必须使得除法的结果唯一（零有序对
的情况除外），这样当我们试图确定未知的有序对 以满足方程时，结果是明确的。 there is one and only one answer, which we can represent(表示)
by () Furthermore the law
involving both addition and multiplication, called the distributive law,
must be satisfied, namely All these conditions (), (), (), (), () can be satisfied by an interpretation
which , though it looks complicated at first, is capable of simple
geometrical interpretation.
This is the definition of the meaning of the symbol when it it written between two
ordered couples. It follows evidently from this definition that the
result of multiplication is another ordered couple, and that the value
of the right-hand side of equation (A) is not altered by simultaneously
interchanging x with x' and y with y'. Hence conditions () and () are evidently satisfied. The proof of
that satisfaction of () , (), () is equally easy when we have given the
geometrical interpretation, which we will proceed to do in a moment. But
before doing this it will be interesting to pause and see whether we
have attained the object for which all this elaboration was
initiated.
We came across equation of the form , to which no solutions could be
assigned in terms of positive and negative real numbers. We the found
that all our difficulties would vanish if we could interpret the
equation , i.e. if we could
so define that .
我们已经证明了 For the future we follow the custom of omitting the sign wherever possible, thus (1,0)
stands for (+1,0) and (0,1) for (0,+1).
对于将来，我们遵循惯例，在可能的情况下省略正号“”，因此 表示 ，而 表示 。
Furthermore we now have
此外，我们现在拥有 Hence both for addition and for multiplication the couple
(0,0) plays the part of zero in elementary arithmetic and algebra;
compare the above equations with , and
Again consider : this plays
the part of 1 in elementary arithmetic and algebra. In these elementary
sciences the special characteristic of 1 is that , for all values of . Now by our law of multiplication
再来看有序对 ：它在基础算术和代数中起到了“1”的作用。在这些基础学科中，1
的特殊性质在于对所有 值都有 。现在，根据我们的乘法法则
Thus (1,0) is the unit couple.
因此， 是单位有序对。
Finally consider (0,1): this will interpret for us the symbol . The symbol must therefore
possess the characteristic property that .Now by
the law of multiplication for ordered couples.
最后，我们来看有序对 ：它将为我们解释符号 。因此，这个符号必须具备这样的特性，即
。现在，根据有序对的乘法法则。 But is the unit
couple, and is the negative
unit couple; so that has the
desired property. There are , however, two roots of -1 to be provided
for , namely .
Consider (0,-1); here again remembering that , we find, .
Thus is the other square
root of . Accordingly,
the ordered couples and are the interpretations of in terms of ordered
couples. But which corresponds to which? Does correspond to and to , or to , and to ? The answer is that it is
perfectly indifferent which symbolism we adopt.
The ordered couples can be divided into three types, (1) the 'complex
imaginary' type , in which
neither x nor y is zero; (2) the 'real' type ; (3) the 'pure imaginary' type
. Let us consider the
relations of these types to each other. First multiple together the
'complex imaginary' couple
and the 'real' type we
find
Thus the effect is merely to multiply each term of the couple by the positive or negative real
number a.
因此，其效果仅仅是将偶对
的每一项乘以正或负的实数 。
Secondly, multiply together the 'complex imaginary' couple and the 'pure imaginary' couple
, we find Here the effect is more complicated, and is best comprehended
in the geometrical interpretation to which we proceed after nothing
three yet more special cases.
在这里，效果更为复杂，并且最好在我们记录了另外三个更特殊的情况之后，进入几何解释来理解它。
Thirdly, we multiply the 'real' couples (a,0) by the imaginary (o,b)
and obtain
Fourthly we multiply the two 'real' couples (a,0) and (a',0) and
obtain Fifthly, we multiply the two 'imaginary couples' (0,b) and
(0,b') and obtain We now turn to the geometrical interpretation, beginning first
with some special cases. Take the couples (1,3) and (2,0) and consider
the equation
image-20241110162010749
In the diagram (Fig.11) the vector represents , and the vector represents , and the vector represents . Thus the product is found geometrically
by taking the length of the vector to be the product of the length of the
vector and , and (in this case ) by producing
to to be of the required length. Again,
consider the product we have
在图（图11）中，向量 表示
，向量 表示 ，而向量 表示 。因此，乘积 从几何上可以通过将向量
的长度视为向量 和
的长度之积来找到，并且（在这种情况下）通过将 延长到 使其达到所需的长度。同样，考虑乘积
，我们得到 The vector ,
corresponds to and the vector
to . Thus which represents the new product is at
right angles to and of the same
length. Notice that we have the same law regulating the length of as in the previous case, namely, that
its length is the product of the lengths of the two vectors which are
multiplied together; but now that we have along the 'ordinate' axis , instead of along the 'abscissa' axis , the direction of has been turned through a
right-angle.
Hitherto in these examples of multiplication we have looked on the
vector as modified by the
vectors and . We shall get a clue to the general
law of the direction by inverting the way of thought, and by thinking of
the vectors and as modified by the vector . The law for the length of the product
of the two vectors. The new direction for the enlarged (i.e ) is found by rotating it in the
(anti-clockwise) direction of rotation from towards through an angle equal to the angle
: it is an accident of this
particular case that this rotation makes lie along the line . Again consider the product of and ; the new direction for the enlarged
(i.e. ) is found by rotating in the anti-clockwise direction of
rotation through an angle equal to the angle , namely, the angle is equal to the angle .
The general rule for the geometrical representation of multiplication
can now be enunciated thus:
用于几何表示乘法的一般法则现在可以这样表述:
The product of the two vectors and is a vector , whose length is the product of the
lengths of and and whose direction is such that the angle is equal to the sum of the angle
and .
Hence we can conceive the vector as making the vector rotate through an angle (i.e. the angle = the angle ), or the vector as making the vector rotate through the angle (i.e. the angle = the angle ).
We do not prove this general law, as we should thereby be led into
more technical processes of mathematics than falls within the design of
this book. But now we can immediately see that the associative law
[numbered () above] for
multiplication is satisfied. Consider first the length of the resultant
vector; this is got by the ordinary process of multiplication for real
numbers; and thus the associative law holds for it.
Again, the direction of the resultant vector is got by the mere
addition of angles, and the associative law hold for this process
also.
同样，结果向量的方向是通过简单的角度相加得到的，结合律对这个过程也成立。
So much for multiplication. We have now rapidly indicated, by
considering addition and multiplication, how an algebra or 'calculus' of
vectors in one plane can be constructed, which is such that any two
vectors in the plane can be added, or subtracted, and can be multiplied,
or divided one by the other.
We have not considered the technical details of all these processes
because it would lead us too far into mathematical details; but we have
shown the general mode of procedure. When we are interpreting our
algebraic symbols in this way, we are said to be employing 'imaginary
quantities' or 'complex quantities'. These terms are mere details, and
we have far too much to think about to stop to inquire whether they are
or are not very happily chosen.
The net result of our investigations is that any equations like or can now always be interpreted
into terms of vectors, and solutions found for them. In seeking for such
interpretations it is well to note that 3 becomes , and -2 becomes , and x becomes the 'unknown'
couple : so the two equations
become respectively , and
We have now completely solved the initial difficulties which caught
our eye as soon as we considered even the elements of algebra. The
science as it emerges from the solution is much more complex in ideas
than that with which we started. We have, in fact, created a new and
entirely different science, which will serve all the purposes for which
the old science was invented and many more in addition. But, before we
can congratulate ourselves on this result to our labours, we must allay
a suspicion which ought by this time to have arisen in the mind of the
student. The question which the reader ought to be asking himself is:
Where is all this invention of new interpretations going to end? It is
true that we have succeeded in interpreting algebra so as always to be
able to solve a quadratic equation like ; but there is an endless
number of other equations, for example, , and so on without
limit. Have we got to make a new science whenever a new equation
appears?
Now, if this were the case, the whole of our preceding
investigations, though to some minds they might be amusing, would in
truth be of very trifling importance. But the great fact, which has made
modern analysis possible, is that, by the aid of this calculus of
vectors, every formula which arises can receive its proper
interpretation; and the 'unknown' quantity in every equation can be
shown to indicate some vector. Thus the science is now complete in
itself as far as its fundamental ideas are concerned. It was receiving
its final form about the same time as when stream engine was being
perfected, and will remain a great and powerful weapon for the
achievement of the victory of thought over things when curious specimens
of that machine repose in museums in company with the helmets and
breastplates of a slightly earlier epoch.
