An Introduction to Mathematics · Alfred North Whitehead
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 succès 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, à propos 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.
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 equations like x^2 + 1 = 3 , x^2 + 3 = 1 , and x^2 + a = b. Exactly the same process is gone through. The equation x^2 + 1 = 3 becomes x^2 = 2 , and this has two solutions, either x = +sqrt(2) , or x = -sqrt(2). The statement that there are these alternative
solutions is usually written x = ±sqrt(2). So far all is plain sailing, as it was in the previous case. But now an analogous difficulty arises. For the equation x^2 + 3 = 1 gives x^2 = -2 and there is no positive or negative number which, when multiplied by itself, will give a negative square. Hence, if our symbols are to mean the ordinary positive or negative numbers, there is no solution to x^2 = -2 , and the equation is in fact nonsense. Thus, finally taking the general form x^2 + a = b , we find the pair of solutions x = ±sqrt((b - a)) , when, and only when, b is not less than a. Accordingly we cannot say unrestrictedly that the "constants" a and b may be any numbers, that is, the "constants" a and b are not, as they ought to be, independent unrestricted "variables"; and so again a host of limitations and restrictions will accumulate round our work as we proceed.
The same task as before therefore awaits us: we must give a new interpretation to our symbols, so that the solutions ±sqrt((b - a)) for the equation x^2 + a = b always have meaning. In other words, we require an interpretation of the symbols so that sqrt(a) always has meaning whether a be positive or negative. Of course, the interpretation must be such that all the ordinary formal laws for addition, subtraction, multiplication, and division hold good; and also it must not interfere with the
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 -c^2 for it, so that c^2 is positive. Then sqrt(a) = sqrt((-c^2)) = sqrt({(-1) × c^2}) ; = sqrt((-1)) sqrt(c^2) = csqrt((-1)). Hence, if we can so interpret our symbols that sqrt((-1)) has a meaning, we have attained our object. Thus sqrt((-1)) has come to be looked on as the head and forefront of all the imaginary quantities.
This business of finding an interpretation for sqrt((-1)) is a much tougher job than 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 occurred, even to the greatest mathematicians, that here a problem existed which was perhaps capable of solution. Equations like x^2 = -3 , 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 sqrt((-1)). 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 sqrt((-1)) and for the addition, subtraction, multiplication, and division of sqrt((-1)) which make the ordinary algebraic rules (e.g. x + y = y + x , etc.) 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 the
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 right angles, as the "axes" from which we start all our measurements. Lengths measured along OX and OY are positive, and measured 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 from 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 to 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 towards Y. Also in (-3, -1) the two lengths 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 (+1, +3), (-1, +3), (-1, -3), (+1, -3), ; (+3, +1), (-3, +1), (-3, -1), (+3, -1). Each of these eight "ordered couples" directs a process of measurement along XOX' and YOY' which is different from 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 OM and ON together correspond
to (+3, +1) , the lengths OM' and ON together correspond to (-3, +1) , OM' and ON' together to (-3, -1) , and OM and ON' together to (+3, -1). But by completing the various rectangles, it is easy to see that the point P completely determines and is determined by the ordered couple
(+3, +1) , the point P' by (-3, +1) , the point P" by (-3, -1) , and the point P"' by (+3, -1). More generally in the previous figure (8), the point P corresponds to the ordered couple (x, y) , where x and y in the figure are both assumed to be positive, the point P' corresponds to (x', y) , where x' in the figure is assumed to be negative, P" to (x' y') , and P"' to (x, y'). Thus an ordered
couple (x, y) , 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 (x, y) 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 "coordinates"
of the point. The idea of determining the position of a point by its "coordinates" 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 1637 A.D. 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 M in fig. 9 is the couple (+3, 0) , the point N is the couple (0, +1) , the point M' the couple (-3, 0) , the point N' the couple (0, -1) , the point O the couple (0, 0).
Another way of representing the ordered couple (x, y) is to think of it as representing the dotted line OP (cf. fig. 8), rather than the point P. Thus the ordered couple represents a line drawn from an "origin," O , of a certain
length and in a certain direction. The line OP may be called the vector line from O to P , or the step from O to P. 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 (x, y) and (x', y'). The interpretation must, (a) make the result of addition to be another ordered couple, (b) make the operation commutative so that (x, y) + (x', y') = (x', y') + (x, y) , (c) make the operation associative so that {(x, y) + (x', y')} + (u, v) = (x, y) + {(x', y') + (u, v)}, (d) make the result of subtraction unique, so that when we seek to determine the unknown ordered couple (x, y) so as to satisfy the equation (x, y) + (a, b) = (c, d), there is one and only one answer which we can represent by (x, y) = (c, d) - (a, b).
All these requisites are satisfied by taking (x, y) + (x', y') to mean the ordered couple (x + x', y + y'). Accordingly by definition we put (x, y) + (x', y') = (x + x', y + y'). 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.
As examples of addition we have (+3, +1) + (+2, +6) = (+5, +7), ; (+3, -1) + (-2, -6) = (+1, -7), ; (+3, +1) + (-3, -1) = (0, 0).
The meaning of subtraction is now settled for us. We find that (x, y) - (u, v) = (x - u, y - v). Thus (+3, +2) - (+1, +1) = (+2, +1), and (+1, -2) - (+2, -4) = (-1, +2), and (-1, -2) - (+2, +3) = (-3, -5).
It is easy to see that (x, y) - (u, v) = (x, y) + (-u, -v). Also (x, y) - (x, y) = (0, 0). Hence (0, 0) is to be looked on as the zero ordered couple. For example (x, y) + (0, 0) = (x, y).
The pictorial representation of the addition of ordered couples is surprisingly easy.
Let OP represent (x, y) so that OM = x and PM = y ; let OQ represent (x_1, y_1) so that OM_1 = x_1 and QM_1 = y_1. Complete the parallelogram OPRQ by the dotted lines PR and QR , then the diagonal OR is the ordered couple (x + x_1, y + y_1). For draw PS parallel
to OX ; then evidently the triangles OQM_1 and PRS are in all respects equal. Hence MM' = PS = x_1 , and RS = QM_1 and therefore OM' = OM + MM' = x + x_1, ; RM' = SM' + RS = y + y_1.
Thus OR represents the ordered couple as required. This figure can also be drawn with OP and OQ in other quadrants.
It is at once obvious that we have here come back to the parallelogram law, which
was mentioned in Chapter VI., on the laws of motion, as applying to velocities and forces. It will be remembered that, if OP and OQ 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 OR. In other words OR is said to be the resultant of the two velocities OP and OQ. 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 OP and OQ is the force represented by the diagonal OR. 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.
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End of part 7 of 19
Continue readingChapter VIII: Imaginary Numbers (Continued)
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