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10. If w is one of the imaginary cube roots of unity, shew that the square of

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hence shew that the value of the determinant on the left is 3 √−3.

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18. If a determinant is of the nth order, and if the constituents of its first, second, third, ...nth rows are the first n figurate numbers of the first, second, third, ...nth orders, shew that its value is unity.

CHAPTER XXXIV.

MISCELLANEOUS THEOREMS AND EXAMPLES.

506. WE shall begin this chapter with some remarks on the permanence of algebraical form, briefly reviewing the fundamental laws which have been established in the course of the work.

507. In the exposition of algebraical principles we proceed analytically: at the outset we do not lay down new names and new ideas, but we begin from our knowledge of abstract Arithmetic; we prove certain laws of operation which are capable of verification in every particular case, and the general theory of these operations constitutes the science of Algebra.

Hence it is usual to speak of Arithmetical Algebra and Symbolical Algebra, and to make a distinction between them. In the former we define our symbols in a sense arithmetically intelligible, and thence deduce fundamental laws of operation; in the latter we assume the laws of Arithmetical Algebra to be true in all cases, whatever the nature of the symbols may be, and so find out what meaning must be attached to the symbols in order that they may obey these laws. Thus gradually, as we transcend the limits of ordinary Arithmetic, new results spring up, new language has to be employed, and interpretations given to symbols which were not contemplated in the original definitions. At the

same time, from the way in which the general laws of Algebra are established, we are assured of their permanence and universality, even when they are applied to quantities not arithmetically intelligible.

508. Confining our attention to positive integral values of the symbols, the following laws are easily established from a priori arithmetical definitions.

I. The Law of Commutation, which we enunciate as follows: (i) Additions and subtractions may be made in any order.

Thus

a+b-c=a-c+b=b-c+a.

(ii) Multiplications and divisions may be made in any order. axb = bxa ;

Thus

a xbx c = bx c xa = ax cx b; and so on.

ab ÷ c = a × b÷ c = (a ÷ c) × b = (b ÷ c) × a.

II. The Law of Distribution, which we enunciate as follows: Multiplications and divisions may be distributed over additions and subtractions.

Thus

(a− b + c) m = am — bm + cm,

(a - b) (cd) = ac — ad — bc + bd.

[See Elementary Algebra, Arts. 33, 35.] And since division is the reverse of multiplication, the distributive law for division requires no separate discussion.

III. The Laws of Indices.

(i)

(ii)

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[See Elementary Algebra, Art. 233 to 235.]

These laws are laid down as fundamental to our subject, having been proved on the supposition that the symbols employed are positive and integral, and that they are restricted in such a way that the operations above indicated are arithmetically intelligible. If these conditions do not hold, by the principles of Symbolical Algebra we assume the laws of Arithmetical Algebra to be true in every case and accept the interpretation to which this assumption leads us. By this course we are assured that the laws of Algebraical operation are self-consistent, and that they include in their generality the particular cases of ordinary Arithmetic.

509. From the law of commutation we deduce the rules for the removal and insertion of brackets [Elementary Algebra, Arts. 21, 22]; and by the aid of these rules we establish the law

of distribution as in Art. 35. For example, it is proved that (a - b) (cd) = ac· ad - bc + bd,

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with the restriction that a, b, c, d are positive integers, and a greater than b, and c greater than d. Now it is the province of Symbolical Algebra to interpret results like this when all restrictions are removed. Hence by putting a = O and c = 0, we obtain (-b) × (− d) = bd, or the product of two negative quantities is positive. Again by putting b=0 and c=0, we obtain a× (−d)=—ad, or the product of two quantities of opposite signs is negative.

We are thus led to the Rule of Signs as a direct consequence of the law of distribution, and henceforth the rule of signs is included in our fundamental laws of operation.

510. For the way in which the fundamental laws are applied to establish the properties of algebraical fractions, the reader is referred to Chapters XIX., XXI., and XXII. of the Elementary Algebra; it will there be seen that symbols and operations to which we cannot give any a priori definition are always interpreted so as to make them conform to the laws of Arithmetical Algebra,

511. The laws of indices are fully discussed in Chapter xxx. of the Elementary Algebra. When m and n are positive integers and m>n, we prove directly from the definition of an index that

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We then assume the first of these to be true when the indices are free from all restriction, and in this way we determine meanings for symbols to which our original definition does not apply. The interpretations for a, a, a" thus derived from the first law are found to be in strict conformity with the other two laws; and henceforth the laws of indices can be applied consistently and with perfect generality.

512. In Chapter VIII. we defined the symbol i or √-1 as obeying the relation=-1. From this definition, and by making i subject to the general laws of Algebra we are enabled to discuss the properties of expressions of the form a +ib, in which real and imaginary quantities are combined. Such forms are sometimes called complex numbers, and it will be seen by reference to Articles 92 to 105 that if we perform on a complex number the operations of addition, subtraction, multiplication, and division, the result is in general itself a complex number,

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