(3) Required the square root of 4+18, where a2-b-16-18-2. Multiplying this by 2, we have 4√/2+√/36=6+4√2, where a2-b =36-32-4;

..√(4+√/18)=√(6+4√/2) _√}(6+2)+√}(6—2) _ 2+√/2

2*+2*=√8+√2.

2

=

=

And a like mode of proceeding will also sometimes succeed when both terms of the binomial surd are irrational; for example:

(4) Required the square root of √/32—√/24.

Multiplying by 2, we have √64-√/48—8—√/48; where a2-b =

c2=16;

..√(√/32—√24) =

√}(8+4)−√ } (8—4)

=

V6 No2

No2

√/2.(√3-1) ✩ 2

= √2.(√3—1)=√2.(√ 9—1)=√ 18—√2.

In each of these two examples, we might have commenced by dividing by 2 instead of multiplying.

EXAMPLES FOR EXERCISE.

Required the square root of each of the following binomial surds:

74. ON IMAGINARY OR IMPOSSIBLE QUANTITIES.

Imaginary or impossible quantities, as already noticed (page 64), are those expressions which indicate an even root of a negative quantity; the arithmetical extraction of such a root being always an impossibility, because no even power of any number, whether positive or negative, can ever be negative. Imaginary quantities thus differ essentially from other surd expressions; the root indicated in each of these latter cannot be accurately exhibited, solely because the quantity under the radical sign differs from a complete power; yet a complete power may always be assigned which shall differ from the in

complete one by a quantity less than any that can be proposed, so that the defect mentioned is never of any practical consequence. But an imaginary quantity admits of no arithmetical representation either accurately or approximately; the bare idea of arithmetical value is altogether excluded from it; the symbol ✔-4 implies an operation upon the 4 of impossible performance; so that if such a symbol were to occur in the answer to any question, we should at once conclude that the solution to that question, in real numbers, is an impossibility; and, consequently, that the conditions to be satisfied are incompatible or contradictory. Imaginary quantities thus subserve a very important purpose: whenever they present themselves as here supposed, they effectually apprise us of concealed absurdities among the conditions upon which our reasoning has been based, or which we are aiming to satisfy, and which might otherwise involve us in bewilderment or error. They are thus necessary to give completeness and certainty to our algebraic results, and on these grounds alone are valuable items in our system of symbols. But independently of this office of imaginary quantities, by which they inform us of the fact when the solution of a question is impossible, algebraists turn them to important account as direct instruments of investigation; frequently introducing them with great advantage into inquiries having reference only to real quantities, and terminating only in real results.

In his first steps in the study of algebra, a learner naturally looks upon the new symbols of quantity to which he is introduced as nothing more than the familiar figures of arithmetic in disguise. It is not easy, nor would it be prudent to correct this too limited notion at the outset; the more comprehensive scope of the symbolic language of algebra gradually unfolds itself to him as he proceeds, till he at length comes to combine his characters and construct his expressions without any thought towards the numerical processes bearing the same names as those which enter into his symbolic combination. In fact, the important truth discovers itself by degrees, that the thing called algebra is a science in which symbols of any interpretation whatever are subjected to certain prescribed laws of combination, in obedience to which various operations may be performed and various results obtained without any reference to the particular characters of arithmetic. This latter science is no doubt suggestive of the symbolical science of algebra; and the learner sufficiently sees that its laws of combination actually become

those of arithmetic, when the particular symbols of the latter replace the more general symbols of the former. In fact, he further sees that, till these general symbols are so replaced by those of arithmetic, many of the so-called operations of algebra are but operations indicated, not operations executed. If we have to multiply a by b, we write ab or a×b, and say that the thing is done; although, in truth, nothing is done, although something, by the sign of operation, is indicated; we have no idea of the actual performance except each symbol, or the multiplier at least, be interpreted by a number; yet if the so-called product ab is to be divided by b, there is no doubt that the result is a, whatever the multiplier b may have been: whether a number or something having no arithmetical meaning. It is true that in the latter case the term "multiplier" might be objected to as not sufficiently significant, but a similar objection might be made to nearly every term introduced from arithmetic into algebra: the terms have a more comprehensive meaning as well as the symbols.

In the instance just adduced, namely

ab
b

a, an operation is

apparently performed on pure symbols irrespective of all interpretation; yet if we really look at what has actually been done, we shall see that it amounts to nothing: when b is written by the side of a a certain operation, called multiplication by b, is directed to be performed on a; except b be a number, we cannot obey this direction; but by writing b underneath, the direction is re-called, so that a, which would be the result of the operations of multiplication and division in arithmetic, is here written down free of these operations, because they neutralise one another: their combined effect being to leave a untouched. It is not necessary that we should be able to assign the effect of the combination ab and then the effect upon this of the combination

ab

b

; it is enough that we know, from the general laws of these combinations, that the second destroys the first and sets a free. It is more especially this recognition of the neutralising influence of certain algebraic combinations on one another, apart from all mere arithmetical considerations, that renders what are here called impossible or imaginary quantities so available in algebra, as instruments of investigation, though not admissible into arithmetic: a whether b be real or imaginary.

ab
Ъ

In the present article our business will be merely to show how imaginary expressions of the more simple and common forms are to be multiplied together and to be divided one by another.

75. MULTIPLICATION.

The multiplication of V-a by itself is, of course, represented by (-a)2; and as the exponent 2 neutralises the radical sign ✔, the result of both operations combined is -a, so that √—a×√—a——-a; or we may say √—a × √—a—√(—a)2 It would not be correct to write a XV. √—ax-ava2±a; the last deduction would be erroneous, because the square root is to have the ambiguous sign only when we are ignorant as to how the power was generated, or when there is nothing, as to its generation, stipulated. In the present case, it is declared that the a2 is generated from —ɑ× —ɑ, and not from +ax+a.

=

If a is to be multiplied by ✔-b, then attending to the V. rule of signs, we must have ✔-a × √ —b=√ —a ×—b— Vab; apparently the same result as we should get by multiplyinga by b; and in each case the root ab, when extracted, would be entitled to the ambiguous sign, because, as it is not from the root that ab is here generated, no restriction, as to generation from a root, is imposed. Nevertheless algebra is not without means of supplying an intelligible distinction between these two apparently identical results, and the distinction is not without some degree of practical importance. Suppose we replace -a and b by ax-1 and bx-1; then —ɑ= ✔a-1 and w—b—√b√ —1; so that √—a × √ —b= (Vav-1) (Vbv-1)=√ ab x-1=-ab. Here the square roots of a and b necessarily imply the ambiguous sign, because we know nothing of the generation of these quantities; if therefore we represent the value of Vab by r, the prefixed minus sign above would convert the result of the preceding multiplication into r, so that the distinction between Vaxb and ✔―ax-b, is that if we write the result of the former operation r, we ought to write the result of the latter r, the plus sign taking the place of the minus and the minus of the plus; and this is quite in accordance with what ought to happen if a and b, which are of course unrestricted, were made equal: for then axa would be +a, and √—a× √——a would be -A

--

Now although it may seem of little consequence whether we write the ambiguous sign before a root or, yet combinations may occur where the using of one for the other would lead to serious error. Thus, suppose we had to add m+Vaxb to n+vax-b; if the above distinction be neglected, we should make the sum m+n+2√ ab; whereas, by taking note of the difference, the correct sum would be found to be m+N.

From what has now been shown, a general rule for the multiplication of imaginary quantities is at once suggested: it is as follows:

RULE.-Multiply the two imaginary quantities together as if the signs under the radicals were plus instead of minus, and then change the sign of the result.

The examples here subjoined exemplify the rule in every case.

(1) (+√—a) (+√—a)=—a; (~√—a) (—√ —a)

(2) (+√−a)(+√ —b)——√ ab; (— √ —a)(— √ —b) =-ab.

(3) (+√—α) (—√ —b)= +√ ab.

The resulting sign being always the opposite of that which would arise if the factors were real instead of imaginary; and in all these instances we see that, although our operations have been with quantities entirely imaginary, yet the results of those operations are all real.*

(4) Multiply 4+√3 by 3+√-5, and 2+V-2 by 3-V-4.

4+ √−3 3+ √−5

12+3√3+4√ —5—√ 15

2+ √-2
3- √ 4
6+3V-2-2-4+√8.

* We have applied the terms imaginary or impossible to the forms treated of in this article, because they have been adopted by general consent, and not because we consider them to be strictly significant of the nature or import of the things meant. An excellent algebraist (James Cockle, Esq. M.A.) has proposed to substitute the designation unreal for imaginary; and the change would be for the better; but it is not easy to give a name to these quantities which shall be at once sufficiently brief and sufficiently descriptive of their character and office. For a disquisition on these, the reader is referred to Chapter II. of the "Theory and Solution of Equations," second edition.