374. This being premised, it is only necessary farther to observe, that the method of adding and subtracting imaginary radicals, is the same as for real quantities.

Thus,

a+24/—a=34/—a; 6+ √ −4+6— √-4=12; and 3 v-ax+-y-(1-ax-y)=21-ax+21/-y.

375. Every imaginary radical quantity of the form .—a, can be reduced to the form Va× √ −1, or a2 V-1.

In order to demonstrate this, let the identical equality be, (c-b)a=(c—b)a; by extracting the root of both sides, we shall have v(c-b) x va= v[(cb)a]; which under the relation, b>c, or in the hypothesis, for instance, b=c+1, becomes 7-1× √α= V-a; and, in general, 2-a=2/ax v1.

2n

It may be demonstrated, in a similar manner, that X √1; and in general, that

C

m

m

21

m

376. Hence, in the calculation of imaginary radicals, it is sufficient to demonstrate the rules for multiplying and involving the imaginary radical -1; since imaginary quantities can be always resolved into factors; so that -1 only shall remain under the radical sign.

377. In the first place, then, it may be observed, when a2 is considered abstractedly, or without any regard to its generation, then Va2 may be either + ora (Art. 295), there being nothing in the nature of the quantity so taken, to denote from which of these two expressions it was derived.

378. But this ambiguity, which, in the abovementioned case, arises from our being anacquainted with the origin of the quantity whose root is to be extracted, will not take place when the sign of the quantity from which it was produced is known; as

there can, then, be only one root, which must evidently be taken in plus or minus, according to the state it existed in before it was involved.

379. Thus [(+a)×(a)], or v√[(+a2)] cannot be of the ambiguous form a, as it would have been if a2 had been conditionally assumed, but it is simply a; and, for a like reason, [(—a)×(—a)], or ✔(-a) is -a, and not a; since the value of the equivalent expression +a2, or a2 in these cases, is determined, from the circumstance of its being known how aa is derived. 380. Hence the product of -1 by ~1, or which is the same, (1-1) is. √1=-1. This is what also appears evident from (Art. 326), since that, in squaring a quantity with the radical sign ✔, we have only to take it away, that is, to pass the quantity from under the radical sign.

381. Also, if the factors, in this case, be both nega→ tive, the result will be the same as before; since -✓ -1) × —(√✓/—1)=+(-1)2=-1; but if one of the factors be positive and the other negative, we shall +(-1)x−(√−1)=—(√−1)2=+1. 382. All whole positive numbers are comprised in one of these four formula;

have

--

4n, 4n+1, 4n+2, 4n+3,

n being a whole positive number; since that, if any whole number be divided by 4, the remainder must be 0, 1, 2, or 3.

If we designate -1 by x, the several powers of 1 shall be therefore represented by one of these four formulæ :

4n

✓✓ − 1 ) 1 ” = x 11 =(x*)”=(+1)"=+1;

1

4 n

( √ − 1 ) + n + 1 = x + n + 1 = x1" • x=x=+√−1;

=x4n+2=x4n.x2 = x2 = 1.

(√ — 1)4n+3=x4n+ 3 = x1n x8 =

-- -1.x=- V-1. Thus, in order to know any given power of V- 1. it is sufficient to divide the exponent of the power proposed by 4, and the power of V-1 indicated by the remainder shall be that which is required.

383. When one imaginary quantity is to be multiplieđ by another, the result, whether they be both positive or both negative, is equal to minus the square root of their product, taking them as real quantities.

Thus, (+1-a) × (+ √ —b)=- Vab; since. (Art. 375), (+√ −a) × (+ √ −b) = √a× √ — 1× √ bxv = vax vb× (√ −1)2=-1× vab v ab. And, in a similar manner, it may be proved that V-a)x(—v—b) = — √ ab.

384. And if one of the imaginary radicals be posilive, and the other negative, the result arising from their multiplication will be plus the square root of their product, taking them as before.

Thus, (+V-a) × (— √ —b)=+vab; since + v―a=+vax V-1, and - √—b=(— √ —1) xvb;.. (vax V-1)x(— v−1)× √b)=[(+v= 1).(V-1)] Vab=+1x vab + vab (Art. 381).

385. When one imaginary radical is to be divided by another, the result, whether they be both positive or both negative, will be equal to plus the square root of their quotient, taking them as real quantities.

386. And if one of the imaginary radicals be positive and the other negative, the result arising from division, will be minus the square root of their quotient, taking them as before.

or

-a

V

a

Thus,

or

=

; and

-α

V -a

387. If an imaginary radical is to be divided by a real radical, or a real radical by an imaginary one, the result will be equal to plus or minus the square root of their quotient, according as the radical is affirmative or negative.

The several powers of imaginary radicals can be readily derived from the formulæ (Art. 382); it only now remains to illustrate the preceding rules by a few practical examples.

Ex. 1. It is required to multiply a-v-b by a→ V-b, or to find the square of aV-b.

a-v-b

a-v-b

a2-av-b

av-b-b

a2-2a (v-b)-b Ans.

Ex. 2. It is required to find the quotient of 1+ √

-1 divided by 1-7-1.

Here

X

1-7-1 1 V -1 1 + V 1-1. Ans.

Ex. 3. It is required to multiply 1+ 1-1 by 1+ -1; or to find the square of 1+1-1.

Ans. 2-1. Ex. 4. It is required to find the product arising from multiplying i+7-1 by 1— √-1.

Ans. 2.

Ex. 5. It is required to find the square, or second power of a+v-b2. Ans. a2b3+2ab v—1. Ex. 6. It is required to multiply 5+2 -3 by 2- V-3. Ans. 16-7 −3. Ex. 7. It is required to find the cube, or third power, of a- √ —b2.

Ans. a3-3ab + (b3 — 3a3b) v-1. Ex. 8. It is required to find the quotient of 3+ √ -4 divided by 3-27-1. Ans. (5+12V-1). Ex. 9. It is required to find the square of v(a+b V-1)+v (a-by-1). Ans. 2a+2 (a2+b2).

CHAPTER VIII.

ON

PURE EQUATIONS.

388. Equations are considered as of two kinds, called simple or pure, and adfected; each of which are differently denominated according to the dimensions of the unknown quantity.

389. If the equation, when cleared of fractions and radical signs or fractional exponents, contain only the first power of the unknown quantity, it is called a simple equation.

390. If the unknown quantity rises to the second power or square, it is called a quadratic equation.

391. If the unknown quantity rises to the third power or cube, it is called a cubic equation, &c.

392. Pure equations, in general, are those wherein ned. only one complete power of the unknown quantity is in creased. These are called pure equations of the first degree, pure quadratics, pure cubics, pure biquadratics, &c., according to the dimension of the unknown quantity.

Thus, x=a+b is a pure equation of the first degree; x2=a2+ab is a pure quadratic;

&c.

x3=a3+a2b+c is a pure cubic ;

x=a+a3b+aca+d is a pure biquadratic;

393. Adfected equations are those wherein different powers of the unknown quantity are concerned, or are found in the same equation. These are called adfected quadratics, adfected cubics, adfected biquadratics, &c., according to the highest dimension or power of the unknown quantity.

Thus, x+ax=b, is an adfected quadratic ;
x2+ax2+bx=c, an adfected cubic;

3

¤1+ax3+bx2+cx=d, an adfected biquad

atic,