58. √x+a+√x+b=√4x+a+5b.

60. √x2+4x+12+√x2-12x-208.

IRRATIONAL NUMBERS.

318. An expression is said to be an Irrational Number when it is not a rational number (Art. 269), and is the result of any finite number of the following operations performed upon one or more rational numbers, provided that, in any indicated root, the number under the radical sign is positive if the index of the root is even :

1. Addition or Subtraction. 2. Multiplication or Division. 3. Raising to any power whose exponent is a rational. or irrational number. 4. Extracting any root.

Note. A surd is one form of an irrational number.

319. A meaning similar to that of Art. 275 will be attached to any irrational number, which does not involve the operation of raising to a power whose exponent is an irrational number.

320. In Arts. 8, 282, and 284, we defined the varics forms of rational exponents; we shall now proceed to define an irrational exponent.

Let a be a positive rational number, and let p, b, and ¿' have the same meanings as in Art. 275.

We shall then define a as being the limit, when r is indefinitely increased, of a'; and a meaning similar to the above will be attached to any form of irrational exponent.

321. It may be shown, as in Art. 276, that every result in Chapter II. holds when any or all of the letters involved represent irrational numbers; and therefore every statement or rule, in Chapters III. to XVI. inclusive, in regard to expressions where any letter involved represents any rational or surd number, holds equally when this letter represents any real number whatever. (Compare Art. 276.)

Also, the theorem of Art. 277 may be proved to hold when any or all of the letters a, b, c, etc., represent irrational numbers which are positive if is even; and the theorem of Art. 278 may be proved to hold when a is any irrational number whose mth power is positive if n is even.

322. We will now prove that (2), Art. 281, holds when a and m are positive rational numbers, and n a positive surd of the form b, where p and b have the same meanings as in Art. 275.

By Art. 320, axa

is the limit, when r is indefinitely increased, of aTM × a3r; and am+5 is the limit of am+b.

But since m and b,' are rational and positive,

and since am × a3r and am+ are functions of r which are equal for every positive integral value of r, by Art. 213 their limits when r is indefinitely increased are equal.

in all cases, not previously considered, where a, m, and n represent any real numbers such that a" and a" are real numbers.

323. It may be proved, as in Art. 322, that

in all cases, not previously considered, where a, m, and n represent any real numbers such that a", a”, aTMn, real numbers.

mn, and b" are

324. It follows from Arts. 321 to 323 that every result in Arts. 291 to 317 inclusive holds when any letter involved, except where used as the index of a root, represents any irrational number such that every expression of the form Va is an irrational number. (Compare Note, page 78.)

XVIII. IMAGINARY NUMBERS.

325. Consider the expression V-a, where a is a positive real number (Art. 318).

It is evidently impossible to find any real number whose square shall be equal to -a; but it is possible to give a definite interpretation to the expression V-a, and this question will be considered in Appendix I.

The expression √a is called an Imaginary Number.

326. An expression is also said to be an imaginary number when it is not a real number, and is the result of any finite number of the following operations performed upon one or more rational numbers (Art. 269):

1. Addition or Subtraction. 2. Multiplication or Division. 3. Raising to any power whose exponent is a real or imaginary number. 4. Extracting any root.

Thus, -5, 2+√-3, and 2 are imaginary numbers.

327. We shall limit ourselves in the present work to imaginary numbers which can be expressed in the form a+bv-1, where a is any real number or zero, and b any real number.

An imaginary number of the form b-1 is called a pure imaginary number; and one of the form a + b√−1, where a is not zero, is called a complex number.

Note 1. The imaginary number V-1 is often represented by the symbol i.

Note 2. It will be shown in Art. 336 that any imaginary number which does not involve the operation of raising to any power whose exponent is an irrational (Art. 318) or imaginary number, can be expressed in the form a+b√ — 1.

See also Appendix I., Art. 751.

328. We shall define √-1 as a number whose square is equal to -1; that is, (√−1)2=-1. (Compare Art. 121.)

-

And in general, we shall define c, where c is a positive real number, and n an even positive integer, as a number whose nth power is equal to -C.

329. The meaning to be attached to the operations of Addition, Subtraction, Multiplication, Division, and Extraction of Roots, when any or all of the numbers involved are pure imaginary or complex (Art. 327), will be discussed in Appendix I.

330. It will be proved in Appendix I. that every result in Chapter II. holds when any or all of the numbers involved are pure imaginary or complex; and therefore every statement or rule, in Chapters II. to XVI. inclusive, in regard to expressions where any letter involved represents any real number whatever, holds equally when this letter represents any imaginary number which can be reduced to the form a+b√-1, where a and b have the same meanings as in Art. 327. (Compare Art. 321.)

331. We will now prove that a, where a is any positive real number, and n an even positive integer, is equal to Va-1. (Compare Art. 277.)

By Art. 328, (a)" = a.

(1)

And since every rule, in Chapters II. to XVI. inclusive, holds when any or all of the numbers involved are pure imaginary,

(VaV − 1)" = (Va)” (V—1)" (Art. 109)

=ax (-1) = a.

From (1) and (2), (V−a)" = (VaV — 1)".

(2)

Whence,

Va= VaV-1.