If the equation contains three or more radicals, no general direction for solving can be given. To facilitate the work, the student must exercise his best judgment in transposing terms before squaring. nature of each equation must be his guide. The special Solve the following radical equations, and by checking determine the roots commonly called extraneous : IMAGINARY AND COMPLEX NUMBERS 325. By the introduction of negative, fractional, and surd numbers we have been enabled to solve many equations that otherwise would have been without solution, but rational numbers and surds do not meet all the demands upon a number system. For instance, if we attempt to solve the equation a2 = 1, we must say that the solution is impossible or we must extend the idea of number to include the new symbol IV-1 to which the equation leads. 326. It is evident that V-1 is neither +1 nor -1, since no number thus far considered gives a negative result when squared. But the even roots of negative numbers give rise to results of the form ±√−1, or ±√−a, and hence, if our number system is to be perfectly general, there is need for a new form of number whose square shall be negative, that is to say, a number whose meaning shall be deduced from the relation (√ − 1)2 = −1, and which, subject to this relation, shall be such that we can operate with it by means of the ordinary laws of algebra. 327. An even root of a negative number is called an Imaginary Number. Thus, √−1, −2, √3√−2, √-a are imaginary numbers. In distinction from imaginaries, all the numbers hitherto described are called real numbers. The designation "imaginary" for this new form of number is somewhat misleading, since these numbers are essentially no less real than are the rational fraction and negative numbers. As we have seen, fractional and negative results may or may not admit of a concrete interpretation in particular problems; imaginaries can have no interpretation in strictly arithmetical problems, as they are not expressible as rational numbers or approximately in terms of such numbers. Fractional, negative, and surd numbers were introduced to enable us to give a formal solution of many problems otherwise impossible. If we extend the system of real numbers by the introduction of imaginaries, we shall be enabled to perform all algebraic operations, including the solution of equations, without the limitations otherwise necessary. 328. We have already seen that positive and negative numbers are conveniently represented as standing on opposite sides of zero on a line laid off into units of length, as in the figure. And since + 1 x − 1 = −1, we may look upon the multiplica 2 tion of +1 by -1 as turning the line 0 A about 0 through 180° to the position 0 A, in the direction indicated by the arrows (counter-clockwise); that is, as swinging the line 0 A from the position in which it represents +1 to the position in which it represents — 1. 329. By a similar method of graphic representation, we are enabled to give a clearer idea of the nature of imaginary -2 Y +2V-1 A1+V-1 A2 0 Y' 0 A represents the positive unit +1, this may be shown as follows: Hence, if we draw the axes XX', YY', intersecting at right. angles at 0 (Art. 244), as in the annexed figure, and regard the symbols +V-1 and -V-1 as denoting direction as well as length (just as + and as signs of quality may indicate direction), we may represent +1 V-1 (or +V-1), +2 V-1, +3√-1, etc., by equal divisions laid off on the perpendicular OY, upward from 0, and 1√-1, -2v-1, −3√-1, etc., by equal divisions laid off on the same line, downward from 0. That is, just as + 2 indicates 2 units to the right of 0 and - 2 indicates 2 units to the left, so +2 √−1 indicates 2 units up from 0 and − 2 √−1 indicates 2 units down. 330. In the above graphic representation of the relation between real and imaginary numbers, XX' is called the axis of real numbers, and YY' the axis of imaginary numbers. It is to be observed that if YY' were taken as the axis of real numbers, then XX' would become the axis of imaginaries. In this representation it is seen that imaginaries have as much reality as do real numbers. 2n 331. In introducing these new numbers into the number system of algebra, we are tacitly assuming that, except so far as the relation (√−1)2 = 1, or in general ( a)2n == α, may determine the quality of the result, they obey all the laws established for other numbers, and as a consequence that they may be used either alone or in combination with real numbers in all algebraic operations. 332. Reduction to the Form a√ - 1. Since Vab√ax √b, it follows that √-2 may be written 2 x −1, or √2 × √ − 1. Hence any number of the form -c may be written in the form av 1, where a is a real number and V-1 the imaginary part, "imaginary unit" as it is called. Thus, 3 = √3x − 1 = √3√−1; √−4 = √‡ x − 1 = √4 × √ − 1, or ±2√1, where only the positive result is used unless otherwise stated. Reduce the following expressions to the form a√-1: 333. The algebraic sum of a real number and an imaginary number is called a Complex Number. Thus, 3+√ 1, 2 V 1, abv-1 are complex numbers. = 334. For brevity the symbol V-1 is often written i Thus, √-1=i, 2√ −1=2 i, √ — 5 = i√5, etc.; also (√ − 1)2=(±i)2 - 1. 335. In addition to the double series of real numbers we now also have the double series of imaginary numbers ...3i, 2i, — i, 0, + i, + 2 i, + 3 i, ... Obviously these series have no number in common except 0. Verify this by reference to the graphic representation given in Art. 329. |