which we obtained when we took a as the initial figure. Thus the only distinct orders in which the n digits can be arranged on a circle are the (n-1)! permutations we obtained by filling the first place with a. EXERCISES 1. In how many orders can 6 men sit around a circular table? Solution : n = 6, n − 1 = 5, (n − 1)! = 5! = 120. 2. In how many ways can 8 men sit around a circular table? 3. In how many ways may the letters of live be arranged on a circle ? 4. In how many ways may the letters of permutation be arranged on a circle ? 5. In how many ways can 4 men and 4 ladies sit around a table so that a lady is always between two men ? 6. In how many ways may 4 men and their wives be seated around a table so that no man sits next his wife? 7. In how many ways can 6 men and their wives be seated around a table so that each man always sits next his wife? 8. In how many ways can 10 red flowers and 5 white ones be planted around a circular plot so that two and only two red ones are adjacent ? 142. THEOREM. The number of permutations of n things of n! which p are alike, taken all together, is p! If all the things were different, we should have n! permutations. But since p of the n things are alike, any rearrangement of those p like things will not change the permutation. For any fixed arrangement of the n things there are p! different arrangements of the p like things. Thus of the n permutations are idenn! p! tical, and there are only distinct permutations of the n things p! 1 COROLLARY. If of n things p are of one kind, q of another kind, r of another, etc., then there are of the n things taken all at a time. n! permutations EXERCISES 1. How many distinct arrangements of the letters of the word Cincinnati are possible? Solution: There are in all 10 letters, of which 3 are i, 2 are c, and 3 are n. Thus the number of arrangements is 2. How many distinct arrangements of the letters of the word parallel can be formed ? 3. How many signals can be made by hanging 15 flags on a staff if 2 flags are white, 3 black, 5 blue, and the rest red? 4. How many signals can be made by the flags in exercise 3 if a white one is at each extreme ? 5. How many signals can be made by the flags in exercise 3 if a red flag is always at the top? 6. Would 3 dots, 2 dashes, and 1 pause be enough telegraphic symbols for the letters of the English alphabet, the numerals, and six punctuation marks? CHAPTER XVI COMPLEX NUMBERS 143. The imaginary unit. When we approached the solution of quadratic equations (p. 52) we saw that the equation x2 = 2 was not solvable if we were at liberty to use only rational numbers, but that we must introduce an entirely new kind of number, defined as a sequence of rational numbers, if we wished to solve this equation. The excuse for introducing such numbers was not that we needed them as a means for more accurate measurement, - the rational numbers are entirely adequate for all mechanical purposes, but that they are a mathematical necessity if we propose to solve equations of the type given. A similar situation demands the introduction of still other numbers. If we seek the solution of 1. we observe that there is no rational number whose square is Neither can we define √−1 as a sequence of rational numbers which approach it as a limit. We may write the symbol √—1, but its meaning must be somewhat remote from that of √2, for in the latter case we have a process by which we can extract the square root and get a number whose square is as nearly equal to 2 as we desire. This is not possible in the case of √-1. In fact this symbol differs from 1 or any real number not merely in degree but in kind. One cannot say √-1 is greater or less than a real number, any more than one can compare the magnitude of a quart and an inch. √-1 is symbolized by i and is called the imaginary unit. The term "imaginary" is perhaps too firmly established in mathematical literature to warrant its discontinuance. It should be kept in mind, however, that it is really no more and no less imaginary than the negative numbers or the irrational numbers are. So far as we have yet gone it is merely a thing that satisfies equation (1). When, however, we have defined the various operations on it and ascribed to it the various characteristic properties of numbers we shall be justified in calling it a number. 3 Just as we built up from the unit 1 a system of real numbers, so we build up from √-1=i a system of imaginary numbers. The fact that we cannot measure √-1 on a rule should cause no more confusion than our inability exactly to measure on a rule. Just as we were able to deal with irrational numbers as readily as with integers when we had defined what we meant by the four operations on them, so will the imaginaries become indeed numbers with which we can work when we have defined the corresponding operations on them. 144. Addition and subtraction of imaginary numbers. We write Also just as we pass from a rational to an irrational multiple of unity by sequences, so we pass from a rational to an irrational multiple of the imaginary unit. Thus we write a √-1, or ai, where a represents any real number. Consistently with § 76 we write ± √ — a2 = ± √a2 · (−1) = ± √a2. √−1=± a√=1=± ai. (II) We speak of a positive or a negative imaginary according as the radical sign is preceded by a positive or a negative sign. We also define addition and subtraction of imaginaries as follows: aibi = (a + b) i, where a and b are any real numbers. (III) ASSUMPTION. The commutative and associative laws of multiplication and addition of real numbers, § 10, we assume to hold for imaginary numbers. 145. Multiplication and division of imaginaries. We have already virtually defined the multiplication of imaginaries by real numbers by formula (I). Consistently with § 76 we define Thus √-a.√b = √a. √bi i = √ab · (− 1) = — √ab. The law of signs in multiplication may be expressed verbally as follows: The product of imaginaries with like signs before the radical is a negative real number. The product of imaginaries with unlike signs is a positive real number. In operating with imaginary numbers, a number of the form should always be written in the form Vai before performing the operation. This avoids temptation to the following error: √=a⋅ √=b=√(− a) · (— b) = √ab. Simplify the following: 1. 8. - 2. Solution: EXERCISES 8. √2 = √8 • i • √2 • i = √2 · 8 · ¿2 = 4 ⋅ (− 1) = − 4. |