Harmonica' An H.P. is a series of numbers such that the series of their reciprocals is an A P. Hence the typical H.P. is of the form , 1 1 1 -- a a+d' a+2d 1 a + (n-1)(1 It is obvious that every problem involving an H.P. is convertible into a problem involving an A.P. If a and b be any two numbers, their arithmetic mean is a number c such that the series a, c, b is an A.P. Hence c a = b c, whence c =1(a+b); i.e., the arithmetic mean of two numbers is half their sum. The geometric mean of a and b is a number c such that the series a, c, b is a G.P. Hence c= b, whence a c c ab; i.e., the geometric mean of two num bers is the square root of their product. The harmonic mean of a and b is c, where c is such 1 , 1 1 that the series a, c, b is an H.P. Hence a c' b is an A.P. Hence a = I ' whence c Denote by A, G and H respectively the arithmetic, the geometric and the harmonic means of a and b. Then A = (a+b), G=V ab, H= a+ 2ab b. It is readily seen that whence G=V All ;i.e., the geometric mean of two numbers is the geometric mean of their arithmetic and their harmonic means.
The Binomial Theorem or Expansion.If a and b are any numbers and n is any positive integer, (a+b)n b 2 1) an . .
? n(n 1) (n 2) . . . (n r+ I) arbnr+ 1 2 3 . . .
? + an expansion containing n+1 terms. For proof of the relationship see article MATHEMATICAL INDUCTION. It can be proved by algebraic means, most readily by Maclaurin's formula (see CALCULUS), that, if a is numerically greater than b and n is any real number, the same expansion as that above given is valid, i.e., (a+b)m = an +loxlb+ , which, however, contains an infinite number of terms, except in the case where n is a positive integer. The equation is called the binomial theorem. It was discovered by Sir Isaac Newton, but its correct ness was not proved by him. One of the sim plest of its countless applications is its appli cation to the problem of finding correct to any required degree of approximation any real root of any real number. For example, sup pose it is desired to know the real cube root of 25 correct to five decimal places. We may proceed as follows: V25 = (25)1 = (27 2)1 = 2)1 = 3. 9 =3 = 2.92402.
8 1 The Number e and the Series for ex. I f n be numerically greater than 1, the fore going theorem yields the equations __I J\ _2\ 2! n 1- k / nj +...; x (1 2! x ) ) n 3! Hence 1 1 2 1 1 + 71) 12) n/I / 2! 3! x \ \ _ _2\ nj vs/ \ nj =1 +x+ + 2! 3! This equation is valid for every value of n numerically greater than 1. The limits ap proached by its members as n increases be yond every finite value are equal, and it may be shown to result from this that 2! 3!
+.., The series on the right is convergent for every finite value of x; in fact, for any given value of x, the series after a certain number of terms converges more rapidly than any decreasing G.P. The series on the left is a special case of that on the right, viz., x-=1. The limit of the sum of the first n terms of the series on the left, i.e., its sum (to infinity) is denoted by e; accordingly the equation may be written: + ... The meaning is that the 2! number e raised to a power indicated by a given value of x is the sum to infinity of the series for that value of x. Since e =1+1+ 2! 3! its approximate value can be readily calculated. That value, correct to 10 decimal places, is e = 2.7182818284. The number e, one of the most important of all numbers, is incommen surable, i.e., not exactly expressible as a rational fraction, and it is transcendental, i.e not a root of an equation =0 where the coefficients a, b, ... are integers (see GENERAL THEORY OF ASSEMBLAGES).
Logarithms. Let a be any positive number greater than I. If x is named loon rithm of N to the base a; symbolically, x = loge N or, if the base is supposed known, simply x = log N. If a be fixed, x and N will vary each with the other, each is a function of the other. Since 1, log 1-0 no matter what the base. But in general the logarithm of a given number will vary with the base; thus, since 2' = 16, 4'= 16, log,16 =4, log.16=2. The general connection can be readily found thus: let ax=N and bv=N, then and log bN bN = y; also ax = bY, a=bx , logba= x og Nwhence logbN=logaN.logba. Calling a the old and b the new base, it is seen that the logarithm of a given number to a new base is equal to the product of the logarithm of the number to the old base and the logarithm of the new base to the old base. Let a' =N, ay = M, then ax+x= NM ; hence the logarithm of a product is the sum of the logarithms of the factors. Again, (ar)k=Nk=akx ; whence it is seen that the logarithm of the kth power of a number is k times the logarithm of the number. Once more, ax -=ax-Y; that is, the logarithm of a fraction is as equal to that of the numerator minus that of the denominator. Logarithms to the base 10 are called common logarithms or Briggsian loga rithms after Briggs, who introduced them in 1615. These are used in practical computation, but in theoretical work logarithms are referred to the number e, the Napiertan base, so called after Napier (1550-1617), the inventor of log arithms.