20 THE SYSTEMS CONCEPT 



Bessel's correction is introduced if the number, n, of samples is small 

 (< 30); then 



a = j/£ Ax 2 /(n - 1) 



The most probable deviation, r, is that value of the deviation such that one- 

 half the observations lies between the limits ±r. 



The relative deviation, usually expressed as a per cent, is the fraction which 

 the deviation is of the observed mean value, i.e., Ax/x. 



Each of these has several names. In the case of random errors, "devia- 

 tion" should read "error," of course; Ax is often called the absolute error of 

 the measurement. Relative error is sometimes called per cent error or proportional 

 error. These are discussed in detail, and examples are given, in Mainland's 

 book. 



Superposition of Errors. In the determination of a quantity, A, af(x, y, z) 

 which requires measurement of x, y, and z, each with an absolute error, the 

 errors must be superimposed one upon the other, or added; the reliability of the 

 value obtained for A is no better than the sum of the errors in x, y and z- 

 That is, the relative error in A is the sum of the relative errors in the meas- 

 urements oix,y, andz- 



9. Indices and Logarithms 



In arithmetic the ancient Greeks devised and used a notation, now called 

 that of indices, to express in shorthand the number of times a number is to be 

 multiplied by itself. Thus, "2 multiplied by itself 5 times" (i.e., 2 x 2 x 

 2x2x2) = 32. This is written in shorthand as 2 5 = 32. The index, 5, is 

 placed as a superscript to the base number 2. 



A number of laws of indices can be shown to exist for the manipulation of 

 such numbers. These laws were observed for cases in which the indices are 

 whole numbers. 



Now there is no reason to suppose that the rules would be different for 

 fractional indices, although to multiply 2 by itself 5 1/2 times would really 

 be tricky! Nevertheless, the rules are assumed to apply to fractional indices, 

 as well as to whole-number ones, and further also to algebraic, unknown 

 indices. In general, the laws of indices are as follows: 



(\)a m = axaxaxaxa m times 



(2) a m a" = a m+n 



(3) a m /a n = a m -" if m > n 



1 .. 

 or a m /a" = it n > m 



