HISTORICAL NOTE The use of exponents to denote powers and roots seem so essential to our present-day work in algebra that it is difficult to imagine how algebra could exist without this notation; but their general use is comparatively recent. Vieta (1591), the founder of modern algebra, had no knowledge of them. In place of our notation he used a system much like that of Diophantos. Thus, for the expression Vieta wrote 66 in which C, Q, and N are the first letters of the Latin words meaning 99 66 cube,' square," and "number." By number is meant the unknown quantity, and by C and Q are meant the square and cube of this number. In the sixteenth century Stifel used integral exponents, and Stevin invented fractional exponents, if we pass over the beginnings made by Oresme (1382), but it remained for Wallis and Newton, in the seventeenth century, to popularize these improvements. John Wallis, born in 1616, was second only to Newton among English mathematicians of the seventeenth century, and became a professor of JOHN WALLIS geometry at Oxford College at the age of 33. He was a charter member of the famous Royal Society of Great Britain, founded in 1663. Exponents, like the binomial theorem and many other principles of algebra, were standardized and made popular through their application to concrete problems, for Wallis, in seeking the areas inclosed by various curves, used the series of powers, x3, x2, x1, x2, x-1, x-2, x-8 in its present meaning, thus placing upon these symbols his stamp of authority. In this way our present definitions of negative exponents and of the zero exponent were established. The fact that these interpretations satisfy the fundamental laws of algebra was shown by Peacock and Hamilton, as noted (p. 36). CHAPTER IV LOGARITHMS MEANING AND USE OF LOGARITHMS 136 Use of Exponents in Computation. By applying the laws of exponents certain mathematical operations may be performed by means of simpler ones. The following table of powers of 2 may be used in illustrating some of these simplifications: 137. Application of Law I, Section 110, p. 65. 262144218 Thus, the process is simply one of inspection. In the above example we merely added 11 and 6 and looked in the table for the number opposite to 217. ORAL EXERCISES State the following products by reference to the table: 138. Application of Law II, Section 111, p. 66. By use of the table determine the value of the following: 139. Application of Law III, Section 112, p. 66. 2. Find: V1024. As above, 1024 = (1024) ✈ = (210) ₫ = 25 = 32, according to the table. 140. The examples and exercises above show that the laws of exponents furnish a powerful and remarkably easy way of making certain computations. In the above illustrations we have used a table based on the number 2, and have limited the table to integral exponents; but for practical purposes a table based on 10 is used and is made to include fractional exponents. For example: 1. It is known that approximately, 2= 1010 or 10-3 (more accurately 10.301). From this we can express 20 as a power of 10, for WRITTEN EXERCISES Given 48 = 101.68; express as a power of 10: 1. 480. 2. 4800. 3. 48,000. Given 649 102.81; express as a power of 10: 4. 4.8. 9. 6.49. 10. 649,000,000. Given 300 = 102.47; express as a power of 10: 141. The use of the base 10 has several advantages. I. The exponents of numbers not in the table can readily be found by means of the table. To make this clear, let us suppose that a certain table expresses all integers from 100 to 999 as powers of 10; then 30, although not in this table, can be expressed as a power of 10 by reference to the table. For, 30= and since 300 is in the supposed table we may find 300 by reference to the table that 300 = 102.47, and hence, 30 = 3.76= Similarly, 3.76 is not in the supposed table, but 376 is and 376 376 Therefore it is necessary only to subtract 2 from 100 102 the power of 10 found for 376 in order to find the power of 10 equal to 3.76. Similarly, 4680 is not in the table, but 468 is, and 4680 = 468.101. Therefore it is necessary only to add 1 to the power of 10 found for 468 in order to find the power of 10 equal to 4680. Such a table would not enable us to express in powers of 10 numbers like 4683, 46.83, and 356,900, but only numbers of 3 or fewer digits, which may be followed by any number of zeros. Similar conditions would apply to a table of powers for numbers from 1000 to 9999, from 10,000 to 99,999, and so on. II. The integral part of the exponent can be written without reference to a table. |