formula (c) could be used when n<r as well as when n>r, and, if we like, we might retain formula (c) and entirely dispense with formula (d). I Again, it may be seen, in a similar manner, that if a = form a9 ula (d) could be used when n>r as well as when n<r, so that we might, if we like, retain formula (d) and entirely dispense with formula (c). If we find that we may use negative exponents upon the above interpretation, then we will for the most part dispense with formula (d), using it only now and then, if at all, when it comes a little handier than formula (c). 18. Again, by the above interpretation formula (c) can be used when one or both of the exponents are negative. First, suppose r negative and equal to -q, then the same result as before, so that formula (c) may be used when r is negative. Second, suppose n negative and equal to -s, then the same result as before if our interpretation of negative exponents be correct, so that formula (c) may be used when ʼn is negative. Third, suppose both n and r negative and let n--s and r=-9, then But by substituting in the formula, a ̄÷÷a ̄2=a ̄s—(—) = a?· the same result as before, hence formula (c) may be used when both n and r are negative. 19. Formula (a) may be used when either or both of the exponents are negative if the above interpretation be correct. First, suppose r=-q, then I a"÷a a"a =a" =a"÷a2=a"-9. But by substituting in the formula, so that formula (a) may be used when n is negative. Third, suppose n=―s and r=· = I I = -9, then I as a? aεa? But by substituting in the formula, = a I I =α (s+9) so that formula (a) may be used where both n and r are negative. 20. Formula (b) may be used when either n or r or both are negative. First, supposer negative and equal to -q, then so that formula (b) may be used when is negative. Second, suppose n negative and equal to -q, then so that formula (b) may be used when n is negative. Third, suppose both exponents are negative and let n=—s and so that formula (b) may be used when both exponents are negative. I a9 21. Thus we see that if we interpret a as being a being any number whatever (not zero), and q being any whole number, the exponents in all our formulas may be any whole numbers, positive or negative, and this makes our formulas considerably more general than they were before. I Now, because the supposition a1= leads to no inconsistency a9 it is permissible, and because it gives greater generality to our formulas it is advantageous. any fraction any factor may be transferred from the numerator to the denominator, or vice versa, by simply changing the sign of the exponent. Hence, if in formula (c) we transfer a" from the denominator to a" the numerator, becomes a"a", which by formula (a) equals ar a"-"; so that formulas (a) and (c) are really identical, but, for the sake of convenience, both are retained. A-3 23. Having now dispensed with formula (d) and extended formulas (a), (b), (c) so that the exponents may be any whole numbers, positive or negative, the question arises, can we give still greater generality to our formulas by using exponents which are fractions? 24. If quantities with fractional exponents have any meaning and if we can use them in our formulas, we must have by formula (b) n being here any positive whole number; i. e. an is a quantity which raised to the nth power equals a. Raise both sides of this equation to the rth power, r being a positive whole number, and we get so that, if we are permitted to use fractional exponents, a denotes. the rth power of the nth root of a. 25. Again, by the definition of a quantity with an ex .26 Thus we have suggestions of meanings for both positive and negative fractional exponents, and if we introduce fractional exponents into our formulas with the meanings suggested, these formulas will be found to give consistent results, as we shall see. 27. Before substituting in our formulas it is necessary to stop and show that, with the meanings suggested, a quantity with a fractional exponent has the same value whether the exponent is in its lowest terms or not. |