Understanding Exponents Eric Rasmusen September 18, 2018

Transcription

1 Understanding Exponents Eric Rasmusen September 18, 2018 These notes are rather long, but mathematics often has the perverse feature that if someone writes a long explanation, the reader can read it much more quickly than if he writes a short explanation, because all the steps are written out and the reader doesn t have to think of them himself. There are several things to know about exponents. The notation 10 3 means , which is 1,000, with being a multiplication symbol, like or * or using (10)(10)(10). More generally, x 3 = x x x. First, here are some rules, presented as examples: = = 10 5 (10 2 ) 3 = = = = ( means about ) 10 1/3 = = = 1 10 = = = 1 1,000 =.001 The paradigm to remember for multiplying exponents is = 10 5 so 1, 000 =, 000 x 2 x 3 = x 5 1

2 2 The best way to remember this is not to memorize it outright, but to try out examples like the one above if you forget. The problem you will run into is that you will ask yourself, Do I add the exponents, or do I multiply them? If you use that paradigm, you will realize that you should add them, because =, 000. Or, you can try a simpler example, because = =, = 10 1 = 10 What s probably even better is to understand the rule, because while you can memorize the multiplication one, there are all those others too, so your memory is bound to fail. Also, if you try to understand the rule, even if you fail to understand it, that acts as a mnemonic. A mnemonic ( nemonic ) is a help to memory. 1 Oddly enough, attaching a few other words to the thing to be remembered, even if only vaguely related, helps us to remember the thing. I think this is probably because the attachments act as extra handles to the thing in the memory, so it is easier to grab onto it and then to haul in the thing itself. Anyway, a logical reason for something acts as a mnemonic for it. Moreover, the logical reason for all those rules is interesting. Here we go. First, the multiplication-of-exponent rule s reason is pretty obvious. We need to have (10 10) ( ) = , because 1 The height of my achievements as a poet is perhaps the following: How do you spell mnemonic It s practically demonic. You put an M before the N; And then it s just phenomic! This poem is itself a mnemonic.

3 that s how multiplication works. But that s exactly the same as saying = The same goes for the exponent-of-exponents rule. We need to have (10 3 ) (10 3 ) = ( ) ( ) = , but that s the same as saying (10 3 ) 2 = What about the rule that says 10 0 = 1? That seems arbitrary, but it s not. We have to have that rule, or we d violate basic arithmetic. Here s why. Our multiplication rule says that That implies 10 a 10 b = 10 a+b = = 10 2 =. So any definition we might have of 10 0 requires that 10 0 =. But there s only one number x that leaves unchanged when you multiply by it to get x, and that is x = 1, x equals one! So the only definition we can use is that 10 0 = 1. How about the rule that says 10 2 = 1? The laws of arithmetic require that too. Our multiplication rule says that 10 a 10 b = 10 a+b 3 That implies = = 10 0 = 1. So any definition we might have of 10 2 requires that 10 2 = 1. But there s only one number x that takes you to 1 when you multiply by it to get x - x = 1, the inverse of! So the only definition we can use is that 10 2 = 1. Note that we need two steps for this, though the exponent rule, and the fact that if we use the exponent rule, we also need to have 10 0 = 1. How about the rule that says 10 1/2 = 10? Let s use an easier example: 9 1/2 = 9 = 3. The laws of arithmetic require that too. Our

4 4 multiplication rule says that That implies 9 a 9 b = 9 a+b 9 1/2 9 1/2 = 9 1/2+1/2 = 9 1 = 9. So any definition we might have of 9 1/2 requires that when multiplied by itself it equals 9 which is exactly the definition of the square root of nine. Of course, this isn t quite right. The number 9 has two square roots, +3 and -3. So we ve got a little leeway, maybe. But think of the implications if we used the negative root. First of all, it would be nice to have f(x) = 10 x be a function, a mapping from x to f(x) that takes only one value, so we need to choose between the positive root and the negative root. More important is that while x 1 2 has two roots, one positive and one negative, x 1 3 has just one root a positive one. Think about , which is 3 27, which is 3, since = 27. But = 27, not 27. Similarly, = 243, not 243. In fact, any odd root will only be positive, not negative. So if we use the negative roots, we won t have any definition for x 1/b for odd values of b! So we really have no reasonable choice but to say that x 1 b equals the positive value of b x. Having decided that x 1 b = b x > 0, it becomes easy to deal with more complicated fractional exponents such as 9 3/2. We just apply the multiplication rule. We need 9 3/2 = 9 1+1/2 = /2 = 9 9 = 9 3 = 27. Our last problem is what to with x a if the number a is neither zero, nor a natural number (1,2,3, etc.), nor a fraction (1/2, 3/4, 5/33, etc.). How can that be? There are certain numbers that are called irrational because they are not fractions of any two natural numbers. Three examples are π (the ratio of the circumference of a

5 circle to its diameter, about 3.1), e (which is the number such that the slope of the function e x is also e x that is, the slope always equals the height of the curve, where e is about 2.7), and the square root of two (about 1.4). I say about because closer approximations are π , e , and , but in all three cases the digits keep on going forever without ever repeating a pattern. For irrational numbers, we can t use the multiplication rule, but again there s only one reasonable way to define x a : to make sure that its value is between the value of nearby smaller and bigger fractions. We want to interpolate. That s much easier to see in an example. What it means is that we want to define 10 π to be in between and , both of which are fractional exponents (because = 31,415 ). That pins 10,000 down the value as finely as we desire. I promised you a graph earlier. If we apply the rules, including using positive roots for fractional exponents and using our interpolation rule for irrational numbers, we can draw the function f(x) = 10 x not just for the natural numbers 1, 2, 3, and so forth, but for any number, any real number, including fractions, zero, negative numbers, and irrational numbers. So we can graph it, as in Figures 1 and 2. 5

6 6 Figure 1: The Function f(x) = 10 x 10 x x Figure 2: A Close-Up of the Function f(x) = 10 x 10 x x In economics, one application of these rules is to finding the marginal product of capital. A typical Cobb-Douglas production function is Q(L, K) = L 2/3 K 1/3 and If K = 8 and L = 27 this works out neatly, for example, because so output is K 1/3 = 8 1/3 = 3 8 = 2 L 2/3 = L 1/3+1/3 = L 1/3 L 1/3 = 27 1/3 27 1/3 = 3 3 = 9, Q = L 2/3 K 1/3 = 9 2 = 18.

7 The marginal product of labor is the derivative of output with respect to labor, the slope of Q plotted against L holding K fixed at some number, e.g., K = 8. That slope is the derivative of Q with respect to L, which, by the usual calculus rule that is d dx xa = ax a 1, dq = d dl dl L2/3 K 1/3 = ( ) 2 3 L 1 2/3 K 1/3 = ( ) 2 3 L 1/3 K 1/3 = ( ) 2 K 1/3 3 L 1/3 If L = 27 and K = 8, that means the marginal product of labor is ( ) ( ) ( ) ( ) 2 K 1/ /3 2 2 MP L = 3 L = 1/ = = 4 1/ , which is to say that adding 1 unit of labor adds about 4/9 to output. More accurately, adding an infinitesimal amount x of labor adds (4/9)x to output, since when you start adding labor you don t have L = 27 any more so the MPL starts to change. 7