Negative Numbers Combined with ExponentialsDate: 03/09/2001 at 01:07:20 From: Dee Ryno Subject: Negative numbers combined with exponentials. I know that -3 ^2 {to the second power} is negative 3 because the order of operations tells us exponentials are done first and then the answer {9} is multiplied by (-1), but why isn't the negative attached to the -3 to say -3 * -3, which would make it positive, since 3 ^2 {to the second power} is 3 * 3? Help. This has confused a lot of teachers too! Dee Date: 03/09/2001 at 15:29:24 From: Doctor Peterson Subject: Re: Negative numbers combined with exponentials. Hi, Dee. You are aware that the order of operations is the key. Because negation is taken as a multiplication, and exponentiation is done before multiplication, we read -3^2 as -(3^2) rather than as (-3)^2 The exponent holds on to the 3 tighter than the minus sign does. I think you're asking why we have this rule, when the minus looks so close to the 3, and it seems so much more natural to think of it as -3 squared. I could just say this is a choice that has been made, and we just follow the convention. But there's an additional reason besides the logic of seeing negation as multiplication by -1. When we get to algebra and want to write polynomials, we find ourselves working with -x^n, which has to follow the same rules as for -3^n: x^3 - 3x^2 - 3x + 5 = 0 There's little question here; we know that 3x^2 is taken as 3(x^2), and then we subtract that. But what about -x^3 - 3x^2 - 3x + 5 = 0 ? Do we take this as (-x)^3 or as -(x^3)? Because this is a polynomial, we know that x is meant to be the base of all the exponents; we don't want to have to write -(x^3) to make it mean what we intend; so we are perfectly happy to follow the logic where it takes us and treat the negation as a multiplication done after the exponentiation, rather than as a part of the base. The rule helps here, rather than seeming odd: In a polynomial we want exponents to come before everything else, because they are the stars of the show. I suspect that polynomials drove much of the development of the order of operations; many of the early examples of algebraic notation in which those rules can be discerned are polynomials. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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