Prove That -(-a) = a
Date: 09/11/2001 at 22:21:28 From: Eduardo Subject: Why does - (-a) = a Why does -(-a) = a? How do you prove this using the properties of real numbers?
Date: 09/12/2001 at 08:26:08 From: Doctor Rick Subject: Re: Why does - (-a) = a Hi, Eduardo. We can start with how -a is defined. It is "the additive inverse of a" - that is, it is the number that, when added to a, gives 0: a + -a = 0 Therefore -(-a) means the number that, when added to -a, gives 0. But applying the commutative property of addition, the equation above becomes -a + a = 0 Therefore the number that, when added to -a, gives 0 is a; or, -(-a) = a A closely related, but different, question is how we can prove that -1 * -1 = 1 The theorem linking these two is this: -1 * a = -a Let's prove this. Start with the fact that zero times any number is zero: 0 * a = 0 Write 0 as (1 + -1), which follows from the definition of -1. (1 + -1)*a = 0 Apply the distributive property: 1*a + -1*a = 0 Use the fact that 1 times any number is the same number: a + -1*a = 0 Now, the number that, when added to a, gives 0 is -a. Therefore -a = -1*a Using this theorem, we can easily prove that -1 times -1 is 1: -1 * -1 = -(-1) = 1 For other approaches to the question of why a negative times a negative is positive (both informal illustrations and formal proofs), see our Dr. Math FAQ, "Negative times a negative": http://mathforum.org/dr.math/faq/faq.negxneg.html - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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