![]() ![]() That’s all it takes it follows immediately from the definition. Therefore the number that, when added to -a, gives 0 is a or, But applying the commutative property of addition, the equation above becomes Therefore -(-a) means the number that, when added to -a, gives 0. ![]() It is "the additive inverse of a" - that is, it is the number that, when added to a, gives 0: Why does -(-a) = a? How do you prove this using the properties of real numbers?ĭoctor Bruce had passed over this lightly we’ll go a little deeper and further here. Next, a slightly different question from 2001: Prove That -(-a) = a In mathematics, you are always entitled to an explanation of WHY things are the way your teacher (or I) say they are. Therefore,Īgain, $$0=(-m)\times 0=(-m)\times(n+(-n))\\=(-m)\times n+(-m)\times(-n)\\=-(m\times n)+(-m)\times(-n)\\ \Rightarrow\ (-m)\times(-n)=-(-(m\times n))=m\times n$$ I hope this doesn't frighten you! The main thing is, keep right on questioning the things that don't make sense. We know -15 has exactly one additive inverse, namely 15. So (-3)*(-5) is doing the job of the additive inverse of -15. We know the first thing, (-3*5) equals -15 because of the law in (4). But by the distributive law, it also equals Negative times negativeīut we can repeat the same process, using the fact just demonstrated: (5) Next, we are forced to accept another new law, that negative times negative equals positive. We can write this in the general case, supposing that m and n are any two positive numbers: $$0=m\times 0=m\times(n+(-n))=m\times n+m\times(-n)\\ \Rightarrow\ m\times(-n)=-(m\times n)$$ So a positive times a negative is a negative (and, by the commutative property, a negative times a positive is negative). There is only one additive inverse of a number anything that does what \(-6\) does must be \(-6\). But the additive inverse of 6 is just -6. So 2*(-3) does the job of the additive inverse of 2*3, and therefore 2*(-3) is the additive inverse of 2*3. This is because we can use the distributive law on an expression like First: (4) Now, we are forced to accept a new law, that negative times positive equals negative. We can show that these facts imply what multiplication of negative numbers has to look like, in two steps. The first and third facts are true of multiplication as we know it in adding the concept of negative numbers (that is, the additive inverse) to the arithmetic we are already familiar with, we don’t want to change these facts. (3) We want negative numbers to obey the distributive law. ![]() Likewise, the additive inverse of -N is N. This means if N is a positive number, then -N is its additive inverse, so that N + (-N) = 0. (2) Every number has exactly one additive inverse. We have to start with known facts about multiplication (assumptions or axioms): Mathematical argument takes a little getting used to. What follows is a proof, presented in a style intended for students who are not familiar with proofs. If someone did just decree this “rule”, then it would be annoying, wouldn’t it? But math is not about arbitrary rules it’s about reasoning from basic assumptions or known facts, to less obvious facts. I want to reassure you that this rule is not just "made up." There is a chain of reasoning - a mathematical "argument" - that shows why the rule *has* to be that negative times negative equals positive. I'll bet it seems like someone just made the rule up out of thin air, with no particular reason why the answer should be positive. I detect in your question a measure of annoyance at having to learn the rule for multiplying negative numbers. How does a negative number times another negative number equal a positive number? But it really requires a mathematical proof, as we’ll explain and demonstrate here, first with a couple different proofs, then with the bigger picture, giving the context of such proofs.įirst, from 1998: Why Does a Negative Times a Negative Equal a Positive? Last time we looked at explanations for the product of negative numbers in terms of various concrete models or examples. ![]()
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