Product of minus z x minus a
Answers
Answer:
Why is the Product of Negative Numbers Positive?
Asked by an anonymous poster on March 18, 1997:
I'm helping a 7th grader with things like: a plus times a plus equals a plus, a minus times a plus equals a minus, and a plus times a minus equals a minus. All OK. But when I tell him a minus times a minus equals a plus he says WHY? (sorry about yelling).
I won't feel bad if you don't answer this. No textbook and nobody has the faintest idea. But just in case you do answer, please remember it's a 7th grader who wants to understand, not to mention yours truly.
The answer has to do with the fundamental properties of operations on numbers (the notions of "addition", "subtraction", "multiplication", and "division"). Your 7th grader's question is an important and fundamental one (which I am both surprised and sorry that he has not been able to find an answer for yet).
Each number has an "additive inverse" associated to it (a sort of "opposite" number), which when added to the original number gives zero. This is in fact the reason why the negative numbers were introduced: so that each positive number would have an additive inverse.
For example, the inverse of 3 is -3, and the inverse of -3 is 3.
Note that when you take the inverse of an inverse you get the same number back again: "-(-3)" means "the inverse of -3", which is 3 (because 3 is the number which, when added to -3, gives zero). To put it another way, if you change sign twice, you get back to the original sign.
Now, any time you change the sign of one of the factors in a product, you change the sign of the product:
(-something) × (something else) is the inverse of (something) × (something else), because when you add them (and use the fact that multiplication needs to distribute over addition), you get zero.
For example, (IMAGE) is the inverse of (IMAGE) , because when you add them and use the distributive law, you get (IMAGE) .
So (IMAGE) is the inverse of (IMAGE) , which is itself (by similar reasoning) the inverse of (IMAGE) .
Therefore, (IMAGE) is the inverse of the inverse of 12; in other words, the inverse of (IMAGE) ; in other words, 12.
The fact that the product of two negatives is a positive is therefore related to the fact that the inverse of the inverse of a positive number is that positive number back again.
The answer to this question is accessible to a 7th grader (and should, in my opinion, be explained as part of every student's arithmetic classes). However, as an aside, he may be interested to know that more advanced versions of this question are studied at a university level: there is a subject called Abstract Algebra (usually only covered in a junior or senior level undergraduate university course) which studies the properties of operations on numbers in complete generality, even in contexts that have nothing to do with numbers at all. Even in such general, non-numerical contexts, the property that the product of two negative things is positive still holds.
Followup Comment by Buzz Breedlove on May 9, 1997:
This is a comment on y