# Extremely Remedial question? Why is a neg. times a neg. a positive?? Proof? **URL:** \ **Category:** Factual Questions\ **Created:** [December 6, 2004, 5:54pm UTC](https://boards.straightdope.com/t/extremely-remedial-question-why-is-a-neg-times-a-neg-a-positive-proof/278389 "2004-12-06T17:54:00Z")\ **Posts on this page:** 5\ **Page:** 2
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [December 7, 2004, 6:20pm UTC](https://boards.straightdope.com/t/extremely-remedial-question-why-is-a-neg-times-a-neg-a-positive-proof/278389/21 "2004-12-07T18:20:23Z")
I see your point, **Mathochist** , and it is legitimate; I’m just not sure it’s relevant to what I was trying to do, which was to answer **Uncommon Sense** ’s quesion (the one I quoted, from post #12), in a way that someone who hasn’t had any substantial math in 20 years could understand. I wasn’t attempting to prove anything, nor to address your previous post, but to try to clear up some of **US** ’s confusion. Actually, I did see **US** ’s question as being sort of asymmetrical, since he was talking about points on a number line and comparing the point -3 with the point -3 \* -3. But I see what bothers you. (FWIW, it’s the same kind of thing that bothers me in the “basic algebra” explanation of adding signed numbers: “For 5 + -3, you start at 5 and move 3 spaces to the left.” I’ve seen this kind of thing in textbooks and maybe said it myself, and it isn’t exactly wrong, but it does make it sound like the two numbers being added are two different kinds of things: one a point, the other a movement/distance/vector.) ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [December 7, 2004, 6:42pm UTC](https://boards.straightdope.com/t/extremely-remedial-question-why-is-a-neg-times-a-neg-a-positive-proof/278389/22 "2004-12-07T18:42:32Z")
> [@Thudlow Boink](#): > > it isn’t exactly wrong The maddening thing from a pedagogical point of view is that it _isn’t_ wrong, just misleading. A clearer example occurs in most courses in multivariable calculus where points and vectors are confounded. Yes: in Euclidean space tangent spaces can be canonically identified with the manifold itself; but in general this doesn’t hold and the student who learned the two concepts as one presents the teacher later on with that much more work. Here, no, the OP is not likely to go on to anywhere the distinction becomes important. However, the question “why?” was asked, not “how can I think about this to make it go away?” I don’t think that the first four one-line proofs in basic ring theory are going to kill anyone who’s made it through high school if they’re willing to read them with an open mind. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [December 7, 2004, 6:46pm UTC](https://boards.straightdope.com/t/extremely-remedial-question-why-is-a-neg-times-a-neg-a-positive-proof/278389/23 "2004-12-07T18:46:37Z")
> [@Mathochist](#): > > I don’t think that the first four one-line proofs in basic ring theory are going to kill anyone who’s made it through high school if they’re willing to read them with an open mind. No, they shouldn’t. But I’m kinda curious as to why you chose to give a proof that’s dependent on the existence of the multiplicative identity when the result holds in arbitrary rings. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [December 8, 2004, 12:01am UTC](https://boards.straightdope.com/t/extremely-remedial-question-why-is-a-neg-times-a-neg-a-positive-proof/278389/24 "2004-12-08T00:01:32Z")
> [@ultrafilter](#): > > No, they shouldn’t. But I’m kinda curious as to why you chose to give a proof that’s dependent on the existence of the multiplicative identity when the result holds in arbitrary rings. Mainly because the steps are (IMHO) clearer to a first-timer if you don’t throw nonunital rings at them at first. Admittedly it can be a fine line sometimes between concrete and abstract, but I thought that a less general proof in the style of modern algebra would be the best course. It isn’t bogged down with artifacts from an interpretation like scaling distances, but keeps it close enough to the familiar (especially to anyone who grew up after the new-new-math) that it should be easily graspable. Bourbaki isn’t _always_ right, you know. ---
**Author:** ![Askance](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/askance/32/8281_2.png) [@Askance](https://boards.straightdope.com/u/Askance)\ **Post date:** [December 8, 2004, 2:59am UTC](https://boards.straightdope.com/t/extremely-remedial-question-why-is-a-neg-times-a-neg-a-positive-proof/278389/25 "2004-12-08T02:59:24Z")
> [@Uncommon Sense](#): > > If you look at a number line and you notice that -3 is three spaces to the left of the zero. You then glance over at the number 9 which is 9 spaces to the right of zero. Common sense would tell you that the nine is twelve spaces from the -3. Yet you only need multiply the -3 by a -3 to get over the twelve spaces? Explain this phenomenon please. How about 3 \* -3 = -9 then? -9 and 3 are 12 spaces away yet simply multiplying by -3 gets you there. Yet you don’t seem to have a problem with that … [Previous page](https://boards.straightdope.com/t/extremely-remedial-question-why-is-a-neg-times-a-neg-a-positive-proof/278389.md?page=1)