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I've read almost all of the previous posts on this question and several webpages/videos and they all talk anecdotally or through some form of example. Such as using the distributive law, via implication or using a real life example. I can't seem to find something that I'm happy with for an explanation.

With addition/subtraction, it is enough for me to accept that in the case of -2 - (2), I'm located on -2 which is 2 units to the left of 0. Subtracting 2 from that is to move in the negative direction by 2. Like wise -2 - (-2) is being located on -2 and moving in the negative direction by -2 units which is to move 2 units in the positive.

But with multiplication and division, I'm failing to appreciate what is occurring. I can follow examples such as defining a positive and negative direction but is there something mathematically there that is convincing? Or are examples using algebraic manipulation as good as it gets, or even is there a definition or set of axioms that could better my understanding?

I appreciate this is asked often but I really could not find anything to understand from the previous posts.

thanks

salman
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    What is your problem with the answers to the above? – Ross Millikan Aug 16 '17 at 20:27
  • Perhaps showing $(-1)\times a = (-a)$ (LHS is multiplying by the additive inverse of $1$, RHS is taking the additive inverse) using the axiomatic approach to the real numbers (the reals are an ordered field) is the way forward to something convincing? https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach – Shuri2060 Aug 16 '17 at 20:31
  • A lot of the proofs out there are axiomatic, but you might think they look like "examples" if you only glance at them. Since you basically just dismissed 48 previous answers without specifically saying how any one of your criticisms applies to any one of those answers, how can anyone guess what you're looking for? – David K Aug 16 '17 at 21:30

2 Answers2

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is there something mathematically there that is convincing? Or are examples using algebraic manipulation as good as it gets, or even is there a definition or set of axioms that could better my understanding?

The usual algebraic manipulations actually do provide what you want: A proof that the axioms of an ordered field, including the distributive law, imply that for all $x,y<0$, we have $xy,x/y>0$. They may not be phrased that way for a lay audience, but that's what they're doing.

Example that gets most of the way there: https://math.stackexchange.com/a/384866/87023

Chris Culter
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I will use your example with addition/subtraction. We have $-2$, so we are 2 units to the left of 0. Subtracting 2 means moving "to the left", in negative direction, by 2 (and it's just the same as adding $-2$).

Similarly, multiplication by 3 means "don't move direction, just expand the distance from 0 so that it is three times the current one". And just the same way as adding 2 is opposite to subtracting 2 (the difference is "use opposite direction"), multiplying by positive number is opposite to multiplying by negative number, it just changes the direction. So $(-5) \times 4$ means "go four times further away from zero" and $(-5) \times (-4)$ means "go four times further from zero in opposite direction" (in positive direction).

As you probably know, division is somehow equivalent to multiplication. The idea is just the same as for multiplication, the only difference is that we reduce the distance from 0.

I mean, you can use exactly the same intuitive logic as used with addition/subtraction.

Puding
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  • I can understand almost all the analogies and number line anecdotes.

    Is it correct that prior to negative x negative multiplication being defined, was it the distributive law in algebra that motivated the definition of -ve x -ve = +ve? @puding

     – salman Aug 17 '17 at 17:59
  • I'm not sure what are you asking. However, I consider these things pretty intuitive (see my answer, also an approach using debts is good), so you don't have to do all algebraic manipulation with distributive laws etc. to use negative by negative multiplication. It's just useful to prove it. @salman – Puding Aug 17 '17 at 18:17
  • well i've read several views of people explaining negative multiplication. But one that I noted was that the distributive law would not work if x + (-x) = 0 wasn't true and that lead to the convention (-1)(-1) = 1. – salman Aug 17 '17 at 18:51
  • Yeah, there are probably more ways how to interpret it. One could also say that definition of $-2$ is "x, for which $2+x=0$", and it is true. You can look at these things from many different angles. – Puding Aug 17 '17 at 19:22
  • ah ok cheers. So how is this property supposed to be learnt both intuitively and abstractly from the start? Is it expected that people just accept the result, and when they reach advanced mathematics that they go on to define/proof it? For example, does (-1)(-1) actually mean anything other than what is written on the paper? – salman Aug 17 '17 at 20:01
  • In real life, yes, this is the way that negative numbers are taught in schools, at least here in my country :) And meaning of numbers is more like a philosphical question (and it shouldn't be debated here in comments under an answer...). – Puding Aug 17 '17 at 20:10
  • Let us continue this discussion in chat. – salman Aug 17 '17 at 20:33