How do I explain to a Y7 why when you multiply two negative numbers it becomes a positive?

[loveyouradvice]

Just that really - I know it does, but can't explain why!

Hoping @noblegiraffe and others might know!!

@LovelyVase

sorry to ask again and I really appreciated how you talked this through with words- but this conclusion part does sound to me like there’s just a rule. That we have to just accept that ( - x - = + ) as the price we pay to have the mathematical system that we use for everything else.

Now you’re getting it.

That makes me feel this task has revealed a glitch in the maths matrix that I just have to accept

It’s not a glitch; it’s a feature.

Underlying the reason that this is difficult to explain is that bare numbers are a different concept to “quantities of things”. Three apples and three bananas are entry points to the idea of “three” but three as a number is really quite different to “a quantity of something”.

I don’t want to have to hold articles of faith in maths, because I have been told how logical and demonstrable maths is.

-3 x-4 = +12 is very demonstrable. You do it by “proof by contradiction”. Assume -3 x -4 = something else and show it leads to a logical fallacy. Ergo your assumption is wrong.

But ion order to do that you have to carefully define how you want the numbers to behave under arithmetical operations. In my opinion it’s easier to start with “these are the numbers, these are the operations (multiplication and addition) and this is how the numbers work under those. If “minus x minus = plus” seems difficult to follow in the context of minus numbers being “debts” or “bad things” the fault is with using those things as analogies. Clearly the 3 that exists in the world of both positive and negative numbers is different to a quantity of bananas because -3 exists as a “thing” in its own right just like +3, and -3 can never be a quantity of bananas.

By the way mathematicians have abstracted the behaviour of the natural numbers under addition and multiplication into a concept called a ring. You might enjoy a scan of the Wikipedia page on it: https://en.wikipedia.org/wiki/Ring(mathematics)

@LovelyVase

And what happens if you divide two negative numbers

Division is the inverse of multiplication. One way to answer your question is to note that dividing a by b is “the same as” multiplying by 1/b. You’ll find that things fall apart pretty quickly if 1 / (-b) isn’t the same as - (1/b) so dividing by a negative number inverts the sign of the answer in the same way as multiplication.

If you enjoyed looking at the Wikipedia page for rings in my previous answer, you could check the page on fields. A field is a commutative ring where every element except “zero” has a multiplicative inverse, i.e. where division is allowed.

https://en.wikipedia.org/wiki/Field_(mathematics)

Unfortunately this proof isnt valid because it involves multiplying by zero, in this line:

(2 x -1) + (-2 x -1) = 0 X -1

it doesn’t mean that your conclusion, that -2 x -1 = 2 is wrong, it just means you haven’t “proved” it.

Why can't I multiply both sides of the equation by -1 where one side is zero?

Isn't this standard alegbra? And isn't algebra just a form of logical argument whereby you know that if the starting relatioship given is true and all rules are followed correctly the output result must also be true?

Yes but the bank correction is to take four lots of minus 1 and take them away from the negative side to get you back to zero

Ok. I will retract my objection. Proofs that involve multiplication by zero are inherently suspicious though - and proofs that involve division by zero are invalid.

Can I frame your demonstration like this?
by definition of (-2) as the additive inverse of +2:

2 + (-2) = 0

By the property of the multiplication operator:

-1 * ( 2 + (-2) ) = 0

Assume multiplication is distributive over addition (and that's a big deal), that a(b+c) = ab + ac.

Then

(-1 2) + (-1 -2) = 0

Therefore

(-1 * -2) = 2

QED.

Here's another take:

Assume -a x -b ≠ a * b (for non zero a, b)

Therefore -a x -b = a * c (c ≠ b)

a x -b + a * c = 0

Using the distributive property of multiplication over addition:
a x (-b + c) = 0

By the definition of 0, therefore either

a = 0 or ( -b + c) = 0

In other words

a = 0 or c=b

However we stated that a was non-zero and that c ≠ b, contradicting our conclusion.

Therefore our initial assumption -a x -b ≠ a x b must be wrong, which proves
-a x -b = a x b

Ok. I will retract my objection. Proofs that involve multiplication by zero are inherently suspicious though - and proofs that involve division by zero are invalid.
Can I frame your demonstration like this?
by definition of (-2) as the additive inverse of +2:
2 + (-2) = 0
By the property of the multiplication operator:
-1 * ( 2 + (-2) ) = 0
Assume multiplication is distributive over addition (and that's a big deal), that a(b+c) = ab + ac.
Then
(-1 2) + (-1 -2) = 0
Therefore
(-1 * -2) = 2

I don't wish to be pedantic but that isn't a reframing of my argument. It literally is exactly the same as my argument and at the exact moment in which you say

By the property of the multiplication operator:

You have multiplied zero by minus one.

I apologise if I am mistaken but this all feels a little bit Chatgptish.

I multiplied both sides of an equation by -1. I didn't "multiply by zero". When I looked more carefully I saw that you didn't either, and retracted my objection to your proof. What you did is fine.

In what way is it like ChatGPT (which I've never used)?

An interesting conclusion is that the distributivity requirement is highlighted. So an answer (not necessarily a very helpful one) to the OP's question is that if minus times minus isn't plus, then three sets of (four + five) sausages might not be the same as three sets of four sausages plus three sets of five sausages.

I'm loving how much activity my "simple" question has generated... I've found my people!! Thank you

It beats discussing the state of the world, that's for sure.