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

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

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.

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.

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

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

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?

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.

@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)

@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)

And what happens if you divide two negative numbers?

Well that is quite intuitive actually. How many lots of -2 would be needed to make -8?

-2 + -2 +-2 +-2 = -8

So the answer is 4. We need need 4 lots of negative 2 to make negative 8.

If you would like a mini logical proof, try this which I posted earlier. You can do it yourself with different numbers.

Step One
2 - 2 = 0
Happy this is true I assume. Most people are.

Step Two
Multiply both sides by negative 1.
(2 x -1) + (-2 x -1) = 0 X -1

Step Three
Assuming you are happy that zero times anything is zero and that a negative times a positive is negative we can simplify to
-2 + (-2 x -1) = 0

Step Four
Add two to both sides
-2 x -1 = 2

We have just demonstrated that negative two x negative one is equal to postive two. You can try this again with whatever numbers you like.

Basically if zero times any number is zero and a negative number times a positive number is negative then it follows logically that a negative number times a negative number MUST be positive.

Imagine a car moving backwards (negative direction) at a certain speed (negative value). If you want to see how far it travels in a certain amount of time (negative value, playing the video backwards), you are essentially multiplying two negative values. The result is a positive distance because the car is moving forward (positive direction) when viewed in reverse.

Try this. Suppose a lovely friend says she’ll lend you £30 and she gives you 3 slips “IOU £10” - they’re ‘-£10 notes’. So, that’s -10 x3, -£30 in total.
So then if she decided to let you off some of the debt, she’d take away (-) one of those -£10 notes from you and leave you with a debt of -£20. Subtracting a negative number makes it less negative, more positive.
Now suppose she decides actually she’s going to let you off the whole amount - she’d take away (-) all 3 IOUs. -3x(-£10) . The net result of all this is of course that she’s given you + £30.

ThreeDeafMice This was great to read and I followed it until:

when we look at the structure of our set under multiplication, to be self-consistent (so as not to be able to prove logically things like 1 = 0, which would make our system useless), it turns out that -1 x -1 has to give the same answer as 1 x 1.
similarly Simon x Simon has to give the same answer as 2 x 2.
So the answer is that it’s implicit in the structure of the number system we invented. If minus x minus wasn’t plus we couldn’t use arithmetic for anything useful as it could “prove” nonsense answers like 1 = 0.

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

So to me that says it’s not madness for me to think as I would do (without prior knowledge of the magic rule) that -3 x -4 = -12
because I’d be making a negative 3, four times more negative eg looking leftwards down the number line gets to -12. That seems logical to me, that negative3 - x negative 4 = negative12 looking up the number line rightward. That mirror image idea is wrong obviously but to me it demonstrates logical consistency if positive3 x positive4 = postive12

And what happens if you divide two negative numbers?

I can grasp that the magic rule here is that it’s like using a grammatical double negative to express something binary , like (‘I’m not NOT curious about maths‘ = same as saying ‘ I AM curious about maths’):

so that -3 x -4 = positive12
I do accept that conclusion but only because I can accept the language rule and I’m being told the maths question follow the same rules.

However (without being told there’s a magic rule that I need to overlay over this) I can’t seem to understand or accept this as a number number sentence that adds up to positive 12. I don’t want to have to hold articles of faith in maths, because I have been told how logical and demonstrable maths is.

Thank you so much for explaining this. I do follow your example and that was kind of you to write off my debt .
Could we use this example and extend it to a negative number multiplied by a negative number? Could you say that in a number sentence and in a language sentence?

I’lll try:
so say I owed you ten pounds (whatever my bank balance that’s not relevant now) and then you phoned me up and said you’d miscalculated and actually I owed you three times more than that- which my brain says is -10 x 3 =—30 then I say OK fine and pay you my debt of £30. And that is also the same as saying as of today’s transaction I have -30 in my previous bank account balance of x. Fine. Here we have multiplied a negative number by a positive numbers.

But let’s try to employ the magic rule for multiplying two negative numbers together in this example ..
if I owed you ten pounds and then you phoned me up and said yes the £10 debt still stood but you’ve done your checking and I DIDNT owe you three times more than that that … then I’d still only owe you £10.

Gahhh I have just multiplied -10 by 1 haven’t I?! I just can’t seem to understand and expressing in words what -10 x -3 = 30 would actually mean in a conversation about my personal balance sheet. I think if I could imagine it as a conversation then I could understand the concept behind it. It’s still escaping me though.

Taking away a negative is the same as adding a positive

You have two hundred pounds in the bank but you owe me ten pounds

So you have +200 but -10
So combining these together you have +190

Now, I say don't worry about repaying that tenner. I take away your debt of ten pounds.

+190 - (-10) = +190 + 10 = 200

I have put + signs in front of the positive numbers.

@loveyouradvice

Doing something nice (+) to a nice person (+) = positive result (+)

Doing something nice (+) to a nasty person (–) = negative result (–)

Being mean (–) to a nice person (+) = negative result (–)

Being mean (–) to a mean person (–) = good result (+)

argh tired brain still struggling! The debt was -£4 though so if the bank repaid £4, you’d be at zero surely?

It’s basic number theory.

We start with the positive integers - the counting numbers, and the operation of addition.

When we “add” two integers we necessarily get a third integer. The set of integers is closed under addition.

we create an ordering on the integers, a relationship so that for each integer we can say which come earlier (are “smaller”) and which come later (are “bigger”).

We note that zero is a special value under addition.It maps each integer back to itself. It’s called the “identity” under addition.

We define an operation inverse to addition and call it subtraction. We note that both addition and subtraction share the same identity element: zero.

Unfortunately our set of positive integers isn’t closed under subtraction. 3 subtract 5 isn’t in our set.

So we extend our set: we invent a new concept of “negative numbers” and include them in our set so now our set of (all) integers is closed under both addition and subtraction.

3 subtract 5 is one of those negative integers. Let’s call it Simon. Simon has the special property that Simon add 2 is the zero. In recognition of this fact we label Simon as “-2” and speak it as “minus two”.

in fact we can pair each positive integer in our original set with a friend in the new set of negative numbers, so that when we add any positive integer to its friend the answer is the additive identity, zero.

This gives a convenient labeling for the negative numbers, each as the inverse under addition from the identity element (zero) of a different positive numbers. Writing them pairwise we have 1 and -1, 2 and -2, 3 and -3 etc.

Now we move on to a new operation called multiplication. What properties does it have that will make it useful? Our set of integers is closed under multiplication, but not under the inverse operation “division” - we’d need to invent the idea of rational numbers (fractions) for that - but we can do that later, not now. To be useful we require that multiplying any number in our set by the additive identity (zero) gives zero.

when we look at the structure of our set under multiplication, to be self-consistent (so as not to be able to prove logically things like 1 = 0, which would make our system useless), it turns out that -1 x -1 has to give the same answer as 1 x 1.

similarly Simon x Simon has to give the same answer as 2 x 2.

So the answer is that it’s implicit in the structure of the number system we invented. If minus x minus wasn’t plus we couldn’t use arithmetic for anything useful as it could “prove” nonsense answers like 1 = 0.

God that seems complicated...if the signs are different it is a negative, if the signs are the same it's a positive...simples

Yes, but also… some people just cant be helped. I think we found one of them.

I don't think your credit card company will accept you ignoring negative numbers.

This has always puzzled me. I googled it and it basically says that a negative number reverses the direction on the number line, therefore multiplying by a negative reverses it twice, resulting in a positive. I think I understand it....... 🤔

As a qualified pedant, I hope you don't mind me pointing out that it should be "An enemy's enemy..." Just the one enemy.