why does (-5)*(-4)=20 ?, eom

[Tiberius]

why does (-5)*(-4)=20 ?

In accounting.

If I remove 5 (-5) debits of 4 dollars (* -4), then I have a net gain of 20 dollars in my account.

That doesn't explain why though.

[Chas]

Edit: Didn't realise Chuck had posted something similar beforehand! However I've expanded on his work and tried to explain it in layman's terms for everyone. :-)

That is, in fact, the proof using the rules of arithmetic that the product of two negative numbers is positive.

I didn't 'perform' the operation - I derived the operation.

It makes it completely clear. It is arithmetic, not metaphysics.

I'm not so sure that is correct. Your task is to prove that two negative numbers, when multiplied, becomes a positive number. However, before your proof is over you have the following:

(1) -4 * 5 = -20

(2) -1 * (- 4 * 5) = -1 * -20

(3) -4 * (-1 * 5) = 20

Everyone agrees with (1). You then multiply both sides of (1) with -1 to get (2). However, in stage (3) you simplify the right (-1 * -20) to just 20...but that is using the rule which you haven't yet proved! How do we know -1 * -20 = 20 when your proof for multiplying negative numbers is not yet over?

The way I think the proof is done is as follows. Let's start with your example calculation:

-4 * -5 = ?

Well, let's come up with a similar calculation for which we know the answer:

-4 * (5 - 5) = 0

We know that the answer to the above must be 0, because 5 - 5 is 0, and anything multiplied by 0 is 0.

However, we can also distribute the -4 to both the 5 and -5 (via the distributive property of arithmetic), like so:

(-4 * 5) + (-4 * -5) = 0

Now, we know that -4 * 5 is -20, so we can solve that part of the equation:

-20 + (-4 * -5) = 0

If we add 20 to both sides, we get:

(-4 * -5) = 20

...and our answer! At no point have we actually multiplied two negative numbers here, but by using a calculation for which we know the answer, and expanding it so that we have two negative numbers multiplied at some point in the calculation, we can find the answer by simply rearranging the calculation.

Q.E.D

FYI, if you want to prove that a negative number multiplied by a positive number is a negative number, you can do a similar thing:

4 * -5 = ?

4 * (5 - 5) = 0

(4 * 5) + (4 * -5) = 0

20 + (4 * -5) = 0

(4 * -5) = -20

Q.E.D

Seriously? You want me to re-derive the already proven Associative and Distributive Laws of arithmetic?

The demonstration was to show there is no mystery or confusion about the OP's question.

[snowtracks]

thanks for the replies. seems it's a logical necessity that the answer would be the opposite of two positive numbers. therefore, a math rule convention is used.
previously was thinking about it straight on without actually considering the answer.

[LastPoet]

There are also numerous algebraic constructions that can be made to the same conclusion. The OP if truly interested, should buy a good algebra or analisys book.

[Tiberius]

Seriously? You want me to re-derive the already proven Associative and Distributive Laws of arithmetic?

No, not at all. The point was, you had to use those laws to show that two negative numbers, when multiplied, become a positive number.

Your demonstration didn't do that. All you did was multiply both sides by -1...twice. The second one of the those multiplications had you multiplying -1 with -20, and producing 20, without showing why that was the case.

The demonstration was to show there is no mystery or confusion about the OP's question.

Ok, but it doesn't answer OP's question, which was why -4 * -5 = 20.

All your answer did was move the focus point of the question, so now it's: why does -1 * -20 = 20?

You can't prove things in mathematics with a single example.

[Chas]

Seriously? You want me to re-derive the already proven Associative and Distributive Laws of arithmetic?

No, not at all. The point was, you had to use those laws to show that two negative numbers, when multiplied, become a positive number.

Your demonstration didn't do that. All you did was multiply both sides by -1...twice. The second one of the those multiplications had you multiplying -1 with -20, and producing 20, without showing why that was the case.

The demonstration was to show there is no mystery or confusion about the OP's question.

Ok, but it doesn't answer OP's question, which was why -4 * -5 = 20.

All your answer did was move the focus point of the question, so now it's: why does -1 * -20 = 20?

You can't prove things in mathematics with a single example.

The OP didn't express a problem with anything but the multiplication of two negative numbers resulting in a positive number.

Your comments have nothing to do with either his question or my answer.

[Tiberius]

The OP didn't express a problem with anything but the multiplication of two negative numbers resulting in a positive number.

Your comments have nothing to do with either his question or my answer.

I'm not entirely sure you're reading my posts correctly. My posts have dealt with the proof that multiplying two negative numbers always results in a positive number, which is the answer to the example the OP gave.

Your posts on the other hand involved multiplying by -1 a lot, without any explanation of what was going on. In addition, you multiplied two negative numbers (producing a positive number) in order to prove that two negative numbers produced a positive number. That kind of proof doesn't work in mathematics.

It would be like saying that all square roots are even, and then showing that the square root of 4 is 2 as proof.

[Chas]

The OP didn't express a problem with anything but the multiplication of two negative numbers resulting in a positive number.

Your comments have nothing to do with either his question or my answer.

I'm not entirely sure you're reading my posts correctly. My posts have dealt with the proof that multiplying two negative numbers always results in a positive number, which is the answer to the example the OP gave.

Your posts on the other hand involved multiplying by -1 a lot, without any explanation of what was going on. In addition, you multiplied two negative numbers (producing a positive number) in order to prove that two negative numbers produced a positive number. That kind of proof doesn't work in mathematics.

It would be like saying that all square roots are even, and then showing that the square root of 4 is 2 as proof.

Well, you caught me there.

[bennyboy]

I think Brakeman's approach makes sense. We have to understand what people mean when they have -X of something. Adding a negative sign is just a way of showing a loss.


So 4 * -5 means we have four groups of losses of $5.

-4 * -5 means we lose four groups of losses of $5.

So all you really need to understand is that a loss is a bad thing, so losing a loss is a good thing.

[Aractus]

There is no real-world example for negative numbers, debt or otherwise. Debts are bank bonds/contracts, they're not negative money, money is not negative bank bonds/contracts. Currency and Bonds don't even behave the same way! You can't go into a store, buy your groceries, and in leiu of payment be given a bond.