(October 12, 2013 at 2:51 am)Aractus Wrote: I'll give you my take on it - mathematics is invented by humans, and is based on rules, which are also invented.

If they're "invented" then they *just happen* to correspond, and predict, and describe perfectly everything that is found to be true in Physics, Chemistry, Biology, and Probability ? Right.

http://www.youtube.com/watch?v=mQYeV4jjKPk

http://www.youtube.com/watch?v=SEiSloE1r-A

(October 12, 2013 at 4:33 am)Bucky Ball Wrote:

(October 12, 2013 at 2:51 am)Aractus Wrote: I'll give you my take on it - mathematics is invented by humans, and is based on rules, which are also invented.

If they're "invented" then they *just happen* to correspond, and predict, and describe perfectly everything that is found to be true in Physics, Chemistry, Biology, and Probability ? Right.

The thing is... mathematics would work equally well regardless of the nomenclature invented to describe it.
Numbers were invented as a means to describe real physical quantities.

(October 12, 2013 at 4:06 am)Faith No More Wrote: Perhaps he/she should have phrased the question that way, then, because the answer to why -5 * -4 = 20 is that 4 * 5 = 20 and when multiplying two negative numbers, you get a positive. If the last part was what he/she was really after, it should have been phrased more clearly. Apparently, I don't have the mind reading powers that you do.

Oh please, I've heard - more than once - the whether lady on TV claim that the temperature will "double" over the course of the day, eg 10deg - 20deg. I've seen, multiple times, people at cash registers have to get out a calculator to do a perfectly simple sum.

We can reliably assume that the OP is already comfortable with the idea that 5 * 4 = 20.

4 * 5 = 20

-1 * 4 * 5 = -20

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

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

-4 * 5 = -20

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

-4 * (-1 * 5) = 20

-4 * -5 = 20 Q.E.D.

Chas, you didn't really help out there, mainly because you perform the calculation -1 * -20 = 20, which is just another version of the problem OP was running into (two negative numbers multiplied becoming a positive number).

(October 12, 2013 at 5:41 am)Aractus Wrote: We can reliably assume that the OP is already comfortable with the idea that 5 * 4 = 20.

Can we? You have more faith in humanity than I do.

(October 12, 2013 at 8:26 am)Tiberius Wrote: Chas, you didn't really help out there, mainly because you perform the calculation -1 * -20 = 20, which is just another version of the problem OP was running into (two negative numbers multiplied becoming a positive number).

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.

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

(-5*-4)=20 Q.E.D.

No need to inject -1*-x=x before deriving -y*-x=yx

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. :-)

(October 12, 2013 at 8:50 am)Chas Wrote: 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

(October 12, 2013 at 12:01 am)snowtracks Wrote: 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.