RE: why does (-5)*(-4)=20 ?, eom
October 12, 2013 at 9:36 am
(This post was last modified: October 12, 2013 at 9:46 am by Tiberius.)
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. :-)
(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 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'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:
I didn't 'perform' the operation - I derived the operation.
It makes it completely clear. It is arithmetic, not metaphysics.
(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
