Realworld examples of x * y = +z
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Realworld examples of x * y = +z
I am not in school, this is not a homework problem.
I am trying to help a friend prepare for a muchencompassing general math test.
I am trying to explain to him that a negative times a negative is a positive. I have come up with realworld examples for positive times positive, and negative times positive, but can't seem to come up with one for this.
Any suggestions? I know this shouldn't be that hard, but I am stumped.
NB: I am not looking for a mathematical proof.
I am trying to help a friend prepare for a muchencompassing general math test.
I am trying to explain to him that a negative times a negative is a positive. I have come up with realworld examples for positive times positive, and negative times positive, but can't seem to come up with one for this.
Any suggestions? I know this shouldn't be that hard, but I am stumped.
NB: I am not looking for a mathematical proof.
Re: Realworld examples of x * y = +z
A Google search found me this, I don't know whether it will help the child.
http://mathforum.org/dr.math/faq/faq.negxneg.html
http://mathforum.org/dr.math/faq/faq.negxneg.html
 jestingrabbit
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Re: Realworld examples of x * y = +z
One reason to believe it is that
0 = (1)*0 = (1)*(1 + (1)) = (1)*1 + (1)*(1) = (1)*(1) 1
So you've got that 0 = (1)*(1) 1, which implies that (1)*(1) = 1.
Its not real world but it is common sense.
0 = (1)*0 = (1)*(1 + (1)) = (1)*1 + (1)*(1) = (1)*(1) 1
So you've got that 0 = (1)*(1) 1, which implies that (1)*(1) = 1.
Its not real world but it is common sense.
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Re: Realworld examples of x * y = +z
When bad things happen to bad people, that's a good thing.
Re: Realworld examples of x * y = +z
You could explain how the negative ends of two magnets repel the same way the positive ones do. It may not be quite what you're looking for but it can at least offer a way to remember it.
Re: Realworld examples of x * y = +z
Two wrongs make a right.

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Re: Realworld examples of x * y = +z
You just gained a 20 dollars debt: + (20)
you are now 20$ less rich.
You just lost that 20 dollars debt:  (20)
you are now 20$ richer.
you are now 20$ less rich.
You just lost that 20 dollars debt:  (20)
you are now 20$ richer.
Re: Realworld examples of x * y = +z
A positive and a negative are negative, so just think of... (a)*(b) as [a*(b)]. The latter is a negative, so the opposite of that will be positive. Also, "losing debt" results in gaining money, as was already pointed out.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.
Re: Realworld examples of x * y = +z
Thanks for the suggestions so far ...
I like the proverby ones, but again, it's not really to help him remember it, it's just he understands things better if he can picture what we're talking about.
And re: the debt (those were the examples I used for  * + and + * + but, for the most part, you're really just subtracting a negative, more than multiplying, I think.
But, again, thanks, still open to ideas, though!
I like the proverby ones, but again, it's not really to help him remember it, it's just he understands things better if he can picture what we're talking about.
And re: the debt (those were the examples I used for  * + and + * + but, for the most part, you're really just subtracting a negative, more than multiplying, I think.
But, again, thanks, still open to ideas, though!
Re: Realworld examples of x * y = +z
Negating is the same thing as reflecting, and reflecting twice gives you the same thing back. (To be more precise, negation is the same thing as reflection through the origin of the real line.) Find your friend a mirror?
Also, subtracting a negative is the same thing as multiplying it by [imath]1[/imath], then adding it.
Also try double negatives in language (the ones that actually mean positive, not the other ones) that don't use the same word twice, such as "I didn't avoid" or "I haven't missed"...
Also, subtracting a negative is the same thing as multiplying it by [imath]1[/imath], then adding it.
Also try double negatives in language (the ones that actually mean positive, not the other ones) that don't use the same word twice, such as "I didn't avoid" or "I haven't missed"...

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Re: Realworld examples of x * y = +z
Have you tried explaining it to him using addition? Or was that exactly what you were trying to avoid? If he understands why subtracting a negative increases the sum, then maybe this will help.
[math]3 * 3 = 3 + 3 + 3 = 9[/math]
So, it follows that when you multiply [math]3 * 3 = 3 + 3 + 3 = 9[/math]
So, when the second number in [math]x * y[/math] is negative, you are subtracting x, y times.
So, [math]3 * 1 = (3) = 3[/math]
or
[math]5 * 4 = (5) (5) (5) (5) = 5 + 5 + 5 + 5 = 20[/math]
Hope that helps.
[math]3 * 3 = 3 + 3 + 3 = 9[/math]
So, it follows that when you multiply [math]3 * 3 = 3 + 3 + 3 = 9[/math]
So, when the second number in [math]x * y[/math] is negative, you are subtracting x, y times.
So, [math]3 * 1 = (3) = 3[/math]
or
[math]5 * 4 = (5) (5) (5) (5) = 5 + 5 + 5 + 5 = 20[/math]
Hope that helps.
Re: Realworld examples of x * y = +z
By the way, writing it as "x * y = +z" is a bad idea. The thinking that anything with a minus sign in front of it must be negative (or that any variable by default represents something positive) is part of what leads to all this confusion in the first place.
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Re: Realworld examples of x * y = +z
Minus times minus is plus, the reason for this we shall not discuss.
Re: Realworld examples of x * y = +z
You're walking backwards (negative speed) and you want to know where you used to be (negative time). Well, negative speed times negative time equals positive distance, and sure enough, you used to be ahead of where you are now.
Alternately, if you just want a memory aid, perhaps this will work: If your buddy can visualize multiplication as finding the area of a rectangle, then consider it as a "cardboard" rectangle. Paint the front red and the back blue. Now when both side lengths x and y are positive, the result is positive and the rectangle xy will be red. For each side that is negative, flip the rectangle over. So positive times negative gives you a blue rectangle, and a negative value. But negative times negative makes you flip it over twice, and you're back to the positive, red side.
More abstractly, every time you multiply by a minus sign, the answer switches sign.
Alternately, if you just want a memory aid, perhaps this will work: If your buddy can visualize multiplication as finding the area of a rectangle, then consider it as a "cardboard" rectangle. Paint the front red and the back blue. Now when both side lengths x and y are positive, the result is positive and the rectangle xy will be red. For each side that is negative, flip the rectangle over. So positive times negative gives you a blue rectangle, and a negative value. But negative times negative makes you flip it over twice, and you're back to the positive, red side.
More abstractly, every time you multiply by a minus sign, the answer switches sign.
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Re: Realworld examples of x * y = +z
z4lis wrote:Also, "losing debt" results in gaining money, as was already pointed out.
Yeah, that's how I convinced myself back whenever I first learned this.
5 means I owe 5.
5*(5) means I have 5 things that say "you owe 5", which means now I owe 25.
(5)*(5) means I owe 5 things that say "you owe 5" on them, which means people have to pay me 25.
Re: Realworld examples of x * y = +z
If you are gaining $5 per day in the stock market, then 6 days from now you will have $30 more, and 6 days from now you will have $30 more.
If you are gaining $5 (losing $5, not unheard of these days) per day in the stock market, then 6 days from now you will have $30 more dollars, and 6 days ago you had $30 more.
If you are gaining $5 (losing $5, not unheard of these days) per day in the stock market, then 6 days from now you will have $30 more dollars, and 6 days ago you had $30 more.
Re: Realworld examples of x * y = +z
antonfire wrote:By the way, writing it as "x * y = +z" is a bad idea. The thinking that anything with a minus sign in front of it must be negative (or that any variable by default represents something positive) is part of what leads to all this confusion in the first place.
Excellent point. In this case, I was just being lazy. When I talk to him, I am using specific numbers to show him what I mean. We are NOT at variables yet.
ETA: Even MORE good suggestions ... thank you, all. I will throw some of these at him and see if they stick. Of course, half the problem is he has spent his whole life hating math (as most (I think) people do) and so now he has an initial instinct that it's hard and complicated.
Re: Realworld examples of x * y = +z
Negation of a number is just swinging the point around to the point on the other side that is equally far away from 0.
When you negate a negative number, you swing it back a second time, back to the positive side of the real line.
When you negate a negative number, you swing it back a second time, back to the positive side of the real line.
 gmalivuk
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Re: Realworld examples of x * y = +z
Uhhuh, but that's not exactly a realworld example of why, for example, (4)*(3)=12.
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Re: Realworld examples of x * y = +z
You have ice cubes.
Each ice cube, when it melts, cools the water in the beaker by 4 degrees.
You can move them in and out quickly enough that they don't melt.
You have three breakers, each with an equal number of ice cubes in them, and have an equal starting temperature.
From beaker A, you take out 3 ice cubes.
Into beaker B, you put in 5 ice cubes.
You then wait for the ice cubes to melt. Beaker B ends up being 33 degrees Celsius.
What is the temperature of beaker B and C?
Temp(B) + 5*(4) = Temp(C) (you added 5 units of 4 temperature to beaker B compared to C).
Temp(A) + (3)*(4) = Temp(C) (you removed 3 units of 4 temperature from beaker A compared to beaker C).
Temp(B) = 33
Temp(C) = 33 + 5*4 = 3320 = 13
Temp(A) + (3)*(4) = Temp(C) = 13
Temp(A) + 12 = 13
Temp(A) = 1
Naturally you can start with adding or taking out a single ice cube. Then try 2. Then try taking out 3, cutting one in half, and putting that half back in.
Each ice cube, when it melts, cools the water in the beaker by 4 degrees.
You can move them in and out quickly enough that they don't melt.
You have three breakers, each with an equal number of ice cubes in them, and have an equal starting temperature.
From beaker A, you take out 3 ice cubes.
Into beaker B, you put in 5 ice cubes.
You then wait for the ice cubes to melt. Beaker B ends up being 33 degrees Celsius.
What is the temperature of beaker B and C?
Temp(B) + 5*(4) = Temp(C) (you added 5 units of 4 temperature to beaker B compared to C).
Temp(A) + (3)*(4) = Temp(C) (you removed 3 units of 4 temperature from beaker A compared to beaker C).
Temp(B) = 33
Temp(C) = 33 + 5*4 = 3320 = 13
Temp(A) + (3)*(4) = Temp(C) = 13
Temp(A) + 12 = 13
Temp(A) = 1
Naturally you can start with adding or taking out a single ice cube. Then try 2. Then try taking out 3, cutting one in half, and putting that half back in.
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Re: Realworld examples of x * y = +z
gmalivuk wrote:Uhhuh, but that's not exactly a realworld example of why, for example, (4)*(3)=12.
Sure it is. All you have to do is interpret the two numbers differently: one is a point on the number line, and one is a scale factor. Negative scale factors are "orientationreversing."

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Re: Realworld examples of x * y = +z
Sure it is. All you have to do is interpret the two numbers differently: one is a point on the number line, and one is a scale factor. Negative scale factors are "orientationreversing."
Yes, you have positive times positive is positive because three groups of four apples is twelve apples. Negative times positive is negative because if you take away three groups of four apples, you've lost twelve apples. negative times negative is positive because you can interpret one as a point and the other as a scale factor
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Re: Realworld examples of x * y = +z
I wish people would take scale factors more seriously. What if I said "change of units" instead? If one of the numbers is a time and another is a velocity, a negative scale factor corresponds to changing units and thinking of what was the negative direction as the positive direction.
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Re: Realworld examples of x * y = +z
People who actually use scale factors regularly might take them more seriously. But it's still definitely not a realworld example the way it is to talk about debt and owing versus earning and such.
I don't care how *you* think of negative numbers, because the fact remains that most of the other examples given here would actually help more with a novice's understanding than the point/scale factor example you gave.
I don't care how *you* think of negative numbers, because the fact remains that most of the other examples given here would actually help more with a novice's understanding than the point/scale factor example you gave.
Re: Realworld examples of x * y = +z
Lets play with vectors.
[math]\vec{V} = \left(\begin{array}{r}4\\3\\10\end{array}\right)[/math]
The same rule with
If I multiply [imath]\vec{V}[/imath] by [imath]2[/imath] i extend it:
[math]\vec{V} \cdot 2 = \left(\begin{array}{r}8\\6\\20\end{array}\right)[/math]
If I multiply it by [imath]2[/imath] I flip it, then extend it:
[math]\vec{V} \cdot 2 = \left(\begin{array}{r}8\\6\\20\end{array}\right)[/math]
Now for a more concrete example:
To multiply by [imath]1[/imath] is the same as substracting from [imath]0[/imath].
[math]3 * 5 = 1 * 1 * 3 * 5 = 1 * 1 * 15 = 0  ( 0  15) = 15[/math]
Now with the logic that if you take away ¤15 of my debt, I am now ¤15 richer (¤ is the universal currency sign, substitude with whatever) it makes sense.
[math]\vec{V} = \left(\begin{array}{r}4\\3\\10\end{array}\right)[/math]
The same rule with
If I multiply [imath]\vec{V}[/imath] by [imath]2[/imath] i extend it:
[math]\vec{V} \cdot 2 = \left(\begin{array}{r}8\\6\\20\end{array}\right)[/math]
If I multiply it by [imath]2[/imath] I flip it, then extend it:
[math]\vec{V} \cdot 2 = \left(\begin{array}{r}8\\6\\20\end{array}\right)[/math]
Now for a more concrete example:
To multiply by [imath]1[/imath] is the same as substracting from [imath]0[/imath].
[math]3 * 5 = 1 * 1 * 3 * 5 = 1 * 1 * 15 = 0  ( 0  15) = 15[/math]
Now with the logic that if you take away ¤15 of my debt, I am now ¤15 richer (¤ is the universal currency sign, substitude with whatever) it makes sense.
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Re: Realworld examples of x * y = +z
Stand in the middle of a field. Positive numbers go east and north.
Draw a rectangle whose opposite vertex is to your south and west. With respect to the coordinate system, the other vertices are say, 3 meters west (3m) and 3 meters south (3m).
The area is positive.
Draw a rectangle whose opposite vertex is to your south and west. With respect to the coordinate system, the other vertices are say, 3 meters west (3m) and 3 meters south (3m).
The area is positive.
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Re: Realworld examples of x * y = +z
How does that example help? The area is the same as if it's 3m south and 3m east, even though that's 3 and +3.
Re: Realworld examples of x * y = +z
One thing I noticed is that adding a zero onto the left of a number (or right, if you have a decimal point) does not change a number's value. For example, 4 = 04 (and 4.9 = 4.90 (unless you're a scientist)). The same applies to negative numbers, as 4 = 04. Therefore you can treat negative numbers as binomials and multiply appropriately:
3 * 4
(0  3)(0  4)
(0  3)0  (0  3)4
(0)0  (3)0  ((0)4  (3)4)
0  0  (0  12)
(12)
12
3 * 4
(0  3)(0  4)
(0  3)0  (0  3)4
(0)0  (3)0  ((0)4  (3)4)
0  0  (0  12)
(12)
12
she/they
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Re: Realworld examples of x * y = +z
Sizik wrote:(12)
12
This is the step we're discussing. I don't see the relevance of the other stuff you wrote; it's neither a proof nor an intuition, since it depends on a quirk of notation.
Re: Realworld examples of x * y = +z
t0rajir0u wrote:Sizik wrote:(12)
12
This is the step we're discussing. I don't see the relevance of the other stuff you wrote; it's neither a proof nor an intuition, since it depends on a quirk of notation.
I sort of skipped a step there. Rather thinking of (12) as (1)(12), think of it as 0  (12). Subtracting a negative gives you a positive.
she/they
gmalivuk wrote:Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one metame to experience both body's sensory inputs.
Re: Realworld examples of x * y = +z
Sizik wrote:Subtracting a negative gives you a positive.
Yes, but the whole point of this thread is basically to try to understand this point.
Re: Realworld examples of x * y = +z
I think the answer is firstly to stress that there is no reason why the rule has to be true; you could define it otherwise, it's just not very useful to do so. By choosing it to be this way it makes lots of real world things work. Money and debt is one but a child might not be too familiar with money and debt (I haven't spotted how old your friend is). I wonder if you could use relative distances as another.
Suppose I have a fixed point (choose something in the room, say a table) and I measure distances from that point along a line. Let distances in one direction be positive and in the other be negative. So you're in front of the table and your position relative to the table is +1m. I'm behind the table and my position is 1m. Having the sign is useful; it tells us more about the situation that just the number. Now what is your position relative to me? It's obvious that we're 2m apart, and you are further along the line in the forward direction so the answer must be +2m. What do we need to do to make the maths work?
To calculate someone's position relative to mine I have to remove myself from the picture; in other words I take my self away. The language is the same in maths; to find someone's position relative to mine I have to take my position away from theirs. So I now have (your position)  (my position) = 2 = 1  (1)
The only way this works is if (1) is the same as +1.
To get from here to (x) * (y) = +xy then remember that (x) * (y) =  (x*(y)) =  (xy) = xy
Suppose I have a fixed point (choose something in the room, say a table) and I measure distances from that point along a line. Let distances in one direction be positive and in the other be negative. So you're in front of the table and your position relative to the table is +1m. I'm behind the table and my position is 1m. Having the sign is useful; it tells us more about the situation that just the number. Now what is your position relative to me? It's obvious that we're 2m apart, and you are further along the line in the forward direction so the answer must be +2m. What do we need to do to make the maths work?
To calculate someone's position relative to mine I have to remove myself from the picture; in other words I take my self away. The language is the same in maths; to find someone's position relative to mine I have to take my position away from theirs. So I now have (your position)  (my position) = 2 = 1  (1)
The only way this works is if (1) is the same as +1.
To get from here to (x) * (y) = +xy then remember that (x) * (y) =  (x*(y)) =  (xy) = xy
Re: Realworld examples of x * y = +z
A couple of people have mentioned age. He's grown. He's just not "mathy" and it has been a while since he has taken ANY math. And I get the feeling (could be wrong) that where he grew up didn't necessarily have the best schools.
I do really like all the examples (even the ones that didn't really address my friend's particular problem), especially since I do some tutoring now and again for people of all different ages, abilities, backgrounds, so I'm sure any one of the examples will help someone out there.
I do really like all the examples (even the ones that didn't really address my friend's particular problem), especially since I do some tutoring now and again for people of all different ages, abilities, backgrounds, so I'm sure any one of the examples will help someone out there.
Re: Realworld examples of x * y = +z
TimC wrote:I think the answer is firstly to stress that there is no reason why the rule has to be true; you could define it otherwise, it's just not very useful to do so.
That's not true at all. [imath](1)(1) = 1[/imath] is a direct consequence of the axioms of any definition of the integers.
Re: Realworld examples of x * y = +z
Sizik binomial interpretation is cute.
The problem arises because people think integers are scalars, but they're not at all; they're vectors, a magnitude and a direction. Any application of sign convention hinges on direction, be it the orientation of components in the integral under a curve or the exchange of commodities. Changing direction (granted only for a paradigm that features one degree of freedom) is multiplying by minus 1. When you one understands that the explanation is as simple as turning around and back again.
Which is why I still can't understand radians being dimensionless quantities. A ratio a lengths? Hell no it isn't, arc length can be negative! That's a vector over a scalar bitches, just another vector! QED.
The problem arises because people think integers are scalars, but they're not at all; they're vectors, a magnitude and a direction. Any application of sign convention hinges on direction, be it the orientation of components in the integral under a curve or the exchange of commodities. Changing direction (granted only for a paradigm that features one degree of freedom) is multiplying by minus 1. When you one understands that the explanation is as simple as turning around and back again.
Which is why I still can't understand radians being dimensionless quantities. A ratio a lengths? Hell no it isn't, arc length can be negative! That's a vector over a scalar bitches, just another vector! QED.
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Re: Realworld examples of x * y = +z
If you walk 3 metres per second for 3 seconds you've walked 9 metres, right?
Okay, if you walk backwards 3 metres per second for 3 seconds, you end up 9 metres before the point you started right. You've just walked 3m/s for 3 seconds.
So, let's say we reversed the first example. we walk for 3 metres per second backwards in time, we end up were we started. we are 9 metres before the point we ended in the first. we travelled at 3 m/s * 3 s = 9.
So naturally, we have to reverse the second. we have to walk backwards and backwards in time, of course we then end up were we started at the second if we do this from were we ended at the second. So we are 9 metres in front of the point we ended in example two. we have travelled 3 m/s * 3 s = 9 metres.
Okay, if you walk backwards 3 metres per second for 3 seconds, you end up 9 metres before the point you started right. You've just walked 3m/s for 3 seconds.
So, let's say we reversed the first example. we walk for 3 metres per second backwards in time, we end up were we started. we are 9 metres before the point we ended in the first. we travelled at 3 m/s * 3 s = 9.
So naturally, we have to reverse the second. we have to walk backwards and backwards in time, of course we then end up were we started at the second if we do this from were we ended at the second. So we are 9 metres in front of the point we ended in example two. we have travelled 3 m/s * 3 s = 9 metres.
^ :/
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Re: Realworld examples of x * y = +z
Reading some of the more convoluted examples here, I'm glad most of you aren't ever going to try and teach basic math...
Re: Realworld examples of x * y = +z
gmalivuk wrote:Reading some of the more convoluted examples here, I'm glad most of you aren't ever going to try and teach basic math...
This. If the student is trying to learn that 1*1 = 1, then you should bring in multiplying vectors. Sorry.
A simple way to think of multiplying with signs is to think:
When a and b are on the same side of 0 (both positive or both negative), then a*b is positive.
When a and b are on different sides of 0 (one positive, the other negative), then a*b is negative.
If a or b is 0, then a*b is 0, no matter what.
Another way to think of this is to use words. Let's consider the sentence, "I am happy," a positive sentence. by adding a "not" to the sentence, we can make it a negative sentence:
"I am not happy."
But what happens if we add another "not" to the sentence, negating it again?
"I am not not happy." = "I am happy."
By negating the positive sentence twice, we make it positive again!
Happy hollandaise!
"The universe is a figment of its own imagination" Douglas Adams
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Re: Realworld examples of x * y = +z
I guess one intuitive way to see it is patterns:
3 x 3 = 9
2 x 3 = 6
1 x 3 = 3
0 x 3 = 0
1 x 3 = 3
2 x 3 = 6
This was brought up in that one link. It might be simple enough to satisfy someone only looking at the surface before they try to ask the reason why the pattern needs to be preserved. Then we're back to square one. But still, this will work as a very striking graphical example.
3 x 3 = 9
2 x 3 = 6
1 x 3 = 3
0 x 3 = 0
1 x 3 = 3
2 x 3 = 6
This was brought up in that one link. It might be simple enough to satisfy someone only looking at the surface before they try to ask the reason why the pattern needs to be preserved. Then we're back to square one. But still, this will work as a very striking graphical example.
Re: Realworld examples of x * y = +z
If you want a way of remembering the rule, try: A minus sign is a single short line. So two minus signs together makes two lines, and sure enough, a plus sign is two lines. I would make a gif animation to show this, but I'm too lazy.
Hmm... the only proof I know of this that hasn't already been given is an engineering one; best not to explain this to your friend, but I'll put it here in case anyone else is interested:
Watts=Volts*Amps. Plug an electric light into a battery (properly, please; no shortcircuits or raptors will eat you). Volts are positive, Amps are positive, and so Watts are positive. And sure enough, the bulb emits energy (light).
Now turn the battery around. The current is now going the other way; Volts become negative, but so do Amps, and the end result is that the light still looks exactly the same; Watts are still positive. So V * A = +W
[trumpet fanfare]
Hmm... the only proof I know of this that hasn't already been given is an engineering one; best not to explain this to your friend, but I'll put it here in case anyone else is interested:
Watts=Volts*Amps. Plug an electric light into a battery (properly, please; no shortcircuits or raptors will eat you). Volts are positive, Amps are positive, and so Watts are positive. And sure enough, the bulb emits energy (light).
Now turn the battery around. The current is now going the other way; Volts become negative, but so do Amps, and the end result is that the light still looks exactly the same; Watts are still positive. So V * A = +W
[trumpet fanfare]
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