Explain it like I am 5: Why does multiplying two negatives give you a positive?

[LiberalArkie]

I give you three $20 notes +3 * +20 = +60 for you
I give you three $20 debts +3 * -20 = -60 for you
I take three $20 notes from you -3 * +20 = -60 for you
I take three $20 debts from you -3 * -20 = +60 for you

the result is the gain or loss from where you started.

[randys1]

1. I dont even understand the question

[Art_from_Ark]

49. In mathematics, if you multiply two negative numbers together, the answer is positive

For example, -4 X -4 = +16

[hifiguy]

2. The negative of a negative is by definition a positive, its opposite.

The negative of a positive is by definition a negative.

At least that's the way I understand it in a purely mathematical and logical sense.

[Rex]

8. I think the question is 'why do two negatives create a positive?' IOW, how can 2 nothings create a

something. All I remember is in multiplication it has to do with grouping. Which means I am no help at all.

[hifiguy]

12. My explanation of this wasn't all that good, probably.

I am hopelessly mathlexic but read tons of theoretical physics and cosmology. My little description is how I get it to make sense for myself.

If I'd only received the math gene, like most other people on the spectrum, I'd have become an astronomer, theoretical physicist, or a cosmologist - that is COSMOLOGIST, not cosmetologist, both useful, but decidedly not interchangeable.

[Rex]

14. At least you have an explanation, all my brain remembers is it has to do

with grouping by association. Which makes no sense to me now that I think about it! ARg...math!

[hifiguy]

15. Thinking of it in terms of electrical charges

is also a good and apt metaphor.

[Dr. Strange]

24. You can think of it in terms of grouping, if you accept that positive times negative is negative.

Theorem: (-3)*(-20) = 60
Proof by "hold muh beer and watch this!"

First, we note that by Euclid's Elements and the US constitution, we have the following three lemmas:
Lemma I. (3)*(20) = 60
Lemma II. (-3)*(20) = -60
Lemma III. (3)*(-20) = -60

We now use the distributive property to find that
0 = 0 * 0
[font color = white]0 [/font]= (3 - 3)*(20 - 20)
[font color = white]0 [/font]= (3)*(20) + (3)*(-20) + (-3)*(20) + (-3)*(-20)
[font color = white]0 [/font]= 60 + (-60) + (-60) + (-3)*(-20)
[font color = white]0 [/font]= -60 + (-3)*(-20).

Adding 60 to both sides yields (-3)*(-20) = 60. QED

(QED is Latin for "Give me back my beer, I'm done mathing.&quot

[valerief]

31. Negatives aren't nothing. If that were the case, my mortgage from day one would be zero. nt

[TexasBushwhacker]

3. Multiplying by a positive by a negative gives you

a negative. It flips the sign of the product. Multiply a negative by a negative also flips the sign - to positive.

Double negatives in speech flip the meaning. "I can't get no satisfaction" means "I can get satisfaction".

[Nevernose]

42. That's not true about language

A double negative in English isn't a positive; you're thinking of math.

In English it indicates an emphasis on the negative or a disregard for grammar, or both.

"I can't get no satisfaction" is grammatically incorrect, but does not indicate that Mick and the boys can, indeed, find satisfaction. It means they're using the vernacular in common usage since at least the time of Chaucer (himself infamous for using double, triple, and even quadruple negatives).

Sorry for being pedantic!

[rjsquirrel]

57. English has both

We often say we re "not unhappy" to mean we are "happy." Not all duplicate negative constructions in English intensify the negation.

We say yeah right" to mean "no way!"

[Chemisse]

4. Good examples.

I don't think anyone ever told me why, or even showed it working like you did; I just got the rule and had to follow it.

[LiberalArkie]

7. Yea I never knew why just that a negative multiplied by a negative gave you a positive.

I just accepted it like you die and go to heaven. Never knew why -1 X 4 = -4 or -1 X -1 = 1.

This was on Reddit and everyone went totally appshit over the answer. I had to share it for all us old farts out there.

[bigwillq]

5. Common Core is so confusing.

[Renew Deal]

11. This is pre-common core

Most of us here probably went to school using old math.

[bigwillq]

16. I know. I was trying to make a joke.

[HereSince1628]

17. I was raised on "Modern Math" my worksheets looked like Jackson Pollack

had visited...

I think modern math' 'thang' was to get us to Algebra before we could do ratios/proportions.

Emphasis on 'Splainin' math had to wait 2 generations. Which was 3 generations after the person teaching me modern math needed to know how to do that.

[Rex]

6. The negatives cancel out.

It has to do with grouping, but I forget why. LOL.

[surrealAmerican]

9. If I take away your debt ...

... that means you don't owe anything, right?

If I take away more debt than you owe, that means giving you money.

[Renew Deal]

10. Does it mean we are subtracting three negatives?

So if we're negative we're getting less negative?

So of we have a $60 debt and we remove 3 $20 debts we are even.

[hifiguy]

13. IIRC, three negatives makes a negative.

Unlike three left turns, which make a right.

[petronius]

18. Because I said so! (And I'm the adult and you're only 5...)

[Jim__]

19. Here's an attempt an explanation.

Mathematically, any negative number, say -10 can also be written as -1 * 10. So, multiplying any 2 negative numbers, -x * -y can be rewritten as -1 * x * -1 * y = -1 * -1 * x * y, and the question can be reduced to what does -1 * -1 equal?

By assumption, we have a distributive law of multiplication: a * (b + c) = ab + ac.

Assume that -1 * -1 = -1.
Then take -1 * (1 + -1) = -1 * 1 + -1 * -1 = -2 by distribution and the assumption.
But, also, -1 * (1 + -1) = -1 * 0 = 0.
So, with our assumption, -1 * (1 + -1) gives us a contradiction: 0 = -2.
You can try the same thing with the assumption that -1 * -1 = 1, and it will work out without a contradiction.

Practically, think of charging 3 $20 items, and let each $20 debt be represented by -20. Then, return the item and represent each returned item as -(-20). The return can be seen as a negative purchase against your debt -1 * -20, -1 purchase against the -20 of debt. Returning the 3 items reduces your debt to 0, or -3 * -20 = 60 which offsets the -60 debt that you had. It's not the clearest explanation, but I think you can see that 3 * -20 = -60 and the offsetting returns amount to -3 * -60 = 60.

[Skittles]

20. that's how Jim__ would explain it to a five-year old

SCHEDULING JIM__ FOR ASS KICKING; yes INDEED

[Flying Squirrel]

46. I didn't see any mention of the square root of negative 1.

[Skittles]

48. I WILL KICK NEGATIVE FLYING SQUIRREL ASS TOO

YES INDEED

[jberryhill]

21. Look at your last sentence - the IRS sees it that way

Yes, debt forgiveness is income.

If I take three $20 debts away from you, that's a gain of $60 on your ledger, and it is taxable income.

[dembotoz]

22. bet they don't have stuff like this on gop boards

[rug]

23. 7 x 13 = 28

[Jim__]

25. The funniest part of that is they rented a room for $13 a week.

[rug]

26. Lol, a long. lomg time ago.

[DavidDvorkin]

27. Try the graphical approach

Positve and negative numbers both lie along the number line:



negative numbers  [ ] positive numbers
-------------------------------0--------------------------------



Putting a minus sign in front of a number is equivalent to flipping it across the zero to the other side of the line. So if a number is negative, hence on the left-hand side of the line, putting another minus sign in front of it flips it over to the positive side.


(The phrase "negative numbers" is supposed to be on the left, and "positive numbers" is supposed to be above line on the right, to the right of the zero. I forgot that what I wrote would get reformatted.)

[jeff47]

28. Your example doesn't actually have any negatives.

While we tend to think of debts as negative, they aren't. Debts can offset assets, but within each category you're only dealing with positive numbers.

As for explaining like you are 5, what the hell are you doing talking about multiplication and negative numbers? You're about two years from learning multiplication, and even further from learning about negative numbers. Have you been stealing your older sister's homework again?

( )

[Jim Lane]

39. Even aside from asking when schools teach multiplication and negative numbers...

I was wondering where he found 5-year-olds who routinely deal in negotiable instruments.

[Mosby]

29. this should help

[Paulie]

37. Of course, it's so obvious

once you write it all out.

[Initech]

30. Bush x Bush = divide by zero error.

[valerief]

33. Brilliant!

[wordpix]

38. or Bush x Bush = Bush squared = 0

[valerief]

32. I dunno if this works...

You owe utility 1 new $10 fee.
Utility 2 is pissed and also wants a new $10 fee, too.
Utility 3 is also pissed and wants their new $10 fee.
You're out $30 or -30.
3 (fees) X your -10 = your -30.

Bernie Sanders comes to the rescue and says this is unfair. They have to give those $10 fees you've paid back to you.
Utility 1 is pissed and gives you $10.
Utility 2 is still pissed and gives you $10.
Utility 3 is still pissed and gives you $10.
You're up $30 or +30.
-3 (fees) X your -10 = your +30.
This is -3 because it's a fee reversal.

[That Guy 888]

55. That works for me

I read the op and couldn't think of an easy way to explain it; word problems always helped make math relevant.

[valerief]

59. It's actually easier to just remember the rule: a negative multiplied by a negative

equals a positive.

Just like it's easier to know how to turn on a teevee than to have to think about how it all works.

[bhikkhu]

34. I had to work out a mental model when I went back to college

Think of a number line with zero in the middle, positive numbers counting forward to the right and negative numbers counting backwards to the left.

Then, say you are multiplying 3 x 3. On the number line you start from zero and make three jumps to the right, each jump going three places. You wind up at nine, and it makes sense - you are counting out three threes.

Say you are multiplying -3 x 3. You start from zero and make three jumps to the left, counting out three negative threes - you wind up at -9. Still makes sense.

Say you are multiplying 3 x -3. You start from zero and make three jumps to the left, counting out three a negative three times (as opposed to a positive three times). Makes little less sense, but its not so hard, and you wind up at -9 still.

Now if you are multiplying -3 x -3, if you understand the steps above, it should make sense that the negative of a negative is a positive. Or that numbers in multiplication are basically positive and proceed to the unless one term is reversed, which reverses the direction of the jumps. The direction reversed once by a negative sign makes the count go left; the direction reversed again by a another negative sign makes the count turn back around and go right. So -3 x -3 is counting out negative three negative threes, which makes nine.

Which makes more sense if you actually write out a number line and do the problems in order, writing out the actual jumps and the logic. I was never great at math myself, but having worked out the above I was able to successfully show my own kids and a few others how and why it works the way it does.

[BlueJazz]

35. Well..

[Liberal_in_LA]

44. Lol

[jberryhill]

58. You have an irrational argument

[Make7]

36. Eleven minutes of YouTube

Why is negative times negative positive? PART I (James Tanton)
Why is negative times negative positive? PART II (JAMES TANTON)

[SheilaT]

40. Here's how I was taught it, as a bit of a thought experiment:

Imagine you have a video camera and a bathtub. Turn on the tap water, and film it filling up. Now run the film backwards. Filling up is the positive quantity, running the film backward is the negative quantity.

Now film the bathtub emptying. That's a negative. Now run the film backward. Another negative, but what you see is the tub filling, a positive, because you're doing two negative things: emptying the bathtub, but running the film backward.

I think that's a LOT easier, because it's a visual, than a more abstract example with money and debts.

[geek tragedy]

41. They cancel each other out

Or, if you do a 180 turn, another 180 cancels that one out

[Snobblevitch]

43. Whn I signed up to DU

I was told there was not a math requirement.

[randome]

60. Ah, but your eye exam requirement was waved. So you missed the fine print.

Besides, it said twice you were not required to know math so, as we can see, two negatives equals a positive.
[hr][font color="blue"][center]There is nothing you can't do if you put your mind to it.
Nothing.[/center][/font][hr]

[Bad Thoughts]

45. Negatives are a reversal of direction

Rather than quantity, they reflect orientation. -5, like 5, is 5 from 0 on a number line. The negative sign reflects the orientation from 0, not necessarily a quantity. We often think of it as the removal or absence of value because we use negative numbers to indicate a a regression on a different scale. If I lose five dollars, it might be said that I have -$5 income. Saying -5 is convenient, but it means obscuring the fact that 5 dollars is itself a real quantity.

Two negatives represent a double reversal, putting the answer on the other side of 0.

[Flying Squirrel]

47. Here's one for you

Three men rent a room for $30, paying $10 each. (This was back in the day) Later, the manager realizes he's overcharged them, the room should be $25, and he sends his bellboy to bring them $5. The bellboy, being dishonest, keeps $2 for himself and gives the men $3. So now each man has paid $9 for the room. 3x9=27, plus 2 for the bellboy = 29. What happened to the other dollar?

[struggle4progress]

50. Because "-" means "turn around and go the opposite way." So doing "-" twice,

you turn around and go the opposite way, and then you turn around and go the opposite way again, which sends you the same direction you were headed originally

[valerief]

61. I like that!!! nt

[Donald Ian Rankin]

51. Turn around and walk backwards, and you'll be going forwards, like in the Gay Gordons. N.T.

[PoliticAverse]

52. So much negativity on DU, is that a net positive? n/t

[MrWendel]

53. I just learned it this way...

2 of the same symbols = a positive (+)
2 of the opposite symbols = a negative (-)

[EdwardBernays]

54. Here's why

Think of it like this though, with a number line:

-3 x -4

Minus three is going towards the left, but it's starting at zero - that is, before you move to the left once, you start at zero. Now I want to do the opposite of that 4 times (x -4)

so the opposite of -3, starting at zero, 4 times = 12.

OR in English:

I have boxes that hold three apples that I use to ship apples from my factory... each box represent -3 apples in my inventory.

Someone returns 4 boxes of -3 apples and viola I have 12 apples.

Another way to consider it is this: a number isn't a number, but a quantity. If I remove a negative quantity repeatedly I will end up with a positive quantity. Another way to think of the term "remove a negative quantity" is add.

I have no kittens. My friend removed a negative quantity of 1 kitten in my house. I have 1 kitten now.

[Travis_0004]

56. Here is how I would explain it

If my mortgage is 200 a month, how much did I pay in 3 months.

3*-200=-600

If instead my employer paid my motgage the math is

-3*-200=600

This means I gained 600 dollars. The 3 is negative since I didnt pay it, somebody else did.

[gollygee]

62. You either have 3 $20 debts (+3 X -20) or forgive (or do away with) 3 $20 debts (-3 X -20)

If you forgive, or do away with, three $20 debts, you are in essence giving the person $60. +$60.

[struggle4progress]

63. If you stop pulling the cat's tail, I'll explain to you some basic facts about ordered rings

No, you can't pinch your little brother

So remember we talked about rings last week

Yes, Mommy has some nice rings, but these rings are different from Mommy's. In these rings you can add, and subtract, and multiply

No, when Uncle Ralph said bacteria multiply by dividing, he was just being funny. Anyway, let's suppose we have a ring R with domain of positivity P

You don't have to pee? Good! Okay, so a domain of positivity is closed under addition and multiplication. And it has trichotomy

Trichinosis is a different word

Yes, it does sound a lot like trichotomy. Trichinosis is why we cook pork

You're right: we don't have to cook Oreos. And you should see immediately that an ordered ring has no zero divisors

I DO think it's funny when Uncle Ralph wears his green visor to play 'Go Fish.'

I think that's a really good idea: let's have some Oreos