# Multiplying negative numbers
**URL:** https://boards.straightdope.com/t/multiplying-negative-numbers/340054
**Category:** Factual Questions
**Created:** [January 15, 2006, 4:34am UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054 "2006-01-15T04:34:32Z")
**Posts on this page:** 17
**Page:** 1
### Author:  [@billthewonderer](https://boards.straightdope.com/u/billthewonderer)
#### Post date: [January 15, 2006, 4:34am UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/1 "2006-01-15T04:34:32Z")
I do not understand the reason that, when 2 negative numbers are multiplied, the result is a positive number.
Any thoughts?
Thanks.
---
### Author:  [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)
#### Post date: [January 15, 2006, 4:46am UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/2 "2006-01-15T04:46:35Z")
This was discussed back in March, 2004. It seems no one could come up with a concrete example of why 2 negative numbers equal a positive when multiplied.
> **[Concrete example for showing that the product of two negative integers is...](https://boards.straightdope.com/sdmb/showthread.php?t=244324)**
>
> I am trying to get teach my kids WHY this works. I don’t want them to just learn the rule for anything we go over, so I am seeking concrete ideas that will enlighten them. It has to be easy enough for 12 year old kids to understand. I have shown...
---
### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [January 15, 2006, 4:49am UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/3 "2006-01-15T04:49:17Z")
It’s pretty much a simple consequence of the way addition and multiplication are defined. Basically, you want the properties we’re used to:
1. a + b = b + a
2. a + (b + c) = (a + b) + c
3. a + 0 = a
4. For every number a, there’s a number (-a) such that a + (-a) = 0
5. a \* b = b \* a
6. a \* (b \* c) = (a \* b) \* c
7. a \* (b + c) = a \* b + a \* c
One of the first things you can deduce from this is that (-a) is unique; that is, if a + (-a) = 0 and a + b = 0, (-a) = b. More generally, if a + b = a + c, then b = c. It’s also easy to show that a \* 0 = 0, which is a useful fact.
Consider a \* a + a \* (-a). By 7), this is equal to a \* (a + (-a)), which is equal to 0. Now consider (-a) \* (-a) + (-a) \* a. By the same properties, this is equal to -a \* ((-a) + a), which is also equal to 0.
So we have now that a \* a + a \* (-a) = (-a) \* (-a) + (-a) \* a. By 5) above, we can write this as a \* a + a \* (-a) = (-a) \* (-a) + a \* (-a), and immediately conclude that a \* a = (-a) \* (-a). Do you follow?
Note that we don’t actually need 5), but it makes the proof much simpler, so I’ve left it in.
On preview: There’s no need for a concrete example, because it’s a _theorem_ that follows from properties that everyone agrees integer arithmetic should have.
---
### Author:  [@bouv](https://boards.straightdope.com/u/bouv)
#### Post date: [January 15, 2006, 4:58am UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/4 "2006-01-15T04:58:19Z")
I always though of it as a negative number is “opposite” a positive. The opposite of an opposite is the normal. Thus, a negative times a negative is a positive.
---
### Author:  [@Monty](https://boards.straightdope.com/u/Monty)
#### Post date: [January 15, 2006, 5:32am UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/5 "2006-01-15T05:32:09Z")
ultrafilter: How would a=0 affect your examples above?
---
### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [January 15, 2006, 5:35am UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/6 "2006-01-15T05:35:47Z")
Not at all, because the proof doesn’t require that a not be 0. I probably should have proved that a \* b = (-a) \* (-b), but the idea’s exactly the same, so there’s no need to walk through that proof.
---
### Author:  [@Monty](https://boards.straightdope.com/u/Monty)
#### Post date: [January 15, 2006, 6:20am UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/7 "2006-01-15T06:20:43Z")
Thanks. I guess I just had the question pop into my head when I saw the word _unique_ in the description. You’re right, of course. Affects it not at all.
---
### Author:  [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)
#### Post date: [January 15, 2006, 9:55am UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/8 "2006-01-15T09:55:13Z")
> [@billthewonderer](#):
>
> I do not understand the reason that, when 2 negative numbers are multiplied, the result is a positive number.
>
> Any thoughts?
>
> Thanks.
An heuristic, geometric answer. Can agree that multiplying a number by a negative number reverses the direction in which the answer is measured on the number line? If so then if I start with a negative number measured to the left of zero on the number line and multiply it by a negative number I reverse the direction and the answer is measured in the positive direction on the number line.
---
### Author:  [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)
#### Post date: [January 15, 2006, 12:45pm UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/9 "2006-01-15T12:45:16Z")
wolf\_meister writes:
\> It seems no one could come up with a concrete example of why 2 negative
\> numbers equal a positive when multiplied.
I thought there were a number of good examples in that thread. The only ones that don’t really work are those that claim that in language, a double negative is a positive. Well, sometimes it is and sometimes it isn’t. It depends on the language, the dialect, and the particular sentence. That isn’t a good example then of the product of two negatives being a positive, so please don’t anyone give such examples, since we’ll pick them apart immediately.
A better concrete example is this: Suppose you had a bunch of credit and debit slips. They say something like:
A owes me $10.
B owes me $15.
I owe C $5.
D owes me $20.
I owe E $25.
So the amount of money you have (for these slips) is 10 + 15 + (-5) + 20 + (-25) = 15. Suppose every slip is doubled in value. This is multiplying by 2. Then the amount of money you would have is (2 \* 10) + (2 \* 15) + (2 \* (-5)) + (2 \* 20) + (2 \* (-25)) = 20 + 30 + (-10) + 40 + (-50) = 30. Suppose every slip is doubled in value and inverted as to who owes who money. This is multiplying by -2. The amount of money you would have is ((-2) \* 10) + ((-2) \* 15) + ((-2) \* (-5)) +((-2) \* 20) + ((-2) \* (-25)) = (-20) + (-30) + 10 + (-40) + 50 = -30.
---
### Author:  [@spingears](https://boards.straightdope.com/u/spingears)
#### Post date: [January 15, 2006, 8:37pm UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/10 "2006-01-15T20:37:49Z")
> [@billthewonderer](#):
>
> I do not understand the reason that, when 2 negative numbers are multiplied, the result is a positive number.
[\*\* Just Sixth Grade Math \*\*](http://www.eduplace.com/math/mathsteps/6/b/)
Fully explains negative numbers and mathematical operations involving same.
---
### Author:  [@cerberus](https://boards.straightdope.com/u/cerberus)
#### Post date: [January 15, 2006, 10:05pm UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/11 "2006-01-15T22:05:27Z")
Think of the real numbers as they are laid out on a real line:
\<-------------------------------0------------------------------------\>
Signed numbers are viewable as very simple vectors, where said vectors can have only two directions: +/Right; -/Left.
In this setting, real numbers have both a magnitude component (absolute distance from 0), and a direction (+,-).
We can then understand the simple product as scaling and flipping:
a_b, a,b\>0 is simply stretching a vector of length b into a vector of length ab.
b_a, a,b\>0 is simply stretching a vector of length a into a vector of length ba.
In simple product notation, -1 can be viewed as a “flip”, as a simple reversing of direction.
Then a_b = a_b when a,b\>0; “Single Stretch”
a_b = (-1)_|a|_b when a\<0 and b\>0; “Single Stretch, Single Flip”
a_b = (-1)_(-1)_|a|\*|b|; “Single Stretch, Double(No) Flip”
Think of a as being a when a\>0 (|a|=a) and as (-1)\*|a| when a\<0.
---
### Author:  [@Steve\_MB](https://boards.straightdope.com/u/Steve_MB)
#### Post date: [January 15, 2006, 10:12pm UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/12 "2006-01-15T22:12:18Z")
> [@billthewonderer](#):
>
> I do not understand the reason that, when 2 negative numbers are multiplied, the result is a positive number.
>
> Any thoughts?
>
> Thanks.
The best explanation-by-example I can think of:
Somebody gives you a $20 check: +20
Somebody gives you a $20 debt: -20
Somebody gives you two $20 checks: (+2) \* (+20) = +40
Somebody gives you two $20 debts: (+2) \* (-20) = -40
Somebody takes two $20 checks away from you: (-2) \* (+20) = -40
Somebody takes two $20 debts away from you: (-2) \* (-20) = +40
---
### Author:  [@GorillaMan](https://boards.straightdope.com/u/GorillaMan)
#### Post date: [January 15, 2006, 10:14pm UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/13 "2006-01-15T22:14:34Z")
I pay £50/month electricity bills.
One month from now, I will owe £50:
1 x -50 = -50
Two months from now, I will owe £100:
2 x -50 = -100
A month ago, I had £50 more than now:
-1 x -50 = 50
---
### Author:  [@RealityChuck](https://boards.straightdope.com/u/RealityChuck)
#### Post date: [January 15, 2006, 11:40pm UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/14 "2006-01-15T23:40:48Z")
Assume multiplying two negative numbers gives a negative result.
If so
-1 x -1 = -1
But -1 x 1 = -1
So it follows that -1 x -1 = -1 x 1
Remove the -1 from both sides of the equation:
-1 = 1
Obviously, this is untrue. Therefore the original assumption is invalid and two negatives multiplied cannot give a negative number. They must give a positive number.
---
### Author:  [@Sunspace](https://boards.straightdope.com/u/Sunspace)
#### Post date: [January 15, 2006, 11:51pm UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/15 "2006-01-15T23:51:24Z")
> [@Steve MB](#):
>
> The best explanation-by-example I can think of:
>
> Somebody gives you a $20 check: +20
>
> Somebody gives you a $20 debt: -20
>
> Somebody gives you two $20 checks: (+2) \* (+20) = +40
>
> Somebody gives you two $20 debts: (+2) \* (-20) = -40
>
> Somebody takes two $20 checks away from you: (-2) \* (+20) = -40
>
> Somebody takes two $20 debts away from you: (-2) \* (-20) = +40
That is brilliant, possibly the best illustration I’ve seen yet of the concept.
---
### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [January 16, 2006, 1:43am UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/16 "2006-01-16T01:43:39Z")
> [@RealityChuck](#):
>
> Obviously, this is untrue. Therefore the original assumption is invalid and two negatives multiplied cannot give a negative number. They must give a positive number.
This is getting technical, but to really conclude that from your argument, you’d need to deal with arbitrary number c rather than a specific one. Otherwise, it’s a nice argument.
---
### Author:  [@alterego](https://boards.straightdope.com/u/alterego)
#### Post date: [January 16, 2006, 2:08am UTC](https://boards.straightdope.com/t/multiplying-negative-numbers/340054/17 "2006-01-16T02:08:02Z")
You can read an explanation of this on Google Book Search in “[Where Mathematics Comes From](http://books.google.com/books?ie=UTF-8&hl=en&vid=ISBN0465037712&id=RxFRRyiyVt8C&dq=where+mathematics+comes+from&vq=negatives&prev=http://books.google.com/books%3Fq%3Dwhere%2Bmathematics%2Bcomes%2Bfrom&lpg=PA93&pg=PA91&sig=D7BVahRwlMJqSB7gZZFs_2bfDI8)” by Lakoff & Nunez. It may not make sense unless you at least read the whole chapter. (I managed to read the whole book. I like pain)