Why do two negatives equal a positive in multiplication?

[Gale]

Ok, so in discussion today we were talking about combinatonics and counting. We were going over binomial coeffients, and i didn't like the explanation, and i related it to "its like, when you're learning to multiply, and they tell you two negatives equal a positive because when you put one minus sign on top of the other it makes a plus sign." But moving the lines around doesn't actually explain why two negatives equal a positive, its just a conveinient way of learning it so you can do the problems right.

...but then someone asked me after class, "why DO two negatives equal a positive?" and i was like "err... cause multiplication is the addition of groups and when you have a negative and you put it into negative groups... Hmmph! i don't actually know dammit!"

So, I've been chewing it over, and i think I've made some sense thinking about it in terms of negative meaning direction, and multiplying meaning you increase the magnitude... but i dunno... i want a better explanation. Please.

 

[whozum]

If a and b are two negative integers they can be written as two positive integers c and d such that

ab = (-1)c(-1)d = (-1)(-1)cd = 1cd = cd [/tex] where a and c are equal in magnitude, as are b and d.<br /> <br /> edit: itex is prettier.

 

[TD]

Perhaps this may help

 

[Gale]

but why is neg one times neg one positive? just an identity? is there a better conceptual way to think of it?

how about some number theory or set theory or something that describes what happens to negative numbers?

 

[whozum]

read the page he sent you.

 

[TD]

It contains multiple examples to make it 'understandable' as well as some mathematical details I believe.

 

[Gale]

ya, ok, i guess that works... i have a better question to ask anyway...

 

[Werg22]

-1x=opposite of x. -1(-x)=opposite of a negative, a positive.

 

[Gokul43201]

The integers are designed so that they form what is known as a ring. A ring must have (among other things) an additive identity and an additive inverse. The additive identity is the number we call 0, which has the property that 0 + x = x + 0 = x. Next we define the additive inverse of 1 as that element which when added to 1, gives the sum 0, or if x + 1 = 0, then x is the additive inverse of 1. We use the symbol "-1" to represent this number. Next we make use of the distributive property and the definition 0 = 1 + (-1) to write
0 = 0*(-1) = {1 + (-1)}*(-1) = 1*(-1) + (-1)*(-1)
Since 1 is the multiplicative identity, we know that 1*x = x*1 = x, for all x, and hence, 1*(-1) = -1. So we have 0 = -1 + (-1)*(-1). But we know that 0 = -1 + 1, therefore, from the uniqueness of addition (-1)*(-1) = 1.

Note : The operation of subtraction is merely a shorthand for adding a negative number.

 

[hypermorphism]

ya, ok, i guess that works... i have a better question to ask anyway...

Or the other way around from the page is to accept the field axioms, then prove yourself that (-1)(-1)=1 using them. You just need to use more precise terms of what a negative refers to and what rules multiplication follows.

 

[eNathan]

We debated this a while back on this thread https://www.physicsforums.com/showthread.php?t=82997.

My approch was to use English grammar exmaples to explain why -- = +. basically, any negative statement such as "not" or "didnt" counts as a -1.

[Quote = eNathan]Well that kinda gets into the very logic of what a negative is. Think of it as logic "not" operations. Think about using these in sentences and you will get the idea.

-1*-1=1 ; No + No = Yes
1*1=1 ; Yes + Yes = Yes
-1*1=-1 ; No + Yes = No
1*-1=-1 ; Yes + No = No

For instance, if I said...
I did not not go to the store. That really means, you DID go to the store. -1 * -1 = 1 ; No + No = Yes ... and so forth.

You kinda see how there is a relationship between mathematics and grammar? lol I am not sure if this is an official way to present it, but I just though it up a while back.[/quote]

 

[Hurkyl]

So, I've been chewing it over, and i think I've made some sense thinking about it in terms of negative meaning direction, and multiplying meaning you increase the magnitude... but i dunno... i want a better explanation. Please.

You've almost got the geometric picture, methinks. It's a nice one, and is important to know when you deal with the complex numbers.

In the real case, multiplication by a positive means you leave the direction unchanged, and multiplication by a negative means that you flip the direction. So, if you have a negative, and multiply by another negative, the result is a positive.

In the complex case, numbers can be seen as having a magnitude and a direction in the complex plane. Multiplying by a complex number that lies at an angle &theta; means rotation by &theta;. (Angles are measured counterclockwise from the positive x axis)

 

[ktoz]

However...

... multiplication is nothing more than a shorthand for repeated additions and in that light, there's no way to justify -1 * -1 = 1. Despite the convoluted logic the conventional explanation for multiplying two negative numbers amounts to this

for negatives
-1^2 = -1 + 2
-2^2 = -2 + 6
-3^2 = -3 + 12
-4^2 = -4 + 20

whereas for positives
1^2 = 1 + 0
2^2 = 2 + 2
3^2 = 3 + 6
4^2 = 4 + 12

So we're supposed to buy the explanation that, through some magical process, the interval between a negative and it's square is always larger then the interval between the positive of the same number and it's square?

Doesn't smell right.

 

[ktoz]

Well that kinda gets into the very logic of what a negative is. Think of it as logic "not" operations. Think about using these in sentences and you will get the idea.

-1*-1=1 ; No + No = Yes
1*1=1 ; Yes + Yes = Yes
-1*1=-1 ; No + Yes = No
1*-1=-1 ; Yes + No = No

For instance, if I said...
I did not not go to the store. That really means, you DID go to the store. -1 * -1 = 1 ; No + No = Yes ... and so forth.

In the sentence examples you stated, the logic follows, but your logic table could also equal this:

-1*-1=1 ; No + No = Emphatic No
1*1=1 ; Yes + Yes = Emphatic Yes
-1*1=-1 ; No + Yes = Terminating Yes
1*-1=-1 ; Yes + No = Terminating No

 

[HallsofIvy]

ya, ok, i guess that works... i have a better question to ask anyway...

Well, what is the better question?? We're waiting!

 

[hypermorphism]

... multiplication is nothing more than a shorthand for repeated additions and in that light, ...

That loses meaning when one looks beyond integers. What does it mean to repeat addition 1/2 times, or Pi times ? i times ?
If you just want to stay within the integers, you first have to define what a negative integer is. In that case, it is the number that when added to the corresponding positive integer, returns 0 (called the additive inverse). Multiplication defined your way makes sense for positive multiples. A negative multiple would have to be translated to a positive multiple multiplied by -1, in which you would apply your repeated addition definition, then have the negative sign applied. A negative multiplied by a negative gives us the following problem then: (-a)*(-b) = (-1)*(-1)*a*b, so we now have the problem of figuring out (-1)*(-1). Now, we know from our definition that 1 + (-1) = 0, so we have (-1) + (-1)*(-1) = 0. It is easy to prove that each integer has only one additive inverse (uniqueness). Thus, (-1)*(-1) must be the additive inverse of -1, which is 1.

 

[SteveRives]

...
for negatives
-1^2 = -1 + 2
-2^2 = -2 + 6
-3^2 = -3 + 12
-4^2 = -4 + 20

whereas for positives
1^2 = 1 + 0
2^2 = 2 + 2
3^2 = 3 + 6
4^2 = 4 + 12

Doesn't smell right.

That's because you are not following the rules!

It would be like me saying,

1 + 1 = 2 + 0

but

2 + 2 = 5 - 1

And then declaring that I have shown something smells fishy.

In reality, however, you have established a false opposition. It is not as if
-1^2 = -1 + 2
in contrast to
1^2 = 1 + 0

To prove it, I could just as easily say:
-1^2 = 1 + 0
and
1^2 = -1 + 2

The fact is, -1^2 = 1^2

Why? Because multiplication by -1 is like 180 degree rotation (flipping) -- note that 90 degree rotation would be multiplication by i. Someone has already pointed us in this direction.

 

[Gokul43201]

So we're supposed to buy the explanation that, through some magical process, the interval between a negative and it's square is always larger then the interval between the positive of the same number and it's square?

Doesn't smell right.

For k>0
neg interval = (-k)² - (-k) = k² + k
pos interval = k² - k

So, neg interval = pos interval + 2k, and since k>0, neg interval > pos interval.

See ? No magic ! No bad smell !

 

[Hurkyl]

I feel the need to comment that -1^2 = -1. Exponents before multiplication.
(Of course, I know both of you meant (-1)^2)

 

[arildno]

a) multiplication is nothing more than a shorthand for repeated additions
b) Doesn't smell right.

a) False
b) Correct, your assertion in a) stinks.

 

[arildno]

Gale17:
Several of the posters have already provided you with sufficient information to get this, so my post is probably redundant:

We'll show this in 3 steps:
1. Proposition: a+(-1)*a=0 ("a" is an arbitrary number)
Proof:
a=a*1.
Hence, a+(-1)*a=a*1+(-1)*a=a*(1+(-1))=a*0=0 Q.E.D.
(The last step, a*0=0 requires some additional justification; I'll skip that for now)

2. Proposition: (-a)=(-1)*a
Proof:
a+(-a)=0 (by definition of the negative)
Hence, by adding (-1)*a to both sides, we get:
a+(-a)+(-1)*a=0+(-1)*a
We shuffle about the left hand side, add together a+(-1)*a (getting 0 from Prop. 1)), and remember that for any number "b", b+0=b.
Thus, we get the desired relation:
(-a)=(-1)*a Q.E.D

3. Proposition: (-1)*(-1)=1
Proof:
1+(-1)=0
Squaring both sides yields:
1+(-1)+(-1)+(-1)*(-1)=0
Identifying 1+(-1)=0 on the left-handside, we have:
(-1)+(-1)*(-1)=0
Adding 1 to both sides yields:
(-1)*(-1)=1 Q.E.D.

Thus, we have as a consequence:
(-a)*(-b)=((-1)*a)*((-1)*b)=((-1)*(-1))*a*b=1*a*b=a*b, where a,b are arbitrary numbers.

 

[Hurkyl]

Werg: that makes no sense at all.

 

[Werg22]

Here is a very simple proof:

Let:

n - a = b
n - b = a

Replace a in the first equation;

n - (n - b) = b
-(-b) = b
-1(-1)b = b
-1(-1) = 1

Any multiplication of two negative number can be written as:

(-a)(-b); where a and b are positive,

(-1)(-1)ab

Since we have prooven that (-1)(-1)=1, and ab is positive, the product of two negative number is positive.

Q.E.D., as they say.

 

[Hurkyl]

Ah, that makes sense. (I peeked at the source before you deleted it)

 

[eNathan]

I feel the need to comment that -1^2 = -1. Exponents before multiplication.
(Of course, I know both of you meant (-1)^2)

As far as I know, when you raise a number to e (or square it), you are mulitplying it itself e number of times. You don't mulitply the absolute value of it. So how is -1^2 = -1 OR -1 \cdot -1 = -1? Unless you are saying that -1^2 is actaully -1 * abs(-1) or something of that sort which is the only way you could derive a product of -1.

Im sure you know what I am saying, there's probally just some sort of misunderstanding as to what you meant to say.

Edit :: Did you mean - (1^2) = -1?

 

[Hurkyl]

Yes, -1^2 means -(1^2).

 

[Hurkyl]

Here is a very simple proof:

I will make one complaint: you've not shown that -b = (-1)b.

 

[Werg22]

This is by definition I beleive. But perhaps you can show it?

 

[motai]

One of my teachers put it this way:

The Friend of my Friend is my Friend | + * + = +
The Friend of my Enemy is my Enemy | + * - = -
The Enemy of my Friend is my Enemy | - * + = -
The Enemy of my Enemy is my Friend | - * - = +

While not exactly a mathematical proof, it makes sense.

 

[hypermorphism]

This is by definition I beleive. But perhaps you can show it?

-b represents the additive inverse of b, ie., the number "-b" such that b + (-b) = 0. -1 is simply the additive inverse of 1. It is not defined that -1*a is the additive inverse of a, but it is trivial to prove.