GAME: A question concerning arithemetics?

[zibi]

Hi all,
I was wondering why if you multiply to negative numbers together you get a positive one.
It is quite intuitive why positive * negative is negative. If I have 9 apples and I eat twice two apple I have two subtract 4.

But it is less intuitive why negative * negative = positive.

The best answer will be rewarded with a thanks!

Luca

[SunnyRabbiera]

Well if you multiply you add numbers instead of remove them, is it THAT simple though?

[tom66]

It's impossible to get a negative square root; try sqrt(-1). Therefore, it is impossible for negative * negative to equal negative.

[SunnyRabbiera]

It's impossible to get a negative square root; try sqrt(-1). Therefore, it is impossible for negative * negative to equal negative.

Kind of figured that crap in, though I am TERRIBLE at math.
I never liked that algebraic stuff, never could get my head into it.

[Barrucadu]

It's because the sign is flipped when multiplying by a negative. Also, you can have the square root of a negative number, it's just not really needed for every day maths :p

[SunnyRabbiera]

It's because the sign is flipped when multiplying by a negative. Also, you can have the square root of a negative number, it's just not really needed for every day maths :p

Well there is where I come in, my math is mainly for every day math and not astrophysicist math.
Its not like I am not aware of stuff like pi, negative numbers and square roots but actually making any real sense of it is where I fail.
The worst part I hated is when I did get into higher maths the teachers were always "its simple" or "a child can do it"
Yeh I like to see my 8 year old son and my 3 year old calculate pi while adding up the square root of 59 or some advanced math crap like that.

[ZankerH]

It's because the sign is flipped when multiplying by a negative.

That's attempting a proof through higher abstract concepts, which is a logical fallacy in mathematical discrete logic. You're supposed to explain a concept with the basic ideas behind it, not with the logically superior concepts it establishes.

As for the OP, it's really simple. A double negative equating a positive can be observed in many situations in real life, even if it's not strictly positive.

For example, if you define small as not(big), you can clearly see that not(not(big)) - a double negative, gets you the positive of the innermost statement:

not(big)=small

therefore

not(not(big))=not(small)=big

I'm sorry if this explanation is confusing, but I didn't learn this in English, and I'm trying my best to explain the concept.

[Idefix82]

The question is actually a good one. This behaviour that neg*neg=pos is forced by two things: the fact that 0*anything=0 and the distributive law, which says that (a+b)*c=a*c+b*c. Now taking this as our starting point, we do this:

-1+1 = 0 by definition of -1
Therefore (-1+1)*x = 0*x=0 for any x
Now we bring in the distributive law: 0=0*x=(-1+1)*x=(-1)*x + 1*x = (-1)*x + x


So we see that x + (-1)*x = 0. So, if x was positive and we add something to get 0 then this something (namely (-1)*x) must be negative. That's what you called intuitively clear. But if x was negative then we must add something positive to get 0. Thus x negative implies (-1)*x positive.

If you now replace 1 and -1 by y and -y throughout then you see that negative*negative=positive.

[Greyed]

But it is less intuitive why negative * negative = positive.

The best answer will be rewarded with a thanks!

Because you're doing negative multiplication. Think of a line of numbers from -4 to +4.

-4, -3, -2, -1, 0, +1, +2, +3, +4

If you subtract -1 from -2 you move right on the above scale from -2 to -1. This is because we're lowering the negative numbers. Negative multiplication is the same. Remember, multiplication is just the same number added the number of times in the multiplier. If the multiplier is negative you're effectively subtracting. So what happens when you subtract a negative number? Look above. You move towards the positive. Since in multiplication the start point is 0 you get positive results. IE:

2 * 2 = 4 because our starting point is 0 and we're adding 2, twice. So 0+2+2 = 4.

2 * -2 = -4 because our starting point is 0 and we're adding -2, twice. So 0 + (-2) + (-2) = -4.

-2 * 2 = -4 because our starting point is 0 and we're subtracting 2, twice. So 0 - (2) - (2) = -4.

-2 * -2 = 4 because our starting point is 0 and we're subtracting -2, twice. So 0 - (-2) - (-2) = 4

[SunnyRabbiera]

The question is actually a good one. This behaviour that neg*neg=pos is forced by two things: the fact that 0*anything=0 and the distributive law, which says that (a+b)*c=a*c+b*c. Now taking this as our starting point, we do this:

-1+1 = 0 by definition of -1
Therefore (-1+1)*x = 0*x=0 for any x
Now we bring in the distributive law: 0=0*x=(-1+1)*x=(-1)*x + 1*x = (-1)*x + x


So we see that x + (-1)*x = 0. So, if x was positive and we add something to get 0 then this something (namely (-1)*x) must be negative. That's what you called intuitively clear. But if x was negative then we must add something positive to get 0. Thus x negative implies (-1)*x positive.

If you now replace 1 and -1 by y and -y throughout then you see that negative*negative=positive.

Of course you can figure it out, you are from Cambridge :D
College boy ;)

[ZankerH]

The question is actually a good one. This behaviour that neg*neg=pos is forced by two things: the fact that 0*anything=0 and the distributive law, which says that (a+b)*c=a*c+b*c. Now taking this as our starting point, we do this:

-1+1 = 0 by definition of -1
Therefore (-1+1)*x = 0*x=0 for any x
Now we bring in the distributive law: 0=0*x=(-1+1)*x=(-1)*x + 1*x = (-1)*x + x


So we see that x + (-1)*x = 0. So, if x was positive and we add something to get 0 then this something (namely (-1)*x) must be negative. That's what you called intuitively clear. But if x was negative then we must add something positive to get 0. Thus x negative implies (-1)*x positive.

If you now replace 1 and -1 by y and -y throughout then you see that negative*negative=positive.

Thank you for an arithmetical clarification. I understand my explanation with discrete logic was too vague?

[Bölvaður]

1*1=1

The problem only exists with integer numbers bigger than 1 or smaller than -1

[ZankerH]

1*1=1

The problem only exists with integer numbers bigger than 1 or smaller than -1

Unfortunately, any integer bunch only limited on one end is infinite, so you still have to solve the problem for a double infinite group of numbers. :lolflag:

[kanterjoe]

the numbers aren't really negative. there opposites. -2 is the opposite of 2. When you multiply -2 * 2 you get the opposite of 4 (-4).

so when you multiply -2 * -2 you get the opposite of -4, which is positive 4.

[shifty2]

Ploughing through the letters:
Its all to do with this property of numbers:
a(b+c)=ab+ac

Establish that (-a)b= -(ab)
(-a)b + ab = [(-a)+a]b
(-a)b + ab=0b
(-a)b + ab=0
(-a)b=-(ab) (subtracting -ab from both sides)

Now,
(-a)(-b) + [ -(ab)] = (-a)(-b)+(-a)(b)
(-a)(-b) + [ -(ab)] =(-a)[(-b)+b]
(-a)(-b) + [ -(ab)] =(-a)0
(-a)(-b) + [ -(ab)] =0

Adding (ab) to both sides we get
(-a)(-b)=(ab)

If we want the basic properties of numbers to be true, then this is forced on us

[fredscripts]

It's because the sign is flipped when multiplying by a negative. Also, you can have the square root of a negative number, it's just not really needed for every day maths :p

Except if you are into electrical engineering,where the surprising case is when we work with real numbers :D

[bapoumba]

If I say "Answer this", you know you should answer.
If I say "Don't answer this", you know you should not answer.

If I say "Dont' don't answer this", you know you should do the opposite of not answering, ie answer.

Multiplying two negative orders (or numbers) is a positive one. The opposite of the opposite of a positive number is that very positive number. Kinda..

[Namtabmai]

It's impossible to get a negative square root; try sqrt(-1). Therefore, it is impossible for negative * negative to equal negative.

Yes you can

sqrt(-1) = i

Imaginary numbers (http://en.wikipedia.org/wiki/Imaginary_numbersImaginary numbers) for the times when real ones just can't do the job.

:)

[zibi]

x

[zibi]

Well, I whish to thank idefix and it's a pleasure to declare that he's the winner.
Shifty2 have given the same answer but, unfortunately too late.

I only wish to make a remark concerning the idea of flipping the sign. The basic intuition we are supposed to share in giving this kind of answer is that the double negation of something is equal to the something itself.

Despite its intuitive appeal there are many cognitive domain in which the intuition fails.
To say "not all x have not a given property" is quite different to say "there is an x that has".
In the sense that often when one claims the latter he is in the position of exhibiting the individual with property in question. On the other hand to claim the former one may only be in possession of evidence showing that it is impossible that no such individual exist.

This ideas are at the basis of intuitionistic logic and in general of constructive reasoning (functional programming etc.).

So in general we need a stronger reason to say that - and - is +. Or at least I guess.

Any way thank you all for having taken part to the game. It was really funny.:KS

[Idefix82]

Thank you, zibi, for this seemingly simple question which hopefully made people think!

[Gizkaguy]

Well there is where I come in, my math is mainly for every day math and not astrophysicist math.
Its not like I am not aware of stuff like pi, negative numbers and square roots but actually making any real sense of it is where I fail.
The worst part I hated is when I did get into higher maths the teachers were always "its simple" or "a child can do it"
Yeh I like to see my 8 year old son and my 3 year old calculate pi while adding up the square root of 59 or some advanced math crap like that.

I just had a XKCD-like vision ....