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Proof a negative times a negative equals a positive


Realintruder
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If we assume that (-1) (-1)=-1 and (-2)(-2)=-4 but know that (0) (0)=0

 

Then

 

(1+-1) (1+-1) must equal 1-1-1-1 =-3 and not the sum of (0) (0) which it should equal 0 unless we have assumed otherwise but have not.

 

Likewise

 

(-1+2) (-1+2) =(-1)(-1) must equal -1+-2+-2+4=-7 and not the product of (-1+2) (-1+2)=(-1)(-1) which should equal -1, which is inconsistent and does not equal -7 if we assume that (-1)(-1)=-1.

 

But if we assume that (-1)(-1)= 1 then (1+-2)(1+-2) should equal 1+-2+-2+4=1 likewise (1+-1)(1+-1)=(0) (0)=1-1-1+1=0 which it does

 

 

 

 

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If we assume that (-1) (-1)=-1 and (-2)(-2)=-4 but know that (0) (0)=0

 

Then

 

(1+-1) (1+-1) must equal 1-1-1-1 =-3

....

 

BTW 1-1-1-1=-2

And

(-1+2) (-1+2) =(-1)(-1) ..

 

(-1+2)(-1+2) = (1)(1)

 

We can see where you are going but two arithmetical errors are a bit much in a proof that relies on arithmetics

Your proof relies on the distributive law being correct - and IMSMC you have refound the same as the standard proof given for the idea.

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BTW 1-1-1-1=-2

And

 

(-1+2)(-1+2) = (1)(1)

 

We can see where you are going but two arithmetical errors are a bit much in a proof that relies on arithmetics

Your proof relies on the distributive law being correct - and IMSMC you have refound the same as the standard proof given for the idea.

 

Thank you. But the software will not let me edit my post.

 

What is the standard proof for the idea?

Edited by Realintruder
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You can simplify your proof to just considering (-1)(-1) as (-x)(-y) = xy (-1)(-1). (Assuming you want to start from here)

 

You know that (-1)0 = 0 = (-1)(-1+1) from the properties of minus being the inverse of addition.

 

You then use distributivity to get

 

(-1)(-1) + (-1)(+1) = -1 + (-1)(-1) =0

 

as we know (-1)(+1) = -1. The question is what is (-1)(-1)? Well, from the uniqueness of the additive inverse we conclude that (-1)(-1)= 1.

 

Mod some typos in your initial proof, you have essentially presented the proof I have given above, which is the standard one for 'high school maths'.

Edited by ajb
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I have never been able to understand why -1 x -1 = 1 it looks wrong in a very basic way to me...

Look at my proof above. If it were -1 then something would have to be wrong with the basic rules we require of numbers.

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I have never been able to understand why -1 x -1 = 1 it looks wrong in a very basic way to me...

It goes like this below

post-19758-0-14218600-1425740515_thumb.jpg

 

Multiplication of positive in the x axis and positive in the y axis give a positive result. That is the yellow up quarter.

Multiplication of positive by negative give a negative result. These are the grey quarters.

 

Now, your problem is about the yellow down left quarter. The correct answer is that multiplication of 2 negatives give a positive result. It corresponds to a rotation of the graph 180 degrees.

 

Or it could be understood as an incapacity to make a distinction between the 2 different positive quarters.

Edited by michel123456
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I have never been able to understand why -1 x -1 = 1 it looks wrong in a very basic way to me...

 

 

Many folks have trouble with this because they fail to distinguish between negative numbers and subtraction as a process.

 

Multiplication is not addition or subtraction they are different.

 

Negative numbers are examples of signed numbers.

That is numbers with a value and a sign.

 

So the number 5 just has a value (5)

The signed number -5 has both a value (5) and a sign -.

 

Again forget any examples involving bank balances for this, they are not helpful.

 

Instead think of your electricity bill.

 

If you have 5 amps flowing we identify the direction by using a sign.

 

So 5 amps flowing from your solar panel to the grid is +5amps; and 5amps flowing into your system is -5amps.

 

Voltage can be reckoned the same way.

 

And power is current times voltage.

 

So the rules multiplication rules are

 

plus times plus or minus times minus makes plus

 

minus times plus or plus times minus makes minus.

 

Using this on our power calculator we have

 

Your voltage is plusV your power is (+5)x(+V) ie positive if you are supplying the grid = +5V watts : So they owe you money

 

and your power is (-5)x(+V) ie negative if the grid is supplying you ie -5Watts : So you owe them money.

 

Sorry did I say forget the bank?

It seems they always get you in the end. :eek:

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  • 2 weeks later...
I have never been able to understand why -1 x -1 = 1 it looks wrong in a very basic way to me...

Expanding on michel's introduction of rotation above: the problem usually is that negative numbers have been introduced as directions on a number line or subtractions from a positive quantity - a better way is to think of the minus sign as a multiplication by unit rotation (-1), a signal that the "amount" or "size" involved has been "rotated" (the -) but otherwise unchanged (the 1) - in this one dimensional case a rotation is a equivalent to a 180 flip on the axis, there being no "in between" states of existence.

 

So multiplying by negative one flips whatever is there on the number line, and - the point - another such multiplication flips it again, which returns it to its initial position.

 

(the rotations are counter-clockwise, unless otherwise specified).

Edited by overtone
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If we have:

[math]x * y = xy[/math]

 

we can divide both sides by y:

[math]x * y/y = \frac{xy}{y}[/math]

 

and get:

[math]x=\frac{xy}{y}[/math]

 

 

f.e.

[math]-5*(-4)=+20[/math]

[math]-5=\frac{+20}{-4}[/math]

[math]-5=-5[/math] (true)

 

 

If you try the same with your assumption

"If we assume that (-1) (-1)=-1 and (-2)(-2)=-4"

it will fail.

Left side of equation, won't match right side.

 

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f.e.

[math]-5*(-4)=+20[/math]

[math]-5=\frac{+20}{-4}[/math]

[math]-5=-5[/math] (true)

 

 

If you try the same with your assumption

"If we assume that (-1) (-1)=-1 and (-2)(-2)=-4"

it will fail.

Left side of equation, won't match right side.

 

 

This reasoning doesn't entirely work. If we assume -1 * -1 = -1, then we have

 

-5 * -4 = -20

-5 = -20/-4 = -20 * -1/4 = -5, as expected.

 

As mentioned earlier in the thread, what we do lose by letting -1 * -1 = -1 is the guarantee that certain rules of arithmetic (perhaps most notably, distributivity) extend to the negative numbers.

 

Edit: I actually should note that the entire notion of doing division involves an element having a multiplicative inverse, so my chain of operations doesn't necessarily work here either, since -4 * -1/4 would be -1 not 1. So we'd need to either give up the field structure of the real numbers or redefine multiplication a bit. Either way, this provides another example of why -1 * -1 = 1 is useful.

Edited by John
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This reasoning doesn't entirely work. If we assume -1 * -1 = -1, then we have

 

-5 * -4 = -20

-5 = -20/-4 = -20 * -1/4 = -5, as expected.

 

And then you have:

-5 * (-4) = -20

-5 * 4 = (also) -20

5 * (-4) = (also) -20

 

Only one would lead to positive result:

5 * 4 = 20

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The problem with the algebraic demos is that they are unintuitive - the new encounter can see that the rules require the definition, but that just punts the question to the next level: why have those rules? They seem messy, arbitrary.

 

So that's one advantage to viewing the negative sign as a flip, or 180 rotation: it's immediately intuitive that two such rotations bring you back, that the absolute value is not affected, and so forth. The picture is clear. The intuitive picture then explains the various obscure algebraic manipulations, fractions, square roots, etc, rather than the other way around.

 

Another advantage is that such an intuitive picture prepares one for complex numbers, in which i is intuitively explained as a 90 degree rotation - (half of -1, which was half of 1) - creating the complex plane and the concept of a root of unity and the beginning of the notion of a "group" and all that good stuff.

Edited by overtone
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The problem with the algebraic demos is that they are unintuitive - the new encounter can see that the rules require the definition, but that just punts the question to the next level: why have those rules? They seem messy, arbitrary.

I don't think the axioms of a field (so real numbers) are particularly messy or unintuitive. Basically you have an addition, a multiplication and you have for each element inverses, except for multiplication by zero. You then require that all these structures satisfy the most natural compatibility conditions you can think of.

 

Other algebraic structures tend to follow this route of some compatibility type conditions.

 

Anyway, more generally in mathematics you do come across some very unnatural definitions and axioms. Often these are not the 'true' definitions and lack clear structure. One hopes that a clearer picture will emerge.

 

 

So that's one advantage to viewing the negative sign as a flip, or 180 rotation: it's immediately intuitive that two such rotations bring you back, that the absolute value is not affected, and so forth. The picture is clear. The intuitive picture then explains the various obscure algebraic manipulations, fractions, square roots, etc, rather than the other way around.

 

 

Another advantage is that such an intuitive picture prepares one for complex numbers, in which i is intuitively explained as a 90 degree rotation - (half of -1, which was half of 1) - creating the complex plane and the concept of a root of unity and the beginning of the notion of a "group" and all that good stuff.

Geometric formulations are usually very helpful, I agree.

Edited by ajb
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