Math question

[Scriptavolant]

Why - multiplied by - gives +?

Mathematical explanations?

[71 dB]

Why - multiplied by - gives +?

Mathematical explanations?

Multiplication by a negative number means a turn of the angle by pii. When you multiply a negative number by another negative number you turn the angle by 2pii (= 360 degrees = 0 degrees) and you have a positive number.

[Topaz]

Why - multiplied by - gives +?

Mathematical explanations?

(i)  Assume that a negative number is a positive number multiplied by -1, i.e. assume that -A = A(-1) where A is positive.  Hence, the product of two negative numbers, -J and -K, may be written as (-1)(-1)JK

(ii) Assume that (-1)(-1) = +1.  Is this assumption reasonable? The alternative is that (-1)(-1) = -1.  Consider the implication of this alternative assumption by looking at the product of -1 and 0.  We would get:

     0=  (-1)(0) =   (-1)(1 + -1) = (-1)(1) + (-1)(-1) = -1 + -1 = -2

Thus, we would get 0=-2, and the "distributive" property of multiplication wouldn't work for negative numbers. Thus (-1)(-1) must = +1.

(iii). It follows that (-J)(-K) = +JK

[head-case]

Multiplication by a negative number means a turn of the angle by pii. When you multiply a negative number by another negative number you turn the angle by 2pii (= 360 degrees = 0 degrees) and you have a positive number.

71 db can't figure out why multiplying two negative numbers results in a positive number, and you assume he or she will understand operations in the plane of complex numbers?  Maybe an intuitive explanation involving group theory or the fully anti-symmetric tensor would be preferable.   

Try the Sherlock Holms method, the process of elimination.  There are only two possibilities, when you multiply two negative numbers the result is either positive or negative.  If the result were negative then this would imply that multiplying negative number by a positive number give the same result as multiplying a negative number by a negative number.  The result of multiplication must depend on the factors, so this can't be.  The other possibility must be true, the result is positive.

[head-case]

(i)  Assume that a negative number is a positive number multiplied by -1, i.e. assume that -A = A(-1) where A is positive.  Hence, the product of two negative numbers, -J and -K, may be written as (-1)(-1)JK

(ii) Assume that (-1)(-1) = +1.  Is this assumption reasonable? The alternative is that (-1)(-1) = -1.  Consider the implication of this alternative assumption by looking at the product of -1 and 0.  We would get:

     0=  (-1)(0) =   (-1)(1 + -1) = (-1)(1) + (-1)(-1) = -1 + -1 = -2

Thus, we would get 0=-2, and the "distributive" property of multiplication wouldn't work for negative numbers. Thus (-1)(-1) must = +1.

(iii). It follows that (-J)(-K) = +JK

That works, why introduce the -1 thing,

0 = (-J)(0) = (-J)(K-K) = (-J)(K) + (-J)(-K)
0 = (-J)(K) + (-J)(-K)
since (-J)(K) is negative, (J)(K) is positive.

[PSmith08]

(i)  Assume that a negative number is a positive number multiplied by -1, i.e. assume that -A = A(-1) where A is positive.  Hence, the product of two negative numbers, -J and -K, may be written as (-1)(-1)JK

(ii) Assume that (-1)(-1) = +1.  Is this assumption reasonable? The alternative is that (-1)(-1) = -1.  Consider the implication of this alternative assumption by looking at the product of -1 and 0.  We would get:

     0=  (-1)(0) =   (-1)(1 + -1) = (-1)(1) + (-1)(-1) = -1 + -1 = -2

Thus, we would get 0=-2, and the "distributive" property of multiplication wouldn't work for negative numbers. Thus (-1)(-1) must = +1.

(iii). It follows that (-J)(-K) = +JK

It's getting unnecessarily complicated for the situation, but I might say that the definitions of a field (i.e., a commutative division ring) answer your question ab initio. If you define a set, R, with necessarily two binary operations, + and *, as the real numbers, you will see that it has all the properties of a ring. If you can show that R has a nonzero identity element (i.e., a*b=1 and a*1=a but 1/=0), then you will have a division ring. You can show this, by the way, for R. If, furthermore, you can show that R is commutative, then you will have shown that R is a field. A quick check of the properties of a field, then, will show that the binary operators are associative and commutative, and it then becomes a triviality to explain how two real numbers with a negative sign multiply to a positive.

[Topaz]

Try the Sherlock Holms method, the process of elimination. 

That's exactly what I set out, viz the "Sherlock Holmes" method, in time-honoured fashion, using symbolism that many folk here should be able to understand, instead of set theory which most folk, I guess, wouldn't.

[PSmith08]

That's exactly what I set out, viz the "Sherlock Holmes" method, in time-honoured fashion, using symbolism that many folk here should be able to understand, instead of set theory which most folk, I guess, wouldn't.

What? They don't teach ring theory in high school anymore? Shock, horror!

But seriously, if you attack it in the abstract realm the problem becomes a matter of simple definitions, rearranged to yield the answer - or make the answer very easy.

[head-case]

Other have done this to, but this is simple:

if (-1)(2) = -2

then  (-2)(-2) = -1(2*2)

...which is incorrect?

[Steve]

Finally, a Math Question, and I'm late to the party. 

[bwv 1080]

...which is incorrect?

Sorry this is what I was trying to think of:

if 2x = x + x
and 2(-x) = (-x-x) = -2x

then -2(-x) = -(-x-x) = -(-2x) = 2x

[Scriptavolant]

Thank you guys for your contributions! Think I'll have to concentrate deeply on your explanations to get to the end of it at last 

[S709]

Already answered, but I like this form of the argument the most:

Let a > 0 be a real number.

We know that:

a + (-a) = 0.

Let b > 0 be another real number.

Assume the distributive property holds and multiply by (-b) :

(-b)(a) + (-b)(-a) = 0.

The left term is negative.

The magnitudes of the two terms are the same.

So the only way that this equation can be true is if the right term is positive.

So then (-b)(-a) = ab.

Then we have:

-ab + ab = 0.



Another way to get more intuition:

For any real a that is not zero:

a(1/a) = 1.

if a = -7 for example:

(-7)(1/(-7)) = 1, NOT -1 so a negative multiplied by a negative must be positive for the concept of multiplicative inverse to make sense.

[Scriptavolant]

Up to now I've seen a lot of most interesting formal and abstract explanations I will (try to) study soon. But what if you should explain it to your child? I mean why the hell multipling - 2 peaches by - 2 peaches you get (+) 4 peaches. There must be a conceptual way to illustrate the thing in pragmatic terms.

[mahlertitan]

Up to now I've seen a lot of most interesting formal and abstract explanations I will (try to) study soon. But what if you should explain it to your child? I mean why the hell multipling - 2 peaches by - 2 peaches you get (+) 4 peaches. There must be a conceptual way to illustrate the thing in pragmatic terms.

for some reason i can not conceptualize -2 peaches multiply by -2 peaches.

[S709]

That is a good question, and the answer is, I think: the whole idea of negative numbers is already abstract.

The way they are introduced is by the 'elevation or temperature below 0' approach, but no one explains (and none of my classmates ever asked, either, or me until later years! ): "why would anyone multiply two negative temperatures or elevations"?

[Steve]

Up to now I've seen a lot of most interesting formal and abstract explanations I will (try to) study soon. But what if you should explain it to your child? I mean why the hell multipling - 2 peaches by - 2 peaches you get (+) 4 peaches. There must be a conceptual way to illustrate the thing in pragmatic terms.

A negative quantity is one that cannot, naturally, be expressed in terms of real quantities.

[Scriptavolant]

A negative quantity is one that cannot, naturally, be expressed in terms of real quantities.

But what if you put it this way.

John has 20 peaches.
Jack has 18 peaches (-2)

And you rehearse the condition for a x positive number of times.

[bwv 1080]

A negative quantity is one that cannot, naturally, be expressed in terms of real quantities.

In $ it can.  If I borrow $100 worth of XYZ stock and sell it then I have a position of -$100 in the stock.  If the stock's return next year is -100% then my gain is
(-100)(-1) = $100

I have made $100

[head-case]

for some reason i can not conceptualize -2 peaches multiply by -2 peaches.

Of course not, what practical problem would have an answer in units or peaches squared.