Negative/Positive & Multiply/Divide

I'll try this one for you.

Let's say that on Monday you have two apples in a bowl on your kitchen table.

Mathematically we say you have +2 apples.

Tuesday morning those two apples have disappeared!!!!

We say you have 0 apples, ie., -2 apples have 'appeared'.

If now you can think of this happening on three separate weeks, you will realise that you provided 6 apples but now have -6 apples.

Mathematically then 3X (-2)=-6

that's fine using analogy, I understand that now better, but can you give me something more rigorous and mathematical? Also, how does that explain why a negative multiplied by a negative is a positive? I'm sorry, I'm not completely satisfied! I'm sure that's not the first reason, like why the ancients said it was so, but thanks for that insight!

I get the feeling you are somewhat hard to please...

The question is what does it mean to multiply by a negative number. The "explanation" (you don't seem to want a proof) that I find most compelling is that multiplication is "communative" (meaning you can reverse the numbers and get the same answer i.e. 2 x 3 = 3 x 2)

I know how to multiply any number by a positive number.

2 x 4 is equivalant to 2 + 2 + 2 + 2.

So I can do the same trick with
-2 x 4 which is -2 + -2 + -2 + -2

Then with the communative property, I now can easily figure out what 4 x -2 is and procede from there.

To figure out what a negative number, you can do a similar thing with the associateve nature of multiplication.

But the easy answer is that mathmaticians finally accepted that negative numbers were real number-- they combined this with their understanding of how operations worked to extend the rules they already understood with the positive numbers. They certainly didn't want to break the laws of communativity and associativity they already knew and loved.

It took mathematicians a long time to accept that negative numbers were a useful concept-- the quite fascinating history of negative numbers can be easily googled.

Now a question for homework....

There were 3 people in a room. Then 5 people left.
Now, how many people have to reenter the room for it to be empty?

Isn't the more interesting question why a negative times a negative is positive?

I mean, I can come up with a fractional argument to make it seem like it must be so...

(-x)*(-y) = ((-1)*x)*(y/(-1)) = ((-1)/(-1))*x*y = 1*x*y = x*y

...but why, fundamentally, must it be so?

lol, sorry if it came off like that....


A proof would be very much welcomed thank-you!


I agree

I
Thanks patiodog anf ebrown_p

Quincy read!
OK the reason is some idiot made it up and so thats kind of teh aciant history i guess but. Its not liek u have to knwo this i mean its some ones invention and so that i say who cares. Math is fun but that doesn't matter man so just stop thinking about it and have some fun! Don't worry its not that important!

When I used to teach the rule of signs (Like signs give positive; unlike signs give negative) to 11 year olds, I used to explain it using " the negation of negation is assertion" (eg, don't not sit down = sit down).

However, we technically have asymmetry from results like the square root of 4 equals + or - 2, whereas the square root of
-4 is undefined. This lead to the invention of "imaginary numbers" to rectify the asymmetry by assigning + or - 2i to the square root of -4 where the product i x i =-1. This in effect reverses all "the rules of signs" when applied to imaginary numbers compared with the results for "real numbers". Note that "imaginary" is a technical term only....these "numbers" are extremely useful when applied to wave mechanics or vector analysis.

...or when you want to whip up a quick Mandlebrot Set before breakfast!

I think Quincy is suffering from what I did at school....nobody was able to explain to me the "why" of maths. They seemed all hung up on the "how".

As an adult, I read one day that the point of mathematics was to model the universe. bing!

If only someone had said that to me when I was in early high school.

That is why the real world "-6 apples" scenario was the best answer all along.

Thanks for your responses everyone. I think I have found the answer, afrter reading chapter 4 of "Introduction to Mathematical Thinking" by Friederich Waismann.
It is a convention, there is no proof, but it follows the principle of preservation.