I have started thinking of the basic mathematical operations again. The thought process began when my father asked me - "why does negative number multiplied with a negative number produce a positive number?".
The basic reasoning on which we are taught multiplication does not seem to work here or if it does I just do not understand how? As per how we are taught multiplication if some one says 'a x b' then it would mean what is the value of adding 'a' 'b' number of times or what is the result of adding 'b' 'a' number of times. This scheme works perfectly when both a and b are positive. This is how we are first taught multiplication and once we understand this concept we start overlooking many things.
It works well even for if one of then is positive and the other is negative. For example 'a x -b' would imply -b added a number of times. As adding a, -b number of times seems absurd (the commutative law of multiplication seems questionable).
What if we have both the numbers negative. We all know because our teachers told that negative number multiplied with a negative number will give us a positive result. But, if I reason the way I have done before I can do no further how do I add something negative number of times. After all you can do a thing once, twice, etc.... or not do at all but how do you do it -1 time?
I am still stuck with this question. Answers and suggestions are welcome.
5 comments:
Interesting indeed.. well, after a lot of thinking when i could not come up with anything convincing, i finally Googled it and found one answer which is convincing to some extent as it appreciates that we have adopted it as a convention and that anything else would have not worked out. Check it out here
They gave convincing reasons/proofs for this in the same article. But, I could not understand why they mentioned "This convention has been adopted for the simple reason that any other convention would cause something to break.". This statement is contradicting the rest of the article.
even I get stumped by things like this at times... and wonder what would happen if simple things like the one you mentioned fail or prove to be wrong... I read a chapter of an ebook which challenged newton's laws of gravitation... will search and send you that link... was an interesting read though not sure what to make of it..
It seems our problems (one at hand) will be solved soon varun seems to have sent (via email) me a very good explanation which is intuitive (with little thought). I shall soon post it in a new post.
I read the answer to this question long back in one of the sixth standard book. Here let me explain. go thru the following
3 x 3 = 9
2 x 3 = 6
1 x 3 = 3
0 x 3 = 0
Here we subtracted each time 3
therefore for one more less that's for -1
-1 x 3 = 0-3 = -3
right? that's product of a -ve no and a +ve no is -ve. Now, lets do the same further
-1 x 2 = -2
-1 x 1 = -1
-1 x 0 = 0
we have increased 1 each time now as we decreased from 3, 2, 1 and 0. So if you decrease one more
-1 x -1 = 0 + 1 = 1
So (-ve no) x (-ve no) = +ve
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