Why in mathematics was a positive number made to be the result of multiplying two negative numbers?

Question

-Where did this first enter into mathematics and why? Is it possible that they adopted it from philosophy and the negation of the negation / absolute negativity (e.g. Hegel / dialectics)?

Answer 1

It seems logical at face value. If you have three fives (3 x 5), you can employ either multiplication or addition:

3 x 5 = 15 or 5 + 5 + 5 = 15.

If you have three negative fives:

3 x -5 = -15 or -5 + -5 + -5 = -15

If you have negative three negative fives:

-3 x -5 = 15 or -(-5) + -(-5) + -(-5) = 15

It all depends on the conventional notion of negative and positive (a pile of dirt or a hole in the ground where -1 +1 = 0) and the conventional manipulation of numbers using the rules (and logic) of math.

Answer 2

The Negation of the Negation would have to be where it comes from because I was trying to think back to when we first learned negative numbers in school and they tried to explain addition of negatives by saying for example that you owe someone $10 then you have -10 and if you get 20 from somewhere else you have +10 but if you owe 10 to 10 different people you are multiplying (-10)X(+10) to get negative 100 so combine this with what is learned in logic to make both negative and you have owe 10 to not 10 people and it is like saying that 10 people owe you 10 therefore making it positive and 100 owed to you and not that you owe to others.

Answer 3

It's not philosophical, it's because nothing else gives you consistent answers.

What is 60 + (-7) x (8 - 2)?

First method: 60 + (-7) x 6 = 60 - 42 = 18.
Second method: 60 + (-7) x 8 + (-7) x (-2) only comes out to the same answer if (-7) x (-2) is 14, not -14.

Answer 4

Here are two good explanations (from several standpoints) of why it's reasonable to make the multiplication of two negatives take a positive value.

http://mathforum.org/dr.math/faq/faq.negxneg.html

http://mathforum.org/library/drmath/view/55717.html

The idea of negative numbers probably dates to about the same time as the 'invention' of zero and decimal notation and the resulting changes in calculation that happened then. In Europe, this seems to have been toward the end of the 15th Century, but read this: http://www.ma.utexas.edu/users/mks/326K/Negnos.html

Answer 5

imagine you have been driving a car in reverse (-5mph)
2 hours from now, you will be 10m "behind" your current position (-5 X 2 = -10).
2 hours ago (0-2=-2hour), you were at 10m "in front" of your current position (-5 X -2 = +10)

Answer 6

because the negative cannot be positive when they met the positive numbers..so in order to give them a turning points..mathematicians around the world has agreed that once they met an equal sign (the negative sign) then they have the chance to be positive..does it sound ok?

Answer 7

Because two wrongs make a right?