Negative numbers were hardly distrusted in Indian mathematics - Brahmagupta in the 6th Century just asked called a negative number as "debt". (see for example, [1])
The real question is why negative times negative is positive - I had this doubt in school. I am a bit ashamed to admit that I still don't know any "from first principle explanation" for this fact.
The real question is why negative times negative is positive - I had this doubt in school. I am a bit ashamed to admit that I still don't know any "from first principle explanation" for this fact.
Don't sweat it, here's a rough proof. First a proof that a negative times a positive is negative:
a*0 = 0 [Assumed but you can prove this too.]
a*(b-b) = 0 [Definition of 0.]
a*b + a*(-b) = 0 [Distributive law.]
For any X, only its inverse -X has the property that X + -X = 0. Therefore a×(-b) is the inverse of a×b [which is -(a×b)].
Then apply the same idea but with a negative a:
(-a)*0 = 0 [Assumed but you can prove this too.]
(-a)*(b-b) = 0 [Definition of 0]
(-a)*b + (-a)*(-b) = 0 [Distributive law.]
-(a*b) + (-a)*(-b) = 0 [From above.]
(-a)*(-b) + -(a*b) = 0 [Commutative addition to rearrange to X + -X = 0 form.]
And as before, only the inverse has this property so -(a×b) must be the inverse of (-a)×(-b) -- which is a×b! So a negative number times a negative number is positive. QED.
In case you're interested, the proof that the additive element is a multiplicative absorbing element (a×0 = 0, for all a) is also fairly trivial[1].
The above does somewhat implicitly assume that 0 and -0 are equal (or at least that (-a)×0 = (a)×0 = 0, which implies that 0 and -0 are equal).
The 'negative' on a number line is just a visualization. If you instead, think of the 'negation' as direction (as in a vector), it becomes easier to understand why multiplying two negative numbers makes the answer positive (You've basically turned around twice).
I actually prefer this visualization rather than the one with a number line with numbers going to the left of 0.
To better understand this, and see how complex numbers can be viewed the same way, go through this article, especially up to the first set of slides: https://acko.net/blog/how-to-fold-a-julia-fractal/
Say you have a $100 debt, so your net worth is -$100. If I take that $100 debt away from you, your net worth goes up by $100. -$100×-1 = +$100: a negative times a negative is a positive.
A debt is a negative quantity, and me taking something from you changes your quantity of whatever it was by a negative amount (if I take one apple from you, you gain -1 apples), so a negative times a negative is a positive.
If you have 3 $100 debts. If I take two of those $100 debts away from you, your net worth goes up by -$100×-2 = $200.
Comments
Negative numbers were hardly distrusted in Indian mathematics - Brahmagupta in the 6th Century just asked called a negative number as "debt". (see for example, [1])
The real question is why negative times negative is positive - I had this doubt in school. I am a bit ashamed to admit that I still don't know any "from first principle explanation" for this fact.
[1] https://nrich.maths.org/5961
Don't sweat it, here's a rough proof. First a proof that a negative times a positive is negative:
For any X, only its inverse -X has the property that X + -X = 0. Therefore a×(-b) is the inverse of a×b [which is -(a×b)].Then apply the same idea but with a negative a:
And as before, only the inverse has this property so -(a×b) must be the inverse of (-a)×(-b) -- which is a×b! So a negative number times a negative number is positive. QED.In case you're interested, the proof that the additive element is a multiplicative absorbing element (a×0 = 0, for all a) is also fairly trivial[1].
The above does somewhat implicitly assume that 0 and -0 are equal (or at least that (-a)×0 = (a)×0 = 0, which implies that 0 and -0 are equal).
[1]: https://en.wikipedia.org/wiki/Additive_identity#The_additive...
The 'negative' on a number line is just a visualization. If you instead, think of the 'negation' as direction (as in a vector), it becomes easier to understand why multiplying two negative numbers makes the answer positive (You've basically turned around twice).
I actually prefer this visualization rather than the one with a number line with numbers going to the left of 0.
Thanks, this was helpful.
To better understand this, and see how complex numbers can be viewed the same way, go through this article, especially up to the first set of slides: https://acko.net/blog/how-to-fold-a-julia-fractal/
Say you have a $100 debt, so your net worth is -$100. If I take that $100 debt away from you, your net worth goes up by $100. -$100×-1 = +$100: a negative times a negative is a positive.
A debt is a negative quantity, and me taking something from you changes your quantity of whatever it was by a negative amount (if I take one apple from you, you gain -1 apples), so a negative times a negative is a positive.
If you have 3 $100 debts. If I take two of those $100 debts away from you, your net worth goes up by -$100×-2 = $200.