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Negative numbers were hardly distrusted in Indian mathematics - Brahmagupta in the 6th Century just asked called a negative number as "debt". (see for example,

by sn41 6y ago

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 https://nrich.maths.org/5961

thunderbong 6y ago
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.
- sn41 6y ago
  Thanks, this was helpful.
  - mkl 6y ago
    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/ https://acko.net/blog/how-to-fold-a-julia-fractal/
deleted 6y ago
[deleted]
cyphar 6y ago
> 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). [1]: https://en.wikipedia.org/wiki/Additive_identity#The_additive_identity_annihilates_ring_elements https://en.wikipedia.org/wiki/Additive_identity#The_additive...
mkl 6y ago
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.

by sn41 6y ago

thunderbong 6y ago

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.

sn41 6y ago
Thanks, this was helpful.
- mkl 6y ago
  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/ https://acko.net/blog/how-to-fold-a-julia-fractal/

deleted 6y ago

[deleted]

cyphar 6y ago

> 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). [1]: https://en.wikipedia.org/wiki/Additive_identity#The_additive_identity_annihilates_ring_elements https://en.wikipedia.org/wiki/Additive_identity#The_additive...

mkl 6y ago

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.