Why is the Product of a Negative and Positive Number Negative?

[Someone502]

i tihnk this is a dumb question but why is -x*x=-y? i tried thinking about it like a arrays on a graph but it was kinda confusing.

 

[honestrosewater]

You mean why the product of a negative and positive is negative? This follows from the axioms.
(-x)y + xy =_{[see note 1]} (-x + x)y =_[2] 0y =_[3] 0. Since (-x)y + xy = 0,_[4] (-x)y = -(xy).
1. by the distributive property
2. additive inverse
3. (0x = 0) is a theorem
4. (If x + y = 0, then x = -y) is a theorem

 

[HallsofIvy]

"-x*x= -y"? Well, that's not always true- it depends on what x and y are!

If you mean "why is a negative number times a positive number negative?" then honestrosewater gave a pretty good answer.
(what would DIShonestrosewater smell like?)

If you are wondering about the distinction between (-x)² and -x², it's a matter of the parentheses: (-2)² means (-2)(-2), a product of two negative numbers, which is positive: 4. -2² means to FIRST square: 2²= 4, THEN make it negative: -4.

 

[honestrosewater]

If you mean "why is a negative number times a positive number negative?" then honestrosewater gave a pretty good answer.

Is there another answer?

(what would DIShonestrosewater smell like?)

Fishy.