Negative times Negative

[AvgStudent]

Hi,
I'm curious why a negative times a negative is equal to a positive. What's the intuition behind it? Is there any concrete proof, or this is something mathematicians agreed upon.

Thanks

 

[R.M.]

This is not something mathematicians "agreed on", and it can be proven.

Start with this fact: -(-a)) = a. In other words, the opposite of the opposite of a number is the original number.

Now using the associative property, commutative property, and the above idea, you can show that the product of two negatives is positive.
(-a)(-b) = -(a)(-b) = -(a)(-(b)) = --(ab) = ab

I'm rusty on my abstract algebra so I may not have every rigorous detail here, but this is the basic idea behind the proof.

 

[Dr.Peterson]

Hi,
I'm curious why a negative times a negative is equal to a positive. What's the intuition behind it? Is there any concrete proof, or this is something mathematicians agreed upon.

Thanks

You can find many explanations of this fact by searching. Here is one source with a number of intuitive answers:

Multiplying Negatives Makes A Positive

Yes indeed, two negatives make a positive, and we will explain why, with examples ... Lets talk about signs. ... is the positive sign, is the negative sign.

www.mathsisfun.com

At a higher level, it is a definition (since no previously existing definition applies), that can be proved to be the only definition such that multiplication will retain the properties it has when applied to positive numbers. To show that, suppose that multiplication is commutative, associative, and distributive. Let a and b be any two real numbers, and consider the number x defined by

x = ab + (-a)(b) + (-a)(-b).​

On one hand

x = ab + (-a)[ (b) + (-b) ] (factor out -a from the last three terms)​

= ab + (-a) * 0​

= ab + 0​

= ab.​

But on the other hand,

x = [ a + (-a) ]b + (-a)(-b) (factor out b from the first two terms)​

= 0 * b + (-a)(-b)​

= 0 + (-a)(-b)​

= (-a)(-b).​

But if x = ab and x = (-a)(-b), then ab = (-a)(-b).

 

[AvgStudent]

Thank you both for the answer. I'm a high school student and doing calculus, and recently I became interested in things that were given as facts since algebra, such as the question of this thread. I have a list of things to ask. Maybe I should make another thread.

 

[Piyush Ranjan Dash]

See to represent a in number line we need to go a units towards the right of 0 on a number line as a is positive(assumption). But to represent -a go a units towards the left of 0 on a number line. Hypothetically speaking,-a is opposite of a and vice versa. That means to get a opposite we should add a minus sign. Thus -(-a) means opposite of -a which is a. Thus,getting that negative times negative results in a positive term.

 

[Deleted member 4993]

Thank you both for the answer. I'm a high school student and doing calculus, and recently I became interested in things that were given as facts since algebra, such as the question of this thread. I have a list of things to ask. Maybe I should make another thread.

you are doing it correctly - one 'dilemma' per thread !