Why is the product of two negative numbers a positive?

[LabRat1]

So a few days ago I was randomly solving some inverse trigonometric problems, and obviously I wasn't paying attention, and this question came on to me. Any clue why? Never really got it.

 

Thanks in advance.

[Function]

I never gave it much thought, truth be told. You could see "negative" as "the opposite of ..."

 

Imagine that in front of all positive numbers is an imaginary (I should be careful here with my terminology, but I don't think using the word 'imaginary' will confound your understanding of basic operations with negative numbers) "plus-sign". But because it is too trivial of writing it down, we leave it away, except when it comes after another number of letter, to distinguish it from a multiplication (ab vs. a + b).

 

So let's take a random positive number, say a.

 

You could write it down as: (+a).

 

Put a minus sign in front of it: -(+a). Which then would mean as much as: the opposite of (+a).

The "opposite" of a number is the absolute value of that number, but with another sign

The opposite of a positive number is, of course, the negative version of its absolute value: -(+a) = -(|a|) = (-a)

 

Now if you were to put a minus sign in front of that:

 

-(-a), you'd want to take the opposite of (-a). The opposite of (-a) is the absolute value of (-a), that is, |(-a)|, which is just (a) (or (+a), as mentioned before). Then take the opposite sign; the original in (-a) was -, so that becomes +|(-a)| = (+a) or just a.

 

---

 

Long story, made over complicated, short:

 

Consensus.

 

a = a
-1 * a = -a

-1 * (-1 * a) = -1 * (-a) = a

-1 * (-1) = 1

[studiot]

So a few days ago I was randomly solving some inverse trigonometric problems, and obviously I wasn't paying attention, and this question came on to me. Any clue why? Never really got it.

 

Thanks in advance.

 

So are you paying attention now?

 

Think of it this way

 

A negative is the not or negation operator operating on a positive.

using any operator is the same as multiplying by something in mathematics.

 

So the negation or not operator operating on say 10 gives not(10) or (-1) x (10) or (-10)

 

If we do this operation a second time that is we negate our original negate we should end up where we started

 

That is (-1) x (-10).

 

this has to be +10 to end up where we started

 

So the product of two negatives is a positive.

[Prometheus]

Recently watched this video giving an explanation.

 

[StringJunky]

Recently watched this video giving an explanation.

 

It's on the same lines as the Khan Academy explanation: it's consistent with the rest of the ways mathematics is done.