# MandysNotes

## Double Negatives Equal A Positive

20 March 2010 By
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The technical rule to remember here is that for any real number a:

(-1)(-a) = a.

Why?

Consider the equation 1= 1.

If we subtract 1 from both sides we obtain:

1 -1 = 1 + (-1) = 0.

Now multiply this equation by (-1):

(-1)(1) + (-1)(-1) = (-1)(0) = 0,

or,

-1 + (-1)(-1) = 0.

Adding 1 to both sides we obtain:

(-1)(-1) = 1.

Now, for any real number, a:

(-a) = (-1)a,

(-1)(-a) = (-1)(-1)a = a.

Therefore: for any real number a:

-(-a) = a

So if you see:

(-1)(-5) = -(-5) = 5.

Two negative signs in this case equal a positive, because the two negatives when multiplied together cancel each other out!

More examples:

-(-10) = 10,

-(-2) = 2,

-(-1000) = 1000.

Furthermore, because for any two real numbers, a, and b:

(-1)a = (-a),

and,

(-1)b = (-b),

therefore:

(-a)(-b) = (-1)a(-1)b = (-1)(-1)ab = ab.

(Note that we have used the commutative property of multiplication of real numbers to obtain this result.)

Examples:

(-2)(-3) = 6,

(-10)(-9) = 90,

(-5)(-3) = 15.