two negs equal a positive?

Question

...how to explain to students the product of two negative numbers is a positive number

Answer 1

Essentially, it boils down to proving the following:

Let x > 0, and let -x denote the additive inverse of x. We wish to show that (-1)*(-x) = x.

To prove this, it suffices to show that (-1)*(-x) is the additive inverse of (-x):

Well, (-x)+(-1)*(-x) = [1 + (-1)](-x) = 0, where the first equality follows from the distributive property.

Hence, (-1)*(-x) is the additive inverse of (-x), so it must be equal to x.

Answer 2

Since multiplication is just a short hand for addition, use that as an intro as well as the idea that positive and negative are just directions along the number line.
So, when you have 3 * -1, that means, start from zero, face the right and then turn around (the negative) and take 3 steps to the left.
For -3 * -1, start at zero, face the left (the first negative) and then turn around (the second negative) and take 3 steps to the right.

Answer 3

Well if you have already convinced them that a positive number times a negative number is negative, try this sequence:

-5 * 3 = -15
-5 * 2 = -10
-5 * 1 = -5
-5 * 0 = 0
-5 * -1 = 5
-5 * -2 = 10
-5 * -3 = 15

And so on -- each step increases by 5. It makes a logical sequence and most students can accept this explanation. Good luck!

Answer 4

The algebraic proof of this is quite tricky. Since the - sign is used for two things, negative numbers and subtraction, we really need to distinguish between them. I will use ' for negative number in the following.
You first need to prove that positive*negative = negative
By definition b + b' = 0
Therefore a(b + b') = ab + ab' = 0
But also by definition ab + (ab)' = 0
Therefore ab' = (ab)' i.e. positive*negative = negative.
Now consider (a + a')(b + b') = ab + a'b + ab' + a'b' = 0
We know that the first two terms sum to zero so second two terms must as well so ab' + a'b' = 0
However by definition ab' + ab = 0
Therefore the last term in each line must be the same
Thus a'b' = ab i.e. negative*negative = positive.

Answer 5

If I owe you $10 ,I write it as -10. If I owe you 3 loans it is 3 x(-10) = -30; I now owe $30.

If I Cancel the debt ( by any means:repaying,excused) that would be -3. Then: (-10) x (-3);meaning 3 cancellations of 3 debts, would bring me back to positive;that the 30 is now in my favor. =+30

Answer 6

multiplycation is adding the sames number, over and over again

2*8=2+2+2+2+2+2+2+2=16
(add 2, 8 times)

now try with one negative
-2*8=
(-2)+(-2)+(-2)+(-2)+(-2)+(-2)+(-2)+(-2)= -16
(add -2, 8 times)

now try with two
-2*-8=
-(-2)-(-2)-(-2)-(-2)-(-2)-(-2)-(-2)-(-2)= 16
(add -2, -8 times, OR subtract -2, 8 times)

show negative and positive addition and subtraction on a number line
start at 0
add going ->
every minus/negative sign, reverse direction

show that this is true with simple problems
2-1=1

-2,-1,0,1,2
____^ <--------- start at 0
-2,-1,0,1,2
____> > ^ <----------- move to 2
-2,-1,0,1,2
_____ ^ < <---------- reverse at the minus move 1

Answer 7

Tell them that two negatives cancel each other, so if there is a third negative then negative overcomes the positive.

Answer 8

Use colors.Use blue color for positive and red for negative.
In a sheet of paper, color 1 red dote and a blue dot.Then explain it to them.Our teacher taught us like this, so they can remember.

Answer 9

I found this website. http://mathforum.org/library/drmath/view/51925.html

Answer 10

tell them they cancel each other out