Multipling two negative integers

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I know the algorithm is (-a) x (-b)= ab; in other words, multiplying 2 negative numbers equals the positive product of the 2 numbers, but why?

Can someone explain this conceptually, and offer a few "real life" examples of this?

Thanks!

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MUS explains a negative as opposite. So if you have one number and put a negative sign in front of it, you have the opposite of that number (on a number line for example) which is -number. If you have two negatives, then it is opposite twice. The first opposite is negative, the next one turns it back to positive.

I'm not sure I explained that well enough to help.

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Imagine it on that "evil" number line. :)

-3 once is still -3. -3 x 2, or -3 twice, is -6. But when you do -3 x -2 you are doing the opposite of -3 x 2, or the opposite of -6, so what you get is +6 instead.

Does that make sense?

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You can also see it as a pattern:

(-3)(3)=-9

(-3)(2)=-6

(-3)(1)=-3

(-3)(0)=0

If we continue the pattern, we continue to decrease our second number and our product continues to increase by three.

So...

(-3)(-1)=3

(-3)(-2)=6

etc.

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We know that:

1 - 1 = 0

Rewrite (it makes a future step easier to see)=>

1 + (-1)*1 = 0

add (-1)*(-1) to both sides =>

1+ (-1)*1 + (-1)*(-1) = (-1)*(-1)

So now we have:

1 + (-1)*1 + (-1)*(-1) = (-1)*(-1)

factor the (-1) out =>

1 + (-1)*(1-1) = (-1)*(-1)

1-1 is 0 so =>

1 + (-1)*0 = (-1)*(-1)

0 times any number is 0 =>

1 = (-1)*(-1)

So we have that (-1)*(-1) is (+1).

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We know that:

1 - 1 = 0

Rewrite (it makes a future step easier to see)=>

1 + (-1)*1 = 0

add (-1)*(-1) to both sides =>

1+ (-1)*1 + (-1)*(-1) = (-1)*(-1)

So now we have:

1 + (-1)*1 + (-1)*(-1) = (-1)*(-1)

factor the (-1) out =>

1 + (-1)*(1-1) = (-1)*(-1)

1-1 is 0 so =>

1 + (-1)*0 = (-1)*(-1)

0 times any number is 0 =>

1 = (-1)*(-1)

So we have that (-1)*(-1) is (+1).

Great Explanation!

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Thanks for all the responses.

On paper, I can understand how to manipulate the numbers and why the product of 2 negative integers is positive. However, it is in the realm of real life that I am trying to figure out when we need to multiply 2 negative numbers together!

So for example, in subtraction, a negative number can represent loss, such as withdrawals from a bank account. So that if I withdraw one hundred dollars, that can be represented as -100. Let's say I withdraw \$100 every month for 3 months. This can be represented mathematically as 3 x (-100) or a net withdrawal of -300.

But then, how do you explain something like -3 X (-100)? If we continue the "month" analogy... could you say that moving forward in time is postive and moving backward is negative, so that 3 months ago is -3. The next part is where I get stuck. What kind of story problem would result in -3 X (-100)? Would you say how much more money did I have at the beginning of 3 months? But then, I would always think in the positive.... just 3X100=300. Why do we even need to multiply 2 negative integers?

Ack!

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The problem with coming up with word problems that result in multiplying two negative numbers is that most people wouldn't solve a problem that way. With the example you started to formulate about bank withdrawals, if you asked:

You withdraw \$100 a month and have been doing so for a year. You currently have \$900 in your bank account, how much did you have three months ago?

Well, most people would not write out "\$900+(-3 months)*(-\$100/month)" to solve the problem. Most people would multiply \$100 by 3 and add it to \$900.

Some examples that aren't word problems that might result in multiplying negative numbers would be graphing an equation like y=-2x+4. But the sort of problem that would require coming up with this formula and using it probably belongs in an algebra course.

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Silver, I think I'm finally understanding... multiplying 2 negative integers might not correspond to a real-world model that makes sense (because most people naturally think in the positive).

I have not taught my kids algebra yet (soon!) but you've helped me to make sense of my dilemma.

Thanks!

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I like the explanation in the Pre-Algebra for Visual Learners demo video.

The program is near the bottom.

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