Negative number multiplied by a negative number

[physio]

For two positive numbers such as 5 and 3 their multiplication results in a positive number because it is 5 times 3 or 3 times 5 (5+5+5) also when we multiply a negative number with a positive number say -5x3 i.e 3 times negative 5 will give us (-5+-5+-5) but now what is the explanation for a negative number multiplied by a negative number? How can you multiply by a negative times of a no.??

Is there an intuitive explanation existing for this kind of a problem?

I found these websites but none offer me an intuitive explanation:
http://mathforum.org/library/drmath/view/57865.html
http://mathlesstraveled.com/2009/09/29/minus-times-minus-is-plus/
http://www.mathsisfun.com/multiplying-negatives.html

 

[castro94]

never thought of it before, its just a fact that u accept as it is i gues, but maybe -5*-3 can be expressed as 5-(-5)-(-5) maybe ?

 

[pwsnafu]

Distance (actually Displacement) = Velocity times Time.

Suppose we have a car set to 10 meters per second, and drive for 3 seconds. Distance is 30 meters.

Suppose we have a car set to -10 meters per second (remember that means drive backwards at 10 m/s), and drive for 3 seconds. The car travels -30 meters, i.e. it travels backwards 30 meters. Video tape this.

Suppose we have a car set to -10 meters per second and drive for -3 seconds. What does this mean? Take the video from above and watch it backwards. You see a car traveling 30 meters forwards. So negative velocity multiplied by negative time gives a positive displacement.

Edit: Another example.

You have a credit account with a banking institution. If the account has a positive amount (e.g. $20) then it means the customer has credit and the bank owes the customer (you) $20. If the account has a negative amount, the customer (you) owes the bank.

So multiply by two. This means there are two customers. So -$20 times 2 customers means a total debt of $40 (this comes up when we are talking about the debt of a married couple).

That's all great and all, but this is from the perspective of the customer. Positive = customer credit doesn't make sense for a bank because for the bank it's actually debt. How do you swap credit and debt? You multiply by a negative number. In the case above, that would be -2, so the bank has total credit $40, and would write so on the tax return.

 

[coolul007]

It's the old double negative. The negation of anything is to do the opposite. If we do the opposite twice we get a positive. It's that simple.

 

[drewfstr314]

Assume A is positive.

By the Additive Identity Law,we have
A - A = 0

From the Definition of Subtraction, we have
A + (-A) = 0

Multiplying by a positive B, we have
B(A + (-A)) = 0B

Distributing,
AB + (-A)(B) = 0B

By the Converse of the Zero Product Property,
AB + (-A)(B) = 0

Now, because A and B are both positive, AB > 0, implying that
(-A)(B) < 0
(recall that A+(-A)=0... one is positive, one is negative.)

Therefore, a [negative] * [positive] = [negative]. No surprise there!

Again, A + (-A) = 0. Multiplying by (-B),
A(-B) + (-A)(-B) = 0

However, A(-B) is negative (see above), and therefore
(-A)(-B) is positive.

QED

This was taken from http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/negative.html

 

[physio]

Thanks to everybody! I understood from the above answers why -x-=+ but now I have another problem that has to do with subtracting from zero, in the sense how can you possibly take away something from nothing? How is 0-5=-5 possible using the above mentioned reasoning? Is there a different interpretation to this then?

Also why is 0-(-5) = 5 (positive)? Does this follow from negative number multiplied by negative number=positive??

Thanks!

 

[arildno]

Thanks to everybody! I understood from the above answers why -x-=+ but now I have another problem that has to do with subtracting from zero, in the sense how can you possibly take away something from nothing? How is 0-5=-5 possible using the above mentioned reasoning? Is there a different interpretation to this then?

Also why is 0-(-5) = 5 (positive)? Does this follow from negative number multiplied by negative number=positive??

Thanks!

Hi, physio!
Your questions are good.
It is perfectly true that when we count apples, zero is our perfect number to signyfy NO apples at all.

But, does it follow from that that the number zero is, always must regarded as "nothing"?
Doesn't that really depend on how we envisage what numbers "are"?

Look at the number line.
Isn't 0 just as much a position on that line as any of the other numbers?

And if so, isn't it rather wrong to say that 0 is.."nothing", since it does exist at a specifyed position on the number line?

 

[Diffy]

Assume A is positive.

By the Additive Identity Law,we have
A - A = 0

From the Definition of Subtraction, we have
A + (-A) = 0

Multiplying by a positive B, we have
B(A + (-A)) = 0B

Distributing,
AB + (-A)(B) = 0B

By the Converse of the Zero Product Property,
AB + (-A)(B) = 0

Now, because A and B are both positive, AB > 0, implying that
(-A)(B) < 0
(recall that A+(-A)=0... one is positive, one is negative.)

Therefore, a [negative] * [positive] = [negative]. No surprise there!

Again, A + (-A) = 0. Multiplying by (-B),
A(-B) + (-A)(-B) = 0

However, A(-B) is negative (see above), and therefore
(-A)(-B) is positive.

QED

This was taken from http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/negative.html

What a great proof.

It makes me think (sorry going OT).

At the beginning of most algebra courses here in the US, we teach these properties of real numbers. But all the quizzes and tests that I have ever seen from high school teachers only test a students ability to recognize and match the properties to examples.

Such as

A = A

What properties is this? Answer: Reflexive.

I would love to see some teachers present an easy to understand proof like this. It has the perfect easy to understand motivation ie why is the product of two negatives a positive?

It really is a lovely proof.

 

[Akshay_Anti]

Hi, physio!
Your questions are good.
It is perfectly true that when we count apples, zero is our perfect number to signyfy NO apples at all.

But, does it follow from that that the number zero is, always must regarded as "nothing"?
Doesn't that really depend on how we envisage what numbers "are"?

Look at the number line.
Isn't 0 just as much a position on that line as any of the other numbers?

And if so, isn't it rather wrong to say that 0 is.."nothing", since it does exist at a specifyed position on the number line?

I completely agree with arildno. As far as i have learned zero is not just a number, it's a symbol to accept and recognise the existence of void that is many a times needed in mathematics. Take for example, the numeral system.. Any base would require a zero for a void. And zero is better defined as intersection of positive and negative number line..

 

[Mark44]

Thanks to everybody! I understood from the above answers why -x-=+ but now I have another problem that has to do with subtracting from zero, in the sense how can you possibly take away something from nothing? How is 0-5=-5 possible using the above mentioned reasoning? Is there a different interpretation to this then?

To continue with the banking analogy in a previous post, if your account balance is 0, and you write a check for $5, your new balance would be -5 dollars. What that means is that you would have to deposit $5 just to get your balance back up to $0. Of course, the bank won't look at it this way, and will charge you a fee for the bounced check.

Also why is 0-(-5) = 5 (positive)? Does this follow from negative number multiplied by negative number=positive??

Not necessarily. You can always rewrite a subtraction expression as an addition by using the opposite (AKA additive inverse) of the thing that is being subtracted.

So 0 - (-5) = 0 + -(-5) = 0 + 5 = 5
The opposite of -5 is + 5.

 

[arildno]

Or, for tht matter:

0-(-5)= 5+(-5)-(-5)=5+0=5, by substituting 5+(-5) for 0

 

[HallsofIvy]

I completely agree with arildno. As far as i have learned zero is not just a number, it's a symbol to accept and recognise the existence of void that is many a times needed in mathematics. Take for example, the numeral system.. Any base would require a zero for a void. And zero is better defined as intersection of positive and negative number line..

No. Interpreting the number 0 as "void" is not mathematics. And, since 0 is NOT either positive or negative, it is NOT the "intersection of positive and negative number lines". The intersection of the sets of positive and negative numbers is empty. Although you start by saying you "completely agree" with arildno, what you say pretty much contradicts what he says.