Why does a negative times a negative equal a positive?

Question

In math class we were learning about negatives (multiplying and dividing them) and my teacher wanted to know Why does a negative times a negative equal a positive?

Answer 1

There are a couple of ways of thinking about this. Using the number line is a good way to visualize it, but it doesn't constitute a legitimate proof. Instead, we can work directly with the definition of "negative" to show why this is true.



So what is a negative number? Well, if x is any positive number, then -x is defined to be the number that when added to x gives us zero. As an example, consider 3. When we add -3 to it, we get



3 + (-3) = 3 - 3 = 0.



Now, let a and b be positive numbers. Then -a and -b are negatives. With a little algebraic manipulation, we can prove that their product is positive. Let's multiply them together and add (-a)(b):



(-a)(b) + (-a)(-b)



I can factor out the -a, giving me



-a(b - b) = -a(0) = 0.



So, that means



(-a)(b) + (-a)(-b) = 0



However, remember that a negative times a positive is a negative, so in that equation, we can replace (-a)(b) with -(ab):



-(ab) + (-a)(-b) = 0



Finally, add ab to both sides to cancel that -(ab) on the left side:



(-a)(-b) = ab



And that's what we wanted to prove.



Now, if you are curious as to why a positive times a negative is a negative, we can use similar techniques. Once again, let a and b be positive numbers. Let's multiply a and -b, then add ab and see what happens:



a(-b) + ab



We can factor out the a:



a(-b + b) = a(0) = 0



So, we have



a(-b) + ab = 0



Now, let's add -(ab) to both sides to cancel with that ab on the left (this is the same thing as subtracting ab from both sides):



a(-b) = -(ab)



So, we've shown that a positive times a negative is a negative.

Answer 2

When you look at multiplication you see that, for example, 2 * 2 = 4....right?



And 3 * 5 = 15 and 3 * 1 = 3.....4 * 5 = 20, and so on.



If you look at what these mean you will notice the following:



2 * 2 means you take two sets of two to get four.

3 * 5 means you take three sets of five to get fifteen.

7 * 1 means you take seven sets of one to get seven.

4 * 5 means you take four sets of five to get twenty.



Now....we take a negative of one of each of the numbers in each example above.



2 * -2

-3 * 5

7 * -1

-4 * 5



Now the meaning isn't as clear....you have two sets of negative two things....say what???



So......here's the deal ------ the negative sign is an instruction of direction essentially. Remember the number line? It's a very valuable tool, but I don't think anyone teaches with it anymore......bummer.



The negatives reside on the LEFT side of zero, and the positives on the right.



So if I have a number like 17....it's 17 units to the right of zero. Everything is measured against zero. And if I had -13, you'd go 13 units to the LEFT. The negative sign tells you wher the number is located ----- left or right or zero.



So....again....as I started to say a minute ago, what happens if you take -3 * 5?



Well...this is what you have: (-1)(3)(5)....or (-1)(3 * 5).



We all know that 3 * 5 is 15....so we end up with (1)*15 = -15.



Not too big a deal right?



So, what happens when we have more than one negative sign? Say we have....-2 * -2



What we have is: (-1)(-1)(2)(2), which is (-1)(-1)(2 * 2) = (-1)(-1) * 4.



NOW we get to it (finally right?) We have two negatives multiplied to each other. Are we back at the beginning? No....all they mean is that each negative sign is just an instruction to reverse the location of the number it precedes.



We have two negatives and a [positive] 4. The 4 is positive, so it resides to the right of zero.....ok fine, but the first negative sign is an instruction that says "no....you are on the *left* of zero at -4, BUT, you have another negative sign that reverses the direction once again, and says no, you must now place yourself to the RIGHT of zero at positive 4!



Each negative sign reverses the direction.



Here's a ridiculous example to make my point.



Where would (- - - - - - -5) be located? There's 7 negatives ina row there! So it's negative negative negative negative negative negative negative 7.....where is it....left or right of zero?



negative ---- left

negative ---- right

negative ---- left

negative ---- right

negative ---- left

negative ---- right

negative ---- left



So...you end up to the left of zero (-7), as it should.



So, in essence odd numbers of consecutive negatives will result in one single NEGATIVE....and an even number results in a POSITIVE.





-2 * -5 = 10 (two consecutive negatives gives you a net result of a positive)



-4 * -7 * -1 * -8 * -3 = -672 (five consecutive negatives gives you a net result of a negative).





Negatives are just instructions of where to put the number, that's all. They're taught like they are some mysterious, mystical and complicated 'thing' that you have to just suck up and accept.



Totally NOT so....if you know what's going on with them, they're as easy to deal with as adding 1 + 1.



Hope that helps.

Answer 3

Take a number line. The opposite of any number is the same number, only on the opposite side of the number line.



For example the opposite of 2 is -2, and the opposite of -2 is 2.



Using that logic, if we have a -(-2), or a minus of a minus 2, we start at the negative side of the number line, and the opposite of the -2 would be a +2.



Funny answer, Barry G. You made me laugh! ;)



Actually, vfbundy, 10º = 1 DOES make sense. For example, take:

10³/10³



The law of division by exponents says if the bases are the same, to subtract the exponent of the numerator by the exponent of the denominator to yield:

10^(3-3) = 10º



Note also that any number over itself = 1 (like 10³/10³), so we can conclude that any number to the zero power = 1



I find that most answers in math have a logical answer including why 1 is not considered a prime number. It's many times NOT just BY DEFINITION. Although it's easier to answer students question that way rather than go through the necessary proofs.

Answer 4

Suppose you have any two positive numbers a and b and multiply

them to get a new number c. Then for the negative

of each of these, we have (-a) times (-b)



(-a)(-b) = c

Then c is the positive number (a)(b).



Proof:



Suppose that c were also negative. We can write (-a) = (-1)(a)

and (-b)=(-1)(b) and we also must have c = (-1)(a)(b) because

we are supposing that c is negative



(-1)(-1)(a)(b) = (-1)(a)(b)



both terms have a common factor of (-1)(a)(b). Cancel this

from both sides. We obtain



-1 = 1



which is a contradiction, proving the contention that

the product of two negatives is a positive.

Answer 5

Let a and b be positive > 0 Assume that -a x -b = -ab We know that a x -b = -ab .... (this is intuitively easy to understand. If you have no money and three outstanding loans of $100 each then you have 3 x -$100 = -$300.) If -a x -b = -ab as per assumption and we know that a x -b also = -ab then it follows that a = -a and therefore 1 = -1. Since this is a contradiction, we know the assumption must be false and we have proved that -a x -b = +ab. In your case: Assume: -5 x -5 = -25 We know 5 x -5 = -25 Therefore -5 x -5 = 5 x -5 ==> -5 = 5 ... a contradiction ==> -5 x -5 = +25

Answer 6

why does a negative times a negative equal a positive?

Answer 7

Negative Times Negative

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Answer 9

Some things are best left a mystery! For example, why does 10º = 1? (Why does everything to the power of zero equal 1? If you REALLY look at that mathematically, does that make any sense?) The reason I got from a teacher when asking this question was, "That's what the answer is BY DEFINITION." My guess is that the answer to your question is the same thing.

Answer 10

if he doesn't know that, aren't you worried that he might not be such a good teacher after all?