Why are the products of multiplied negatives, positives?

Answer 1

You're looking for an answer, which come from our daily life, I think. Sorry, I don't know such a one.

But in mathematics it is like that: The numbers we are talking about, the "real numbers" forms a so called field. The field has one (1) neutral element regarding the multiplication. This is the 1. Also there is to any number a inverse number, so that multiplied they become the 1. We know that the inverse to x is 1/x, e.g. to 4 it is 1/4, to -2 it is -1/2. -2 * (-1/2) = 1.

Of course we could define -1 as the neutral element. But it would result just into the equally strange question, why 2 * 1/2 = -1.

So out of x * 1/x = 1, we can (by multiplying on both sides) we can follow, that any multiplication of two negative numbers results in a positive number.
Ex.: we like to proof: -3 * (-11) = 33; do :33 => -3 * (-1/3) = 1.

I know its not a vivid explanation, but mathematics can be sometimes dry, sorry.

Answer 2

It is because of four properties of the real numbers:

First, if A and B are positive numbers, then their product, AB is also a positive number. This is an axiom of the real numbers.

Second, given any real number A, you either have -A <= 0 <= A or A <= 0 <= -A. This means that if A is negative, then -A is positive, and vice versa. This is an easy property to demonstrate from the real number axioms.

Third, given any real numbers A and B, we have (-A)B = A(-B) = -(AB). To take this further, (-A)(-B) = A(-(-B)) = AB. This can also be shown easily using the real number axioms.

Taking these three properties into account, if A and B are both negative, then -A and -B are positive. Then AB = (-A)(-B) > 0.

For the curious, this Wikipedia article provides a complete list of axioms of the Real Numbers in the "Synthetic approach" section that you may find useful:

http://en.wikipedia.org/wiki/Construction_of_real_numbers

Answer 3

Well think on line of numbers say
... -3 -2 -1 0 +1 +2 +3 ...
let find the result of (-2 * -2)
we know the result well be +4
To show how we get this actually you know the definition
of multiplying which is adding
at -2 add on it 4 so
-2 +1 +1 +1 +1 +1 +1 [(why 6) because number+adding two numbers 2+4=6]

Answer 4

Think of it like this. Positive numbers are a measure of presence. Negative numbers are a measure of absence.
When we multiply we use the word of so 3 of 5 can be thought of as '3 of those groups of 5'. Forgetting the size though and just looking at the signs.
The presence of presence will give you presence right? Okay.
The presence of absence is absence and so is the absence of presence.
But now back to your question, the absence of absence is presence. If I were to say there is no nothing then there is something see?
This is how I visualised it back when I was trying to understand it, hope this helps.

Answer 5

A negative sign means that you reverse your direction of counting on the number line. The first negative sign sends you off to the left, counting, while the second negative sign reverses that direction (back positive). It works for both addition and multiplication, but in multiplication you are counting multiple times (not just one time for addition), so you always end up back on the positive side of the number line.

.

Answer 6

lol that just made me think and it doesnt sound like ur struggling with it so i'll just give my opinion on it

i see negative numbers as very small numbers, so when u multiply two small numbers it NEEDS to make it bigger
and that when it crosses the line between negative and positive.
lol

Answer 7

i think its because a negative is the opposite of a number (ex: opposite of 3 is -3) so when you have 2 opposites they cancel out (ex:the opposite of 3 is negative 3 and the opposite of negative 3 is 3) so theyy becaome positive again

Answer 8

Consider -1 x -3
Divide both sides by -1
That equals 1 x -3 / -1
But -3 / -1 = 3
So 1 x -3 / -1 = 3

Answer 9

It's like you have a hole in each of your two front pockets. If you put the two holes together, that makes one positive pocket. Pretty good and I dropped out of the third grade.

Answer 10

If you mean say -1x-3, then ,yes, the answer is positive, 3. Think of it as the two - joining up to form a +