Why is the product of two negative numbers always a positive number?

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Can somebody explain as I have been unable to find a satisfactory argument?
« Last Edit: December 01, 2024, 08:54:00 AM by E E K »

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markjo

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Re: Why is the product of two negative numbers always a positive number?
« Reply #1 on: December 01, 2024, 03:20:23 PM »
Really?  I didn't have any trouble finding this:
Why do we get a positive number when we multiply two negative numbers?
When we multiply or divide two negative numbers, the result is a positive number. This might seem strange at first, but it's important to remember that a negative sign in math is really just an instruction to change the direction of a number on a number line. So when we multiply or divide two negative numbers, we're reversing the direction twice, which brings us back to a positive number.
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Re: Why is the product of two negative numbers always a positive number?
« Reply #2 on: December 01, 2024, 08:58:18 PM »
Really?  I didn't have any trouble finding this:
Why do we get a positive number when we multiply two negative numbers?
When we multiply or divide two negative numbers, the result is a positive number. This might seem strange at first, but it's important to remember that a negative sign in math is really just an instruction to change the direction of a number on a number line. So when we multiply or divide two negative numbers, we're reversing the direction twice, which brings us back to a positive number.
But I have a trouble in finding here is how

Negative sign on a number line tells us to go in negative direction while positive sign tells us to go in positive direction.

The sum of +3 and -8 is -5.

On a number line, count 3 in a positive direction from zero. At the same point, count 8 but in negative direction (opposite to positive) till we reach to the point of -5. OR

On the same number line, count 8 in a negative direction from zero. At the same point, count 3 but in positive direction (opposite to negative) till we reach at the point of -5

So the REVERSE (opposite) of NEGATIVE is POSITIVE and vice versa.

Multiplication reduced the laborious work of addition / subtraction

For example

The product of +3 and +5 means

The sum of +3 five times in the direction positive sign = +3 + 3 + 3 + 3 + 3 = +15 or the sum of +5 three times in positive direction = +5 + 5 + 5 = +15

No negative sign is seen which tells to go in opposite direction in the above example therefore the answer is a positive number

Similarly, the product of - 3 and -5 must also means

The sum of - 3 five times in the direction of negative sign = - 3 - 3 - 3 - 3 - 3 = -15 or the sum of -5 three times in negative direction = -5 - 5 - 5 = -15

No positive sign is seen which guide us to go in REVERSE/positive direction in the product of two negative numbers.

This means if + x + = +, then – x - must also = - , but, not
« Last Edit: December 02, 2024, 07:36:31 AM by E E K »

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Atam-Or

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Re: Why is the product of two negative numbers always a positive number?
« Reply #3 on: December 02, 2024, 01:58:15 AM »
What's the interest here? Numbers are fun. Multple negatives have a positive result, someone say's? *bats eyelashes* hmmm

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Re: Why is the product of two negative numbers always a positive number?
« Reply #4 on: December 17, 2024, 11:11:36 AM »
If we multiply a by -b, the result should be -(a⋅b)
Now, if we multiply two negative numbers, say (-a) and (-b), and distribute:
(-a)(-b)=-(a(-b))

a(-b)=−(a*b), so we can substitute :
-(a(-b))=-(-(ab))=ab

This shows that multiplying two negative numbers results in a positive number.
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Re: Why is the product of two negative numbers always a positive number?
« Reply #5 on: December 21, 2024, 12:43:32 AM »

But I have a trouble in finding here is how

Negative sign on a number line tells us to go in negative direction while positive sign tells us to go in positive direction.

The sum of +3 and -8 is -5.

On a number line, count 3 in a positive direction from zero. At the same point, count 8 but in negative direction (opposite to positive) till we reach to the point of -5. OR

On the same number line, count 8 in a negative direction from zero. At the same point, count 3 but in positive direction (opposite to negative) till we reach at the point of -5

So the REVERSE (opposite) of NEGATIVE is POSITIVE and vice versa.

Multiplication reduced the laborious work of addition / subtraction

For example

The product of +3 and +5 means

The sum of +3 five times in the direction positive sign = +3 + 3 + 3 + 3 + 3 = +15 or the sum of +5 three times in positive direction = +5 + 5 + 5 = +15

No negative sign is seen which tells to go in opposite direction in the above example therefore the answer is a positive number

Similarly, the product of - 3 and -5 must also means

The sum of - 3 five times in the direction of negative sign = - 3 - 3 - 3 - 3 - 3 = -15 or the sum of -5 three times in negative direction = -5 - 5 - 5 = -15

No positive sign is seen which guide us to go in REVERSE/positive direction in the product of two negative numbers.

This means if + x + = +, then – x - must also = - , but, not

Summing in a positive or negative direction isn’t a thing.  Only adding   numbers together, which can be positive or negative.

Summing -3 five times gets you -15.  That’s equivalent to -3 x 5, not -3 x -5

But multiplication isn’t really repeated addition, it’s a fundamentally different function.  It only works like repeated addition when multiplying by positive integers (natural numbers).
It doesn’t work like that when multiplying by fractions, variables, units of measurement, etc and it doesn’t work like that when multiplying by a negative number.

On a number line it’s better to think of multiplication as scaling or magnification.  Rather than adding more line, you are stretching (or shrinking) the line.

Multiplying by 0.5 is equivalent to dividing by 2.  As the number you are multiplying by approaches 0, so does the product.  As you pass 0 into negative numbers, so does the product.  That works both ways.

Make sense?  I’ve never tried to explain it before, so might not be that good.


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Aera23

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Re: Why is the product of two negative numbers always a positive number?
« Reply #6 on: January 02, 2025, 12:40:53 AM »
>>This means if + x + = +, then – x - must also = - , but, not

The example that led to that 'conclusion' is incomplete... running the calculation -2 x -2 will give the answer 4, rather than -4.
this is because multiplying a negative number by a negative number flips the sign each time... odd amount of flips will give negative numbers, even ones will be positive.

Lets say, -2 people owed -2 dollars, the amount of dollars that the bank would have got was 4, because the negative amount of people (suggesting they paid), cancels out the negative number of dollars.
(there are better explanations online though, tho the conclusion is the same)
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Re: Why is the product of two negative numbers always a positive number?
« Reply #7 on: January 06, 2025, 08:37:46 AM »
>>This means if + x + = +, then – x - must also = - , but, not

The example that led to that 'conclusion' is incomplete... running the calculation -2 x -2 will give the answer 4, rather than -4.
this is because multiplying a negative number by a negative number flips the sign each time... odd amount of flips will give negative numbers, even ones will be positive.

Lets say, -2 people owed -2 dollars, the amount of dollars that the bank would have got was 4, because the negative amount of people (suggesting they paid), cancels out the negative number of dollars.
(there are better explanations online though, tho the conclusion is the same)
The opposite of + is – and vice versa. We flip negative sign into positive but not positive into negative sign - why?  There are tons of explanation on internet but they are satisfactory that’s why I asked.

Re: Why is the product of two negative numbers always a positive number?
« Reply #8 on: January 06, 2025, 05:16:25 PM »
Graph it and all will be revealed.