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

Can somebody explain as I have been unable to find a satisfactory argument?

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.

Really?  I didn't have any trouble finding this:

Quote from: https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-negative-numbers-multiply-and-divide/x6b17ba59:multiply-with-negatives/a/negative-numbers-multiplication-and-division-faq

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

What's the interest here? Numbers are fun. Multple negatives have a positive result, someone say's? *bats eyelashes* hmmm

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.

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.

>>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)

>>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.

Graph it and all will be revealed.

Quote from: Aera23 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)

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.

Wow, didn't see this post for months!

For anyone who reads... negative sign can be seen as changing the number to be the opposite of positive. Two opposites cancel each other out. (Or somethign similar, idk)

Who is the evil twin of your evil twin? It's you.

Who is the evil twin of your evil twin? It's you.

Who is the Good twin of your evil twin?  Also you.

Bet your mind is blown now. 

Quote from: Torve on September 30, 2025, 04:33:46 PM

Who is the evil twin of your evil twin? It's you.

Who is the Good twin of your evil twin?  Also you.

Bet your mind is blown now. 

Not blown, just disappointed.

If you know anything about Evil Twin Science (ETS) then you know that to your evil twin, you are evil, not good (assuming you are good in your universe).

If you know anything about Evil Twin Science (ETS) then you know that to your evil twin, you are evil, not good (assuming you are good in your universe).

Sounds like your ETS knowledge is pretty outdated.  Why am I not surprised?  It is now known that twins can exist in states of both good and evil simultaneously.  I could take you through the math, but somehow I suspect it would be wasting time for both of us.

Quote from: Torve on October 03, 2025, 09:31:00 AM

If you know anything about Evil Twin Science (ETS) then you know that to your evil twin, you are evil, not good (assuming you are good in your universe).

Sounds like your ETS knowledge is pretty outdated.  Why am I not surprised?  It is now known that twins can exist in states of both good and evil simultaneously.  I could take you through the math, but somehow I suspect it would be wasting time for both of us.

You are referring to a heretical quantum interpretation that is discredited by Occam's Razor. Twins are in fact in an eternal state of quantum entanglement. If one is good, the other can only be evil at that time.

Do identical twins share a soul?

Do identical twins share a soul?

A pair of shoes have each their own sole.

Quote from: E E K on January 06, 2025, 08:37:46 AM

Quote from: Aera23 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)

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.

Wow, didn't see this post for months!

For anyone who reads... negative sign can be seen as changing the number to be the opposite of positive. Two opposites cancel each other out. (Or somethign similar, idk)

We switch to + when multiplying - x - (two negative) but don't switch to - when multiplying + x + (two positive).

Quote from: JimmyTheLobster on October 04, 2025, 09:49:45 AM

Quote from: Torve on October 03, 2025, 09:31:00 AM

If you know anything about Evil Twin Science (ETS) then you know that to your evil twin, you are evil, not good (assuming you are good in your universe).

Sounds like your ETS knowledge is pretty outdated.  Why am I not surprised?  It is now known that twins can exist in states of both good and evil simultaneously.  I could take you through the math, but somehow I suspect it would be wasting time for both of us.

You are referring to a heretical quantum interpretation that is discredited by Occam's Razor. Twins are in fact in an eternal state of quantum entanglement. If one is good, the other can only be evil at that time.

My mother is a twin and I can confirm that she was good and her twin was evil but now both are evil.  Sorry, but their entangled state can be overwritten so they are duplicate pairs instead of complimenting.

Quote from: Aera23 on September 28, 2025, 10:34:06 PM

Quote from: E E K on January 06, 2025, 08:37:46 AM

Quote from: Aera23 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)

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.

Wow, didn't see this post for months!

For anyone who reads... negative sign can be seen as changing the number to be the opposite of positive. Two opposites cancel each other out. (Or somethign similar, idk)

We switch to + when multiplying - x - (two negative) but don't switch to - when multiplying + x + (two positive).

Ok, so what about that do you find difficult?

Ok, so what about that do you find difficult?

We know

+ x + = + , - x - = + , + x - = - , - x + = -

Plus sign on the number line indicates going away from zero on +x-axis or +y-axis while negative sign on the number line indicates going away from zero on -x-axis or -y-axis.

If + x + = +, then - x - = - (must also), but it’s not.

Since I’m unable to find the real logic behind - x - = + therefore I find it difficult why - x - = +? Science doesn’t believe in hocus-pocus. I believe - x - = + was made up w/o any logic behind it. I don’t understand why they assume that + x + = + (just because + numbers were discovered first and they were used to with + x + = +) when trying to prove - x - = +

If negative number were discovered first, then - x - = - would have been fine.

People told you the logic. Multiplication by -1 flips the sign, so doing it twice flips it back. Which part of that do you find difficult?

If you want more groups (postiv) of a poative you get more postives.
Ans if you went netgative gorups of a postive you get more negafive.
And of you went more geoups (postive) of a negative you get more negative.

You went negative groups of negative sp it must be postivie.

People told you the logic. Multiplication by -1 flips the sign, so doing it twice flips it back. Which part of that do you find difficult?

Multiplication by -1 flips the sign and doing it twice flips it back. The logic is perfectly fine on +x-axis or +y-axis or if we multiply positive number by -1 i.e. [(+!)(-1) = (-1)]. I find it difficult when applying the same logic on -x-axis or -y-axis or if we multiply negative number by +1. Here multiplication by +1 doesn’t flip the sign but remain the same i.e. [(-!)(+1) = (-1)]. Mustn’t it be [(-!)(+1) = (+1)] and doing it twice flips it back to -1 if your quoted logic is true.

Quote from: Pezevenk on May 24, 2026, 12:30:45 PM

People told you the logic. Multiplication by -1 flips the sign, so doing it twice flips it back. Which part of that do you find difficult?

Multiplication by -1 flips the sign and doing it twice flips it back. The logic is perfectly fine on +x-axis or +y-axis or if we multiply positive number by -1 i.e. [(+!)(-1) = (-1)]. I find it difficult when applying the same logic on -x-axis or -y-axis or if we multiply negative number by +1. Here multiplication by +1 doesn’t flip the sign but remain the same i.e. [(-!)(+1) = (-1)]. Mustn’t it be [(-!)(+1) = (+1)] and doing it twice flips it back to -1 if your quoted logic is true.

Minus is what flips the sign of the other number, not plus. Multiplying anything by +1 almost by definition doesn't change it, so I don't understand why you would assume (-1)(+1) = (+1), especially since you accept that (+1)(-1) = (-1), and the order of operations doesn't matter in multiplication...

Quote from: E E K on May 24, 2026, 10:12:13 PM

Quote from: Pezevenk on May 24, 2026, 12:30:45 PM

People told you the logic. Multiplication by -1 flips the sign, so doing it twice flips it back. Which part of that do you find difficult?

Multiplication by -1 flips the sign and doing it twice flips it back. The logic is perfectly fine on +x-axis or +y-axis or if we multiply positive number by -1 i.e. [(+!)(-1) = (-1)]. I find it difficult when applying the same logic on -x-axis or -y-axis or if we multiply negative number by +1. Here multiplication by +1 doesn’t flip the sign but remain the same i.e. [(-!)(+1) = (-1)]. Mustn’t it be [(-!)(+1) = (+1)] and doing it twice flips it back to -1 if your quoted logic is true.

Minus is what flips the sign of the other number, not plus. Multiplying anything by +1 almost by definition doesn't change it, so I don't understand why you would assume (-1)(+1) = (+1), especially since you accept that (+1)(-1) = (-1), and the order of operations doesn't matter in multiplication...

I’m not assuming that multiplication by +1 flips sign and doing it twice flips it back. I thought foregoing is also conceivable if the argument of “multiplication by -1 flips the sign and doing it twice flips it back” is possible but no idea why it was ruled out or ignored while making rules.

I believe, your replied would be like the following if the rule was “multiplication by +1 flips the sign and doing it twice flips it back”. [And it could possible as the order of operations doesn't matter in multiplication, but they choose the other way].
.
“Plus is what flips the sign of the other number, not minus. Multiplying anything by -1 almost by definition doesn't change it, I don’t understand so why you would assume (-1)(+1) = (-1), especially since you accept that (+1)(-1) = (+1), and the order of operations doesn't matter in multiplication.”

RIGHT?

But it doesn’t solve the problem either. Then one would ask if rule of “multiplication by +1 flip the sign and doing it twice flips it back” is possible then “multiplication by -1 flips the sign and doing it twice flips it back” is also reasonable.

I know no one will believe but it seems order of operation matters in multiplications.

I don't understand what you are saying. What do you think the result of negative multiplications is, if not the standard one?

multiplication is groups of a thing.

4groups of 2things is 8.

because you want negative more groups of a negative number of things.

it's the opposite of negative more groups of a postive number of things.


-4groups of your 2$ = you give me 8$
-4groups of -2$ = you owe me 2$ already, but in -4groups means I owe you 8$

I don't understand what you are saying. What do you think the result of negative multiplications is, if not the standard one?

If negative number were discovered first, then (- x - = -) would be the standard rule. As this was also possible but rule makers chose (+ x + = +) either ignorantly or deliberately. This was my original statement. As I was deviated therefore deleted my last post in which I messed up and made mistakes and hence unable to explain what I wanted to. Anyway

If (+ x + = +) indicates increase in absolute values of + numbers on right on a number line, then by analogy (- x - = -) must also shows increase in absolute values of numbers (decrease with - sign) on left on a number line.

Similarly, if + sign of the absolute value of numbers on right tells us to stay on right on a number line then - sign of the absolute value of numbers must also tell us to stay on left on a number line.

Following are the standard rules taught in school (multiplying by -1 changes the sign)
(+ x + = +) ; (- x - = +) ; (+ x - = -) ; (- x + = -)

Example:
                 (2)^{^2} = (+2)(+2) = (+4)
                 (-2)^{^2} = (-2)(-2) = (+4)
This means
                 (-2)^{^2} = (2)^{^2}
We get the following after taking square on both sides (or since powers are equal on both sides therefore their basis are also equal)
                 (-2) = (2)

Is above correct or I made a mistake? If correct then IMPOV, the order of operation for multiplications of (- x - = +) ; (+ x - = -) needs revising.

Explanation:
(+2)(+1) = +2
From L to R; +2 means go two steps right on a number line, one time. + sign of the multiplier tells us to stay on right direction. Therefore, we get +2.


(-2)(-1) = -2
From L to R; -2 means go two steps left on a number line, one time. - sign of the multiplier tells us to stay on left direction. Therefore, we get -2.

(+2)(-1) = -2
From L to R; +2 means go two steps right on a number line, one time but - sign of the multiplier tells us to flip direction. Therefore, go left on a number line instead of going right from zero. So, we get -2.

(-2)(+1) = +2
From L to R; -2 means go two steps left on a number line, one time but + sign of the multiplier tells us to flip direction. Therefore, go right on a number line instead of going left from zero. So, we get +2.

The square root is typically defined to give the positive of the two roots. So, the square root of x^2 is technically the absolute value of x, not x. So the square root of (-2)^2 is the absolute value of -2, which is 2. This resolves your complaint.

(-2)(+1) =+2 is inconsistent with the properties we expect multiplication to obey. First of all, by definition anything you multiply by 1 doesn't change. Furthermore if you tried to force it to be that way, here is what you would get:

(2-2)*1 = (2+(-2))*1 = 2*1+(-2)*1

You are saying (-2)*1=2. Therefore according to you the result of (2-2)*1 should be 4, which is obviously wrong.

Interestingly your version of multiplication is essentially n°m = |n|*m, where ° symbolizes your weird version of multiplication, and |n| is the absolute value of n. This is associative. However it doesn't even have a multiplicative identity on both sides (only one side) and it breaks the distributive property (doesn't work properly with addition) and the commutative property (order of operations matters, and we don't want that).

The 4 main axioms of multiplication are 1 as the multiplicative identity, associativity, commutativity and the distributive property. Your version breaks half of the first (1 is only the multiplicative identity when it is n, not when it is m) and the final two, and only maintains intact associativity. So even though you can make it internally consistent, it's basically a useless version of multiplication... But it is interesting that it's internally consistent at least.