# Sign rule for multiplication

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Hello everyone,

Why  -2 .(-3) = 6

Why is it positive?

Why we accept that (minus sign) times (minus sign) is positive ? What is its origin?

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Posted (edited)
2 hours ago, neonwarrior said:

Why  -2 .(-3) = 6

Let's inverse equation and use division:

6 / (-3) = -2

or

6 / (-2) = -3

2 hours ago, neonwarrior said:

Why is it positive?

Division is the inverse of multiplication. And vice versa. (With a little exception)

In the simplest case:

1 / (-1) = -( 1 / 1 ) = -1

Left and right side of the equation must be equal.

Edited by Sensei
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Because , if you take away a debt, from someone, they end up in credit.

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4 hours ago, neonwarrior said:

Hello everyone,

Why  -2 .(-3) = 6

Why is it positive?

Why we accept that (minus sign) times (minus sign) is positive ? What is its origin?

Because if you are not not a rabbit, then you are a rabbit. It's just logic.

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5 hours ago, neonwarrior said:

Hello everyone,

Why  -2 .(-3) = 6

Why is it positive?

Why we accept that (minus sign) times (minus sign) is positive ? What is its origin?

I note you have posted this in Linear Algebra and Group Theory, not simply general Mathematics.
So I assume you know what these terms are and that Groups are connected with symmetry.

Well signed multiplication is an operation with a symmetry.

The full list of possibilities is

positive times positive makes positive

positive times negative makes negative

negative times positive makes negative

As you see this list is symmetrical: there are two ways for a positve result and two ways for a negative one.

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Because the property (a + b)*c = a*c + b*c requires it. Or, to slightly modify what John Cuthber said: if you cancel someone else's debt, you are giving them money.

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A rigorous mathematical proof would require using the precise words used in defining the product of two numbers.  What definition do you have?

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Posted (edited)
On 5/10/2021 at 6:04 AM, neonwarrior said:

Hello everyone,

Why  -2 .(-3) = 6

Why is it positive?

Why we accept that (minus sign) times (minus sign) is positive ? What is its origin?

I might have not remember well, but preferably you may check some very basic notations from group teory & algebra such as : product (relevant) theorems and prime product separation/differentiation region (relevant theorems).

Edited by ahmet
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Posted (edited)

$(-1)(-1) - 1 = (-1)(-1) + (-1) = (-1)(-1) + (-1)(1)$

$= (-1)(-1 + 1) = (-1)(0) = 0$

So that

$(-1)(-1) - 1 = 0$ and therefore $(-1)(-1) = 1$.

Then $(-3)(-2) = (3)(-1)(2)(-1) = (3)(2)(-1)(-1) = (3)(2)(1) = 6$.

All this follows from the associative, distributive, and commutative properties of a ring, which are satisfied by the integers.

Edited by wtf
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• 4 months later...
On 5/9/2021 at 11:04 PM, neonwarrior said:

Hello everyone,

Why  -2 .(-3) = 6

Why is it positive?

Why we accept that (minus sign) times (minus sign) is positive ? What is its origin?

It's good that you asked in Linear Algebra And Group Theory, because we're going to need some algebra you likely have never seen (unless you went to college) to answer it.

Multiplication isn't one thing. What multiplication is depends on what you're multiplying. Algebra is how we define this.

In Algebra, there are a handful of different kinds of structures. Here, we're interested in Groups, Rings, and Fields. Rings and Fields are kind of made of Groups, so we'll start there.

Say we have a set (the lay concept of set will work fine for our purposes), and we'll call it S.

On this set, we need to define a rule called a "binary relation" that takes any two things in the set and gives some output. We want this set to be closed under this relation, so the relation can only give us things that are already in the set. For this combination of set and relation (for now, we'll use ? to denote the relation) to be a Group, they need to have the following properties:

1) Associativity: for any three things a, b, and c in the set, the relation doesn't care about where the parentheses go. a?(b?c)=(a?b)?c

2) Identity: there is a special thing in the set (traditionally denoted by e when talking abstractly) where, for any other thing a in the set, a?e=e?a=a

3) Invertability: for any thing a in the set, there is another thing in the set a* where a?a*=e=a*

That's enough to be a Group. But we want a special kind of Group, called an Abelian Group. That's just a regular Group that has an extra property:

4) Commutativity: for any two things a and b in the set, a?b=b?a

Tradition dictates that for Abelian Groups, + is used in place of ? and 0 is used in place of e and -a in place of a*. If a Group is not Abelian, we often use × (or nothing at all) in place of ? and 1 in place of e and 1/a in place of a*.

If S is the set, we write (S, +) or (S, ×) for the group, but we often just write S if it is clear from the context that we're talking about a group.

This is enough to let us build a Ring.

With Rings, we still have a set S, but we have two relations. (S, +) is an Abelian Group, but × is a bit more lax. × only has to satisfy two properties:

1) Identity, and

2) Distributivity: for any three things a, b, and c in S, a×(b+c)=(a×b)+(a×c)

Like how (S, ?) is a Group, (S, +, ×) is a Ring.

If a Ring is commutative and has inverses for each relation, then the Ring is called a Field.

There are four particularly important facts mentioned above that are important to why a negative times a negative is a positive:

1) a+(-a)=0,

2) a+0=a,

3) a×0=0 (not mentioned above, but still important), and

4) a×(b+c)=(a×b)+(a×c)

Proof -a×-b = a×b:

Let a and b be positive numbers in our field (S, +, ×).

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

-a×-b = (-a×-b) + a(b+(-b)) = -a×-b + a×b + a×-b = -a×-b + a×-b + a×b = -b×a + -b×-a +a×b = -b(a+(-a)) + a×b = a×b

TL;DR:

It's because Fields are commutative, have identities, and are distributive

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4 minutes ago, ydoaPs said:

It's good that you asked in Linear Algebra And Group Theory, because we're going to need some algebra you likely have never seen (unless you went to college) to answer it.

Multiplication isn't one thing. What multiplication is depends on what you're multiplying. Algebra is how we define this.

In Algebra, there are a handful of different kinds of structures. Here, we're interested in Groups, Rings, and Fields. Rings and Fields are kind of made of Groups, so we'll start there.

Say we have a set (the lay concept of set will work fine for our purposes), and we'll call it S.

On this set, we need to define a rule called a "binary relation[/]" that takes any two things in the set and gives some output. We want this set to be closed[/] under this relation, so the relation can only give us things that are already in the set. For this combination of set and relation (for now, we'll use ? to denote the relation) to be a Group, they need to have the following properties:

1) Associativity: for any three things a, b, and c in the set, the relation doesn't care about where the parentheses go. a?(b?c)=(a?b)?c

2) Identity: there is a special thing in the set (traditionally denoted by e when talking abstractly) where, for any other thing a in the set, a?e=e?a=a

3) Invertability: for any thing a in the set, there is another thing in the set a* where a?a*=e=a*

That's enough to be a Group. But we want a special kind of Group, called an Abelian Group. That's just a regular Group that has an extra property:

4) Commutativity: for any two things a and b in the set, a?b=b?a

Tradition dictates that for Abelian Groups, + is used in place of ? and 0 is used in place of e. If a Group is not Abelian, we often use × (or nothing at all) in place of ? and 1 in place of e.

If S is the set, we write (S, +) or (S, ×) for the group, but we often just write S if it is clear from the context that we're talking about a group.

This is enough to let us build a Ring.

With Rings, we still have a set S, but we have two relations. (S, +) is an Abelian Group, but × is a bit more lax. × only has to satisfy two properties:

1) Identity, and

2) Distributivity: for any three things a, b, and c in S, a×(b+c)=(a×b)+(a×c)

Like how (S, ?) is a Group, (S, +, ×) is a Ring.

If a Ring is commutative and has inverses for each relation, then the Ring is called a Field.

There are four particularly important facts mentioned above that are important to why a negative times a negative is a positive:

1) a+(-a)=0,

2) a+0=a,

3) a×0=0 (not mentioned above, but still important), and

4) a×(b+c)=(a×b)+(a×c)

Proof -a×-b = a×b:

Let a and b be positive numbers in our field (S, +, ×).

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

-a×-b = (-a×-b) + a(b+(-b)) = -a×-b + a×b + a×-b = -a×-b + a×-b + a×b = -b×a + -b×-a +a×b = -b(a+(-a)) + a×b = a×b

TL;DR:

It's because Fields are commutative, have identities, and are distributive

That's a somewhat spectacular return ydoaPs  +1

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3 minutes ago, studiot said:

That's a somewhat spectacular return ydoaPs  +1

Well thank you. It took some doing to fix my bbtex mistakes on the phone with autorendering

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• 2 weeks later...

-2(3+ (-3))= -2(0)= 0

So, by the distributive rule

-2(3)+ (-2)(-3)= 0

(-2)(-3)= 2(3).

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