Commutative & Associative property of negative numbers

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In summary, the commutative and associative properties of addition hold true for integers, regardless of their signs. This is because subtraction is not a binary operation, but rather an abbreviation for addition with the additive inverse. The sign of a number does not affect these properties, and in some mathematical structures, such as the integers modulo 3, there is no concept of "positive" or "negative".
TL;DR Summary
Commutative & Associative property of addition of negative numbers.

If a & b are integers then,

a+b = b+a
2+3 = 3+2
5.

Does not work for subtraction.
2-3 = -1
3-2= 1

Having said that, what about the special case with negative numbers (when we also move their respective signs)
-5 + 7 = 2 & 7 + (-5) = 2.
15 -7 = 8 & -7 + 15 = 7.

If a, b & c are integers then,

a + (b+c) = (a+b) + c
2 + (3+4) = (2+3) + 4
2+7 = 5+4
9.

I tried 5 scenarios for the above,
a= - b = + c= -
a= + b= - c=+
a=+ b=+ c=-
a=- b=- c=+
a=- b=- c=-

And they all seem to work. It also seems to work for negative numbers in multiplication as well.

Is there a special case for commutativity & associativity for negative numbers?

The only reason it does not work is the wrong understanding of subtraction. On the level of group axioms which you used as language here, subtraction does not exist as binary operation. It is the unary operation of inversion: ##x \longmapsto x^{-1} := -x##. What you call subtraction is actually an addition: ##(x,y)\longmapsto x+ (-y)##. It is commonly written as ##x-y##, but this is only an abbreviation which causes confusion if used as in your question.

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fresh_42 said:
The only reason it does not work is the wrong understanding of subtraction. On the level of group axioms which you used as language here, subtraction does not exist as binary operation. It is the unary operation of inversion: ##x \longmapsto x^{-1} := -x##. What you call subtraction is actually an addition: ##(x,y)\longmapsto x+ (-y)##. It is commonly written as ##x-y##, but this is only an abbreviation which causes confusion if used as in your question.
Makes perfect sense! Thanks.

Last edited by a moderator:
And they all seem to work. It also seems to work for negative numbers in multiplication as well.
Besides the group axioms that fresh_42 mentioned, there are other mathematical structures such as rings, integral domains, and fields, all of which have two binary operations: addition and multiplication. Subtraction isn't included as one of the operations.

However, we can define ##a - b## as ##a + (-b)##, where ##-b## is the additive inverse of ##b##. Then ##a + (-b) = (-b) + a##, and ##a + (-b + c) = (a + (-b)) + c##, so we have commutivity and associativity
Is there a special case for commutativity & associativity for negative numbers?
The sign of the numbers doesn't enter into things.

Mark44 said:
The sign of the numbers doesn't enter into things.
Further, the fact that you happen to put a minus sign in front of something does not make it negative. -(-1) is positive.

In some domains there is not even a notion of "positive" or "negative". For instance, the integers modulo 3 where 2 = -1.

1. What is the commutative property of negative numbers?

The commutative property of negative numbers states that the order in which negative numbers are added or multiplied does not affect the result. In other words, when adding or multiplying two or more negative numbers, the order in which they are added or multiplied does not change the final result.

2. Can the commutative property be applied to all operations with negative numbers?

Yes, the commutative property can be applied to addition, subtraction, multiplication, and division with negative numbers. This means that the order of the numbers does not change the result, regardless of the operation being performed.

3. What is the associative property of negative numbers?

The associative property of negative numbers states that the grouping of numbers being added or multiplied does not affect the result. This means that when adding or multiplying three or more negative numbers, the result will be the same regardless of how the numbers are grouped.

4. Can the associative property be applied to all operations with negative numbers?

Yes, the associative property can be applied to addition, subtraction, multiplication, and division with negative numbers. This means that the grouping of numbers does not change the final result, regardless of the operation being performed.

5. How are the commutative and associative properties related to each other?

The commutative and associative properties are related in that they both involve the order of operations with negative numbers. While the commutative property deals with the order of numbers, the associative property deals with the grouping of numbers. Both properties allow for numbers to be rearranged without changing the final result.

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