Commutative & Associative property of negative numbers

[sonadoramante]

TL;DR Summary

Commutative & Associative property of addition of negative numbers.

Commutative property of addition.

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.

Associative property of addition.

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?

 

[fresh_42]

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.

 

[sonadoramante]

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.

 

[Mark44]

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.

 

[jbriggs444]

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.