# Can commutativity of multiplication and addition under real numbers be assumed?

• sapnpf6
In summary, the operation * defined by x*y = x + y + xy for every x,y ∈ S, where S = {x ∈ R : x ≠ -1}, is commutative because addition and multiplication are commutative in the real numbers. This is sufficient for showing that * is an abelian group.
sapnpf6

## Homework Statement

so.. let the operation * be defined as x*y = x + y + xy for every x,y ∈ S,
where S = {x ∈ R : x ≠ -1}. Now i have proven associativity, existence of an identity and inverses, all without commutativity, but i must show that this is an abelian group, so now i have to show commutativity. I know that multiplication is not always commutative, but i am not using matrices, just reals/{-1}.

## Homework Equations

the operation * is defined by x*y = x + y + xy.

## The Attempt at a Solution

i simply say that first, addition is commutative on the reals, so x+y=y+x, by the commutative law of addition of real numbers, and second that xy=yx, by the commutative law of multiplication of real numbers. this would show that x*y=y*x. Is this allowed or do i have to show commutativity without assuming addition and multiplication are commutative on real numbers? and if i have to show commutative without the assumptions above, can someone point me in the right direction on the proof?

in case i did not show enough of an attempt at a solution, here is a more detailed attempt summary...

the operation * is commutative if for every x,y∈{x∈R:x≠ -1} such that x*y = y*x.
But, x*y = x + y + xy, and
y*x = y + x + yx,
= y + x + xy (by commutative law of multiplication of real numbers)
= x + y + xy (by commutative law of addition of real numbers)
So x*y = y*x, and thus, * is commutative.

this is my original solution but i am not entirely sure whether it holds up. it seems like using the law of commutativity of multiplication/addition of real numbers might be too "flimsy" to use in this kind of proof. it would sure be great if i could hear some feedback.

You are correct in your reasoning. If x*y = x+y + xy, then y*x = x+y +yx, which holds since addition and multiplication are commutative in the reals.

so it is not necessary to prove commutativity of addition or multiplication in real numbers here? just making sure...

No, you do not have to "reinvent the wheel" for every problem. You do not have to prove the properties, that have already been proven, for the "usual" operations on the real numbers. if you are given some new operation, defined using the "usual" operations, you can assume all of the properties of the "usual" operations in order to prove (or disprove) those properites for the new operation.

## 1. What does it mean for multiplication and addition to be commutative under real numbers?

When operations such as multiplication and addition are commutative, it means that the order in which the operations are performed does not affect the final result. In other words, switching the order of the numbers being multiplied or added will not change the outcome.

## 2. Can we assume that multiplication and addition are commutative under real numbers?

Yes, we can assume this because it is a fundamental property of real numbers. This property is known as the commutative property of addition and multiplication, and it holds true for all real numbers.

## 3. Are there any exceptions to the commutativity of multiplication and addition under real numbers?

No, there are no exceptions to this property. It holds true for all real numbers, regardless of the values being multiplied or added. This property is a fundamental part of the real number system and is essential in various mathematical operations.

## 4. How is the commutativity of multiplication and addition useful in mathematics?

The commutative property of multiplication and addition is useful in simplifying mathematical expressions and solving equations. It allows us to rearrange the terms in an expression without changing the final result, making it easier to work with and manipulate equations.

## 5. Is the commutativity of multiplication and addition unique to real numbers?

No, this property also holds true for other mathematical sets, such as complex numbers, rational numbers, and integers. However, it may not hold true for all mathematical operations, such as division or subtraction.

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