Why do we care about the identity property of an operation?

[musicgold]

Homework Statement

This is not a homework problem. I am trying to understand why a mathematical operation need to have an identity property to be an useful operator.

Relevant Equations

1. I see that we often highlight that 0 is the identity property of addition / subtraction and 1 is the identity property of multiplication and division? Why do we care so much about the identity property?

2. Are there some essential properties an operation has to have to be a successful / widely used operation?

3. Does the modulo operation have an identity property?

I am reading a lot of stuff on advanced algebra and running into these questions.

Thank you

 

[PeroK]

Homework Statement: This is not a homework problem. I am trying to understand why a mathematical operation need to have an identity property to be an useful operator.
Homework Equations: 1. I see that we often highlight that 0 is the identity property of addition / subtraction and 1 is the identity property of multiplication and division? Why do we care so much about the identity property?

2. Are there some essential properties an operation has to have to be a successful / widely used operation?

3. Does the modulo operation have an identity property?

I am reading a lot of stuff on advanced algebra and running into these questions.

Thank you

You don't see why 0 and 1 are useful numbers?

 

[member 587159]

1. Lots of modern mathematics is about classifying structures. For example, when one is studying a mathematical object and one identifies this object as a vector space, then we get many of the properties of this mathematical object for free: any property that a vector space has, has this object too.

Identifying if a mathematical object has an identity is crucial to see that an object is a group or a ring, and once an object is identified as a ring, we have acces to the full power of ring theory to get more information about this object.

2. You might want to look into the definitions of group, ring, vector space and module.

3. Yes, the modulo operation has $0$ as additive identity and $1$ as multiplicative identity.