What does "properties of number systems are abstracted out" actually mean?

[Jignesh77]

"Roughly speaking, abstract algebra is the study of what happens when certain properties of number systems are abstracted out; "

Abstract Algebra | Brilliant Math & Science Wiki

Abstract algebra is a broad field of mathematics, concerned with algebraic structures such as groups, rings, vector spaces, and algebras. Roughly speaking, abstract algebra is the study of what happens when certain properties of number systems are abstracted out; for instance, altering the...

brilliant.org

What does "when certain properties of number systems are abstracted out " mean?
Thanks.
I have never learned "abstract algebra" at school. I did Masters in Organic Chemistry (23 years ago). I work in healthcare but love learning mathematics.
Thanks

 

[Dr.Peterson]

"Roughly speaking, abstract algebra is the study of what happens when certain properties of number systems are abstracted out; "

Abstract Algebra | Brilliant Math & Science Wiki

Abstract algebra is a broad field of mathematics, concerned with algebraic structures such as groups, rings, vector spaces, and algebras. Roughly speaking, abstract algebra is the study of what happens when certain properties of number systems are abstracted out; for instance, altering the...

brilliant.org

What does "when certain properties of number systems are abstracted out " mean?
Thanks.
I have never learned "abstract algebra" at school. I did Masters in Organic Chemistry (23 years ago). I work in healthcare but love learning mathematics.
Thanks

I would just read on to see what examples they give that illustrate this summary statement. Often in reading mathematics, you have to read through something several times in order to understand all that is being said.

But a property is something like the commutative property, that numbers can be added in either order with the same result. Abstracting this means taking the concept away from the concrete setting of numbers and addition. You consider some unknown objects with an unknown operation, and ask what you could know about them if some of these properties are known to be true, but nothing else.

 

[Steven G]

Suppose you have a set with just two numbers, 7 and 12.

You are told that 7*7=0, 7*12=0, 12*7=0 and 12*12=12.
* is some new definition (it need not be regular addition and it not be regular multiplication).

The question is those this set, with *, obey the usual properties of the real numbers?
(that is, does * have the commutative property, associative property, identity number and does each number in the set have an inverse?)

 

[Steven G]

Suppose you have a set with just two numbers, 7 and 12.

You are told that 7*7=0, 7*12=0, 12*7=0 and 12*12=12.
* is some new definition (it need not be regular addition and it not be regular multiplication).

The question is those this set, with *, obey the usual properties of the real numbers?
(that is, does * have the commutative property, associative property, identity number and does each number in the set have an inverse?)

I changed 0 to 12 (in my mind), but not everywhere.
I meant to say that 7*7=7, 7*12=7, 12*7=7 and 12*12=12.

 

[Jignesh77]

What does "abstract" mean in the phrase "abstract algebra"?
I thought even algebra is abstract because numbers, functions, etc, are just ideas. They aren't concrete.
Thanks.

 

[Dr.Peterson]

What does "abstract" mean in the phrase "abstract algebra"?
I thought even algebra is abstract because numbers, functions, etc, are just ideas. They aren't concrete.
Thanks.

True. Math in general is abstract (we "abstract out" the sheep, and just talk about the number 5 rather than "5 sheep"). Abstract algebra just takes it one step further, abstracting out the numbers themselves.

The idea is that many different things have similar behavior, with objects (instead of numbers) and operations on them; for example, you can "add" (or "multiply") motions by doing one after another (slides, or rotations) and observe that this operation is associative. Then you can ask yourself, what if we didn't know whether the objects we are combining are numbers or motions or something else entirely? What could we prove, knowing only the basic properties of an unknown operation? Then what we learn can be applied to many situations, some of them not even imagined yet.

 

[Jignesh77]

Thank you for your help.