Can someone answer this doubt I have on Set theory?

[Rishabh Narula]

"The fact that the above eleven properties are
satisfied is often expressed by
saying that the real numbers form a
field with respect to the usual addition and
multiplication operations."

-what do these lines mean?
in particular the line "form a field with
respect to"?
is it something like real numbers make up
a particular field or category of subject
in which you can perform addition,subtraction,
division and multiplication
in different ways(assosicative,commutative,distributive etc.)
on the various elements?

 

[PeroK]

A Field has a mathemtical definition:

https://en.wikipedia.org/wiki/Field_(mathematics)

 

[Stephen Tashi]

is it something like real numbers make up
a particular field or category of subject
in which you can perform addition,subtraction,
division and multiplication
in different ways(assosicative,commutative,distributive etc.)
on the various elements?

That's the basic idea, but you should understand that the word "field" is being used in a technical sense, not in the generic sense of a "field of study". There are mathematical definitions and terms for a variety of mathematical structures which have associated operations. For example, there are monoids, semigroups, groups, rings, and fields. Although there are common language meanings for words like "group" and "field", the mathematical use of such terms involves technical definitions. As @PeroK points out, the mathematical use of the word "field" has a specific definition. I'm not sure how the use of the word "field" came about in the history of mathematics, but "field" in mathematics is more specific that the common language use of the word to indicate an area of study.

If you glance at the mathematical definitions for "group", "ring", "field" etc., you'll understand that the mathematical definitions for such terms differ from the common language use of them. There is even a highly technical mathematical definition for "category".