What is the real number system as a field?

Answer 1

The set of real numbers, R, is a mathematical field. In order for it to be a field, it must satisfy the following.
For any three real numbers x, y and z and the operations of addition and multiplication, · x + y belongs to R (closure under addition) · (x + y) + z = x + (y + z) (associative property of addition) · There is an element, 0, in R, such that x + 0 = 0 + x = x (existence of additive identity) · There is an element, -x, in R, such that x + (-x) = (-x) + x = 0 (existence of additive inverse) · x + y = y + x (Abelian or commutative property of addition) · x * y belongs to R (closure under multiplication) · (x * y) * z = x * (y * z) (associative property of multiplication) · There is an element, 1, in R, such that x * 1 = 1 * x = x (existence of multiplicative identity) · For every non-zero x, there is an element, 1/x, in R, such that x * 1/x = 1/x * x = 1 (existence of multiplicative inverse) · x * (y + z) = x*y + x * z (distributive property of multiplication over addition)