Enumerate the properties of real numbers?

Answer 1

The real numbers form a field. This is a set of numbers with two [binary] operations defined on it: addition (usually denoted by +) and multiplication (usually denoted by *) such that:

• the set is closed under both operations. That is, for any elements x and y in the set, x + y and x * y belongs to the set.
• the operations are commutative. That is, for all x and y in the set, x + y = y + x, and x * y = y * x.
• multiplication is distributive over addition. That is, for any three elements x, y and z in the set, x*(y + z) = x*y + x*z
• the set contains identity elements under both operations. That is, for addition, there is an element, usually denoted by 0, such that x + 0 = x = 0 + x for all x in the set. For multiplication, there is an element, usually denoted by 1, such that y *1 = y = 1 * y for all y in the set.
• for every x in the set there is an additive inversewhich belongs to the set, and for every non-zero element x there is a multiplicative inverse which belongs to the set. That is for every x, there is an element denoted by (-x) such that x + (-x) = 0, and for every non-zero element y in the set, there is an element y-1 such that y*y-1 = 1.