Axioms for the Real Numbers1
STA 4321 – Fall, 2010
We assume as given the set R of real numbers, and the functions ‘+’ and ‘·‘, each of which maps
an ordered pair of real numbers (x, y) to another real number z. These functions satisfy the
following axioms. The first group of axioms describes the algebraic properties and the second
the order properties of the real number system.
A. The Field Axioms: For all real numbers x, y, and z, we have:
A1. Commutativity of addition
A2. Associativity of addition
A3. Additive identity
A4. Additive inverse
A5. Commutativity of multiplication
A6. Associativity of multiplication
A7. Multiplicative identity
A8. Multiplicative inverse
A9. Distributivity
x + y = y + x.
(x + y) + z = x + (y + z).
There exists a real number 0, such that x + 0 = x, for
every x.
For any x, there exists a real number w such that
x + w = 0. We usually write w as -x.
x · y = y · x.
(x · y) · z = x · (y · z).
There exists a real number 1, such that x · 1 = x, for
every x.
For any x, there exists a real number w such that
x · w = 1. We usually write w as x-1 or as 1/x.
x · (y + z) = (x · y) + (x · z)
Any set that satisfies these axioms is called a field under the operations of ‘+’ and ‘·’, which we
will call addition and multiplication.
The second class of properties possessed by the real numbers have to do with the fact that the
real numbers are ordered.
B. Axioms of Order: For any real numbers x and y, we have:
B1. (x < y & z < w) ⇒ x + z < y + w.
B2. ( 0 < x < y & 0 < z < w) ⇒ x · z < y · w.
B3. A number x cannot be both greater than and less than a number y.
B4. Of any two different numbers, one must be greater than the other.
Any system satisfying both the field axioms and the order axioms is called an ordered field. The
set R of real numbers, with the operations of addition and multiplication, constitutes an ordered
field.
1
Royden, H. L. (1988). Real Analysis. Macmillan Publishing Company, New York.