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 Math Help > First Principles > Axioms > Axioms of Real Numbers

In the list, below, the * asterisk means multiplication, and ^= means not equal.

The real numbers satisfy, and are defined by, the following 15 axioms

a + (b + c) = (a + b) + c (associativity of +)

a + b = b + a (commutativity of +)

a + 0 = a (identity of +)

a + (-a) = 0 (inverse of +)

a * (b * c) = (a * b) * c (associativity of *)

a * b = b * a (commutativity of *)

a * 1 = a (identity of *)

a ^= 0 ==> a * (1/a) = 1 (inverse of *)

a * (b + c) = a * b + a * c (* distributes over +)

a ≤ b or b ≤ a (≤ is total)

(a ≤ b and b ≤ a) ==> a = b (≤ is antisymmetric)

(a ≤ b and b ≤ c) ==> a ≤ c (≤ is transitive)

a ≤ b ==> a + c ≤ b + c (substitution principle of addition)

(a ≤ b and 0 ≤ c) ==> a * c ≤ b * c (substitution principle of multiplication)

And finally the completeness axiom:

Any non-empty set of real numbers that is bounded above has a least upper bound.

Some authors also point out "closure" as a separate axiom -- addition, multiplication, and the least upper bound all result in real numbers.  One author (Keisler) also points out the "root axiom", that for every real number a>0 there is a real number b>0 such that bn=a.

### Internet references

Elementary Calculus: An Approach Using Infinitesimals, by H. Jerome Keisler.

Test Preparation Review: CBEST Test, HOBET Test

### Related pages in this website

Arithmetic for sixth through ninth graders -- a less rigorous (that is, more practical) lesson on the rules of arithmetic.

Proving the basic facts about arithmetic for the natural numbers -- the Peano Postulates

Axioms of Arithmetic -- the reflexive, symmetric, and transitive properties of equality; the identity, commutative, and associative properties of addition; and of multiplication; the distributive property.

Upper Bound -- definition of "upper bound" and "least upper bound" of either sets or functions

Definition of Interval -- a subset satisfying certain properties of a totally connected set such as the set of real numbers.  "Totally connected" means there exists a "≤" operator that is reflexive (a property in all sets), and antisymmetric, reflexive, and total (three of the thirteen properties listed above).

The webmaster and author of this Math Help site is Graeme McRae.