Real Numbers and Completeness

Flashcards covering the defining properties of real numbers, the Archimedean axiom, and the axiom of completeness.

Real Numbers (R)

The set of numbers including rationals and irrationals, characterized as an ordered field with the least upper bound property.

Natural Numbers (N)

N := {1, 2, 3, 4, . . . , }

Integers (Z)

{. . . , −3, −2, −1, 0, 1, 2, 3, . . .}

Rational Numbers (Q)

{ n / m | n ∈ Z, m ∈ N}

n/m = k/l if and only if what?

nl = km

What is assumed about the relationship between Q and R?

Q ⊆ R, operations of addition and multiplication on Q extends to all of R

Commutativity Laws

a + b = b + a and ab = ba for all a, b ∈ R

Associativity Laws

(a + b) + c = a + (b + c) and (ab)c = a(bc) for all a, b, c ∈ R

Additive Inverse

For all a ∈ R there exists an element b ∈ R such that a + b = 0

Multiplicative Inverse

For all a ∈ R there exists an element b ∈ R such that ab = 1

Distributive Law

a(b + c) = ab + ac for all a, b, c ∈ R

What do axioms (F1)-(F5) say that R is?

A structure that mathematicians call a field.

What is R assumed to be?

An ordered field containing Q as a subfield.

Archimedean Axiom

For every x ∈ R there exists a natural number n ∈ N such that n > x.

Density of Q in R

For every two real numbers a and b with a < b, there exists a rational number q ∈ Q satisfying a < q < b.

Bounded Above

A set A ⊆ R is bounded above if there exists a number M ∈ R such that a ≤ M for all a ∈ A.

Upper Bound

A number M ∈ R

Bounded Below

A set A is bounded below if there exists a number L ∈ R satisfying L ≤ a for every a ∈ A.

Lower Bound

A number L ∈ R

Least Upper Bound / Supremum

A real number s is the least upper bound or supremum for a set A ⊆ R if it meets the two criteria: s is an upper bound for A; if M is any upper bound for A, then s ≤ M.

How do you write the least upper bound / Supremum?

Denoted by sup A.

Infimum for A

Greatest lower bound

Maximum of Set A

Amax is an element of A and amax ≥ a for all a ∈ A.

Minimum of Set A

Amin ∈ A and amin ≤ a for every a ∈ A.

Axiom of Completeness

Every nonempty set of real numbers that is bounded above has a least upper bound.

What is c + A

c + A := {c + a | a ∈ A}.

What does Lemma 1.8 state?

For A ⊆ R and c ∈ R one has sup(c + A) = c + sup A

What property is equivalent to the Axiom of Completeness?

Nested Interval Property

How do you prove Theorem 1.11 (Existence of square roots)?

Consider the set T = {t ∈ R | t^2 < 2} and set α = sup T, which exists by the Axiom of Completeness.