Economic

Number Systems as Algebraic Structures

Dr Damien S. Eldridge

Australian National University

25 January 2021

D. S. Eldridge (ANU) Number Systems as Algebraic Structures 25 January 2021 1 / 17

Reading Guide Part 1

Anton, H (1987), Elementary Linear Algebra (fifth edition), John Wiley and Sons, USA: Chapters 4 and 6.

Corbae, D, MB Stinchcombe, and J Zeman (2009), An introduction to mathematical analysis for economic theory and econometrics, Princeton University Press, USA: Chapter 2 (pp. 72-105).

Coroneos, J (Undated), A higher school certificate course in mathematics: Year eleven three unit course, Coroneos Publications, Australia: Chapter 1 and the Appendix (on mathematical induction).

Herstein, IN (1975), Topics in algebra (second edition), John Wiley and Sons, Singapore.

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Reading Guide Part 2

Simon, CP, and L Blume (1994), Mathematics for economists, WW Norton and Company, USA: Appendix A1 (pp. 847-858).

Spivak, M (2006), Calculus (corrected third edition), Cambridge University Press, The United Kingdom: Chapters 1, 2, 28, 29 and 30 (pp. 3-35 and 571-596).

Sundaram, RK (1996), A first course in optimization theory, Cambridge University Press, USA: Chapter 1, and Appendices A, B, and C.

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Some Common Number Sets

The set of natural numbers: N = {1, 2, 3, · · ·}; The set of non-negative intergers: Z+ = {0, 1, 2, · · ·}; The set of integers: Z = {· · · ,−2,−1, 0, 1, 2, · · ·}; The set of rational numbers: Q =

{ m n

: m ∈ Z, n ∈ N }

;

The set of non-negative rational numbers: Q+ = {x ∈ Q : x > 0}; The set of positive rational numbers: Q++ = {x ∈ Q : x > 0};

The set of real numbers: R = (−∞, ∞); The set of non-negative real numbers: R+ = {x ∈ R : x > 0}; The set of positive real numbers: R++ = {x ∈ R : x > 0}; and

The set of complex numbers:

C = { a + bi : a ∈ R, b ∈ R, i =

√ −1 }

.

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Some Types of Algebraic Structures

There are many types of algebraic structures that are studied in mathematics. Some of these include the following.

Groups. Abelian (or Commutative) Groups. Rings. Associative Rings. Associative Rings with a Unit Element. Commutative Rings. Commutative Rings with a Unit Element. Integral Domains. Division Rings. Fields. Vector Spaces.

In this subject, the main algebraic structures that you will encounter are fields and vector spaces.

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Number Systems as Algebraic Structures

Some examples of number systems as algebraic structures include the following.

(Z, +,×) is a commutative ring with unit element. (Q, Q++, +,×) is an ordered field. (R, R++, +,×) is a complete ordered field.

The relationship between some algebraic structures is as follows.

A field is a commutative division ring. A division ring is an associative ring whose non-zero elements form a group under multiplication. An associative ring is an Abelian group under addition that is both associative and closed under multiplication. An Abelian group is a group that is commutative.

We will discuss the “completeness” and “ordered” properties later.

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Groups

A set paired with a binary operation, (X , ~), is a group if the following five axioms are satisfied.

(G 1) (Non-Empty): X 6= ∅. (G 2) (Closed): If x ∈ X and y ∈ X , then x ~ y ∈ X . (G 3) (Associative): If x ∈ X , y ∈ X and z ∈ X , then x ~ (y ~ z) = (x ~ y) ~ z. (G 4) (Existence of an Identity Element): There exists some a ∈ X such that a ~ x = x ~ a = x for all x ∈ X . (G 5) (Existence of Inverse Elements): For every x ∈ X , there exists an element x̂ ∈ X such that x ~ x̂ = x̂ ~ x = a, where a ∈ X is the identity element for the group.

Note the following.

In a group under addition, ~ = +. In a group under multiplication, ~ = × and 0 /∈ X (because 0 has no multiplicative inverse). (Z, +), (Q, +) and (R, +) are each groups in which ~ = + (addition), a = 0 is the identity element and (−x) is the inverse element for x.

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Abelian Groups

A set paired with a binary operation, (X , ~), is an Abelian group if the following two axioms are satisfied.

(AG 1) (Group): (X , ~) is a group. In other words, (X , ~) satisfies axioms (G 1)–(G 5).

(AG 2) (Commutative): If x ∈ X and y ∈ X , then x ~ y = y ~ x. Note that (Z, +), (Q, +) and (R, +) are each Abelian groups in which ~ = + (addition), a = 0 is the identity element and (−x) is the inverse element for x.

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Associative Rings

A set paired with two binary operations, (X , +,×), is an associative ring if the following four axioms are satisfied.

(AR1) (Abelian Group Under Addition): (X , +) is an Abelian group. In other words, (X , +) satisfies axioms (AG 1) and (AG 2).

(AR2) (Closed Under Multiplication): If x ∈ X and y ∈ X , then x ×y ∈ X . (AR3) (Associative Under Multiplication): If x ∈ X , y ∈ X and z ∈ X , then x × (y ×z) = (x ×y)×z. (AR4) (Distributive): If x ∈ X , y ∈ X and z ∈ X , then x × (y + z) = (x ×y) + (x ×z) and (y + z)×x) = (y ×x) + (z ×x).

Note that (Z, +,×), (Q, +,×) and (R, +,×) are each associative rings.

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Division Rings

A set paired with two binary operations, (X , +,×), is a division ring if the following two axioms are satisfied.

(DR1) (Associative Ring Under Addition): (X , +) is an associative ring.

In other words, (X , +) satisfies axioms (AR1)–(AR4).

(DR2) (Group Under Multiplication): (X�{0} ,×) is a group. In other words,(X�{0} ,×) satisfies axioms (G 1)–(G 5).

Note the following.

The additive identity element will typically be denoted by 0, while the multiplicative identity element will typically be denoted by 1. If x ∈ X , then the additive inverse for x will typically be denoted by (−x). If x ∈ X�{0}, then the multiplicative inverse will typically be denoted by either x−1 or 1x . (Z, +,×) is not a division ring because it does not contain multiplicative inverses for every non-zero element. (Q, +,×) and (R, +,×) are each division rings.

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Fields

A set paired with two binary operations, (X , +,×), is a field if the following two axioms are satisfied.

(F 1) (Division Ring): (X , +,×) is a division ring. In other words, (X , +,×) satisfies axioms (DR1) and (DR2).

(F 2) (Commutative Under Multiplication): If x ∈ X and y ∈ X , then x ×y = y ×x.

Note that (Q, +,×) and (R, +,×) are each fields.

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Ordered Fields

A set X , along with a subset of that set P ⊂ X , paired with two binary operations, (X , P, +,×), is an ordered field if the following four axioms are satisfied.

(OF 1) (Field): (X , +,×) is a field. In other words, (X , +,×) satisfies axioms (F 1) and (F 2).

(OF 2) (Classification): If x ∈ X , then exactly one of the following statements is true: (i) x ∈ P; (ii) x = 0; or (iii) (−x) ∈ P. (OF 3) (P is Closed Under Addition): If x ∈ P and y ∈ P, then x + y ∈ P as well. (OF 4) (P is Closed Under Multiplication): If x ∈ P and y ∈ P, then x ×y ∈ P as well.

Note thae following.

The phrase “exactly one” in axiom (OF 2) means “both at least one and at most one” or, if you prefer, “one and only one”. (Q, Q++, +,×) and (R, R++, +,×) are each ordered fields.

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Inequalities

If (X , P, +,×) is an ordered field, then we can define four binary inequality relations for the elements of X as follows.

Strictly Greater Than (>): If x ∈ X , y ∈ X and (x + (−y)) ∈ P, then x > y.

Strictly Less Than (<): If x ∈ X , y ∈ X and (y + (−x)) ∈ P, then x < y.

Greater Than Or Equal To (>): If x ∈ X , y ∈ X and either (i) (x + (−y)) ∈ P or (ii) (x + (−y)) = 0, then x > y.

Less Than Or Equal To (6): If x ∈ X , y ∈ X and either (i) (y + (−x)) ∈ P or (ii) (y + (−x)) = 0, then x 6 y.

We can use these inequality relations to define two concepts of boundedness for sets contained within an ordered field.

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Upper Bounds for Sets in Ordered Fields

Suppose that (X , P, +,×) is an ordered field and A ⊆ X . The set A is said to be bounded above if there exists some element x ∈ X such that x > a for all a ∈ A.

The element x is called an upper bound for the set A in such cases. There may well be many elements of X that are upper bounds for the set A. Suppose that x̂ is an upper bound for A and that x̂ 6 x for any x that is an upper bound for A. In other words, suppose that x̂ is the smallest upper bound for A. The element x̂ ∈ X is called the supremum, or least upper bound, for the set A in such cases.

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Lower Bounds for Sets in Ordered Fields

Suppose that (X , P, +,×) is an ordered field and A ⊆ X . The set A is said to be bounded below if there exists some element x ∈ X such that x 6 a for all a ∈ A.

The element x is called a lower bound for the set A in such cases. There may well be many elements of X that are lower bounds for the set A. Suppose that x̂ is a lower bound for A and that x̂ > x for any x that is a lower bound for A. In other words, suppose that x̂ is the largest lower bound for A. The element x̂ ∈ X is called the infemum, or greatest lower bound, for the set A in such cases.

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Complete Ordered Fields

A set X , along with a subset of that set P ⊂ X , paired with two binary operations, (X , P, +,×), is a complete ordered field if the following two axioms are satisfied.

(COF 1) (Ordered Field): (X , P, +,×) is an ordered field. In other words, (X , P, +,×) satisfies axioms (OF 1)–(OF 4).

(COF 2) (Existence of Suprema and Infema): If A ⊆ X , A 6= ∅ and A is bounded above, then there exists a supremum for A. If B ⊆ X , B 6= ∅ and B is bounded below, then there exists an infemum for B.

Note the following.

(Q, Q++, +,×) is not a complete ordered field. (R, R++, +,×) is a complete ordered field. Indeed, in a sense, (R, R++, +,×) is the only complete ordered field. All other complete ordered fields are, in a precise way known as an “isomorphism”, essentially the same as (R, R++, +,×).

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The Problem with Rational Numbers

The reason that (Q, Q++, +,×) is not a complete ordered field relates to the existence of irrational numbers.

The term “irrational number” simply means that the number cannot be written as the ratio of an integer and a natural number. Examples of irrational numbers include

√ 2, π and e.

The existence of irrational numbers means that there are “holes” in the set of rational numbers. These holes mean that some non-empty bounded subsets of Q either do not have a rational supremum or do not have a rational infemum (or do not have either of these).

Example: A = { x ∈ Q : x2 < 2

} .

In a sense, the set of real numbers can be constructed from the set of rational numbers by “filling in these holes”. The set of real numbers can in some sense be viewed as the “completion” of the set of rational numbers.

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