Expanding Set Theory: Building Integers, Rational Numbers, and Coordinate Systems

In the previous post, we explored fundamental concepts of set theory, Peano's axioms, and equivalence relations. Now, let’s dive deeper and see how these ideas are applied to construct the integers and rational numbers. We’ll also connect these constructions to coordinate systems by treating coordinates as pairs of rational numbers.

From Natural Numbers to Integers

Peano's axioms provide a foundation for constructing the natural numbers N\mathbb{N}. To extend this framework to integers Z\mathbb{Z}, we use equivalence classes to define negative numbers and zero.

Constructing Integers

  1. Representation as Ordered Pairs:
    Each integer can be represented as a pair of natural numbers (a,b)(a, b), where:

    • aa represents the "positive part,"
    • bb represents the "negative part."

    For example, (3,1)(3, 1) represents 22, and (1,3)(1, 3) represents −2-2.

  2. Equivalence Relation:
    Define an equivalence relation ∼\sim on these pairs such that:

    (a,b)∼(c,d)  ⟺  a+d=b+c.(a, b) \sim (c, d) \iff a + d = b + c.

    This ensures that pairs like (3,1)(3, 1) and (5,3)(5, 3) are treated as the same integer.

  3. Definition of Integers:
    The set of integers Z\mathbb{Z} is the set of equivalence classes of N×N\mathbb{N} \times \mathbb{N} under ∼\sim:

    Z={[(a,b)]∣(a,b)∈N×N}.\mathbb{Z} = \{[(a, b)] \mid (a, b) \in \mathbb{N} \times \mathbb{N}\}.

  4. Operations on Integers:

    • Addition: [(a,b)]+[(c,d)]=[(a+c,b+d)][(a, b)] + [(c, d)] = [(a + c, b + d)].
    • Multiplication: [(a,b)]⋅[(c,d)]=[(ac+bd,ad+bc)][(a, b)] \cdot [(c, d)] = [(ac + bd, ad + bc)].

From Integers to Rational Numbers

The next step is to construct the rational numbers Q\mathbb{Q}, which extend the integers by including ratios of integers.

Constructing Rational Numbers

  1. Representation as Ordered Pairs:
    A rational number is represented as a pair (a,b)(a, b), where a∈Za \in \mathbb{Z} and b∈Z∖{0}b \in \mathbb{Z} \setminus \{0\}. The pair (a,b)(a, b) represents the fraction ab\frac{a}{b}.

  2. Equivalence Relation:
    Define an equivalence relation ∼\sim such that:

    (a,b)∼(c,d)  ⟺  ad=bc.(a, b) \sim (c, d) \iff ad = bc.

    This ensures that pairs like (2,4)(2, 4) and (1,2)(1, 2) are treated as the same rational number.

  3. Definition of Rational Numbers:
    The set of rational numbers Q\mathbb{Q} is the set of equivalence classes of Z×(Z∖{0})\mathbb{Z} \times (\mathbb{Z} \setminus \{0\}) under ∼\sim:

    Q={[(a,b)]∣a∈Z,b∈Z∖{0}}.\mathbb{Q} = \{[(a, b)] \mid a \in \mathbb{Z}, b \in \mathbb{Z} \setminus \{0\}\}.

  4. Operations on Rational Numbers:

    • Addition: [(a,b)]+[(c,d)]=[(ad+bc,bd)][(a, b)] + [(c, d)] = [(ad + bc, bd)].
    • Multiplication: [(a,b)]⋅[(c,d)]=[(ac,bd)][(a, b)] \cdot [(c, d)] = [(ac, bd)].

Rational Numbers as Coordinates

Rational numbers play a critical role in defining coordinate systems, which are foundational in geometry and algebra. Here’s how:

Cartesian Coordinates

  1. Definition of Coordinates:
    A point in the Cartesian plane is represented as a pair (x,y)(x, y), where x,y∈Qx, y \in \mathbb{Q}. This means every coordinate is a rational number.

  2. Applications in Geometry:

    • Distance Formula: The distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

      d=(x2−x1)2+(y2−y1)2.d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.

      When x1,x2,y1,x_1, x_2, y_1, and y2y_2 are rational, the squared distance is also rational.

    • Midpoint Formula: The midpoint of two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

      Midpoint=(x1+x22,y1+y22).\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).

      This guarantees that the midpoint’s coordinates are rational if the inputs are rational.

  3. Graphing Lines:
    The equation of a line, y=mx+by = mx + b, where mm and bb are rational, ensures that all points on the line with rational xx-values have rational yy-values.

Building Mathematical Structures from Sets

By starting with the foundational natural numbers defined through Peano's axioms, we have built up to integers, rational numbers, and their applications in coordinate geometry. This process demonstrates how set theory provides the framework for constructing and understanding mathematical systems.

In the next post, we’ll explore how these ideas extend to constructing real numbers R\mathbb{R}, and how irrational numbers fit into the coordinate plane. Stay tuned!

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