2.2 Integers, Rationals, and Real Numbers

Number systems are the foundation of mathematics. We'll explore integers, rationals, and real numbers, building on natural numbers. These systems have unique properties and structures that allow for increasingly complex mathematical operations.

We'll dive into algebraic structures, division in integers, and the density of rationals. We'll also examine the completeness of real numbers, a crucial concept in advanced math. Understanding these systems is key to grasping more complex mathematical ideas.

Extending the Number System

Constructing the Integers, Rationals, and Real Numbers

Top images from around the web for Constructing the Integers, Rationals, and Real Numbers

• 

  Real Numbers: What are Real Numbers ? ~ I Answer 4 U View original

  Is this image relevant?

• 

  Number Sets View original

  Is this image relevant?

• 

  Number Sets View original

  Is this image relevant?

• 

  Real Numbers: What are Real Numbers ? ~ I Answer 4 U View original

  Is this image relevant?

• 

  Number Sets View original

  Is this image relevant?

1 of 3

Top images from around the web for Constructing the Integers, Rationals, and Real Numbers

• 

  Real Numbers: What are Real Numbers ? ~ I Answer 4 U View original

  Is this image relevant?

• 

  Number Sets View original

  Is this image relevant?

• 

  Number Sets View original

  Is this image relevant?

• 

  Real Numbers: What are Real Numbers ? ~ I Answer 4 U View original

  Is this image relevant?

• 

  Number Sets View original

  Is this image relevant?

1 of 3

• Form the integers by extending the natural numbers to include their negatives and zero
  • Creates a set closed under addition and subtraction
• Form the rational numbers by taking ratios of integers
  • Creates a set closed under addition, subtraction, multiplication, and division (except by zero)
• Form the real numbers by including all limits of Cauchy sequences of rational numbers
  • Creates a complete ordered field
• Represent the real numbers as points on a number line, with the integers and rationals as subsets

Properties and Structures of the Extended Number Systems

• The integers, rationals, and reals all form commutative rings under addition and multiplication
  • The rationals and reals also form fields
• The integers have a well-defined division algorithm
  • Allows for the concepts of divisibility, greatest common divisors, and prime factorization
• The rationals are the smallest field containing the integers
  • Allows for the solution of linear equations
• The reals are a complete ordered field
  • Allows for the solution of polynomial equations and the definition of continuous functions

Properties of Number Systems

Algebraic Structures and Operations

• Commutative property: $a + b = b + a$ and $a \times b = b \times a$ for all elements $a$ and $b$
• Associative property: $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$ for all elements $a$, $b$, and $c$
• Distributive property: $a \times (b + c) = (a \times b) + (a \times c)$ for all elements $a$, $b$, and $c$
• Identity elements: 0 for addition and 1 for multiplication
  • $a + 0 = a$ and $a \times 1 = a$ for all elements $a$
• Inverse elements: additive inverses (negatives) and multiplicative inverses (reciprocals) for non-zero elements

Division and Factorization in the Integers

• Division algorithm: for any integers $a$ and $b$ with $b \neq 0$, there exist unique integers $[q](https://www.fiveableKeyTerm:q)$ (quotient) and $[r](https://www.fiveableKeyTerm:r)$ (remainder) such that $a = bq + r$ and $0 \leq r < |b|$
• Divisibility: an integer $a$ divides an integer $b$ if there exists an integer $k$ such that $b = ak$
  • Notation: $a | b$ (read as "$a$ divides $b$")
• Greatest common divisor (GCD): the largest positive integer that divides both $a$ and $b$
  • Notation: $gcd(a, b)$ or $(a, b)$
• Prime numbers: integers greater than 1 that have exactly two positive divisors, 1 and itself
  • Examples: 2, 3, 5, 7, 11, 13, 17, ...
• Fundamental Theorem of Arithmetic: every integer greater than 1 can be uniquely represented as a product of prime numbers (up to the order of factors)

Density of Rational Numbers

Existence of Rational Numbers Between Any Two Real Numbers

• For any two distinct real numbers $a$ and $b$, there exists a rational number $r$ such that $a < r < b$
• Constructive proof: given real numbers $a < b$, choose an integer $n > \frac{1}{b - a}$. Then there exists an integer $k$ such that $an < k < bn$. The rational number $r = \frac{k}{n}$ satisfies $a < r < b$

Implications of the Density Property

• The rational numbers are infinite and countable
  • They can be put into a one-to-one correspondence with the natural numbers (e.g., using a diagonal enumeration)
• The real numbers are infinite and uncountable
  • They cannot be put into a one-to-one correspondence with the natural numbers (proof by Cantor's diagonal argument)
• Rational numbers can be used to approximate real numbers with arbitrary precision
  • For any real number $x$ and any $\epsilon > 0$, there exists a rational number $r$ such that $|x - r| < \epsilon$

Completeness of Real Numbers

The Completeness Property and Its Equivalences

• Completeness property: every non-empty subset of the reals with an upper bound has a least upper bound (supremum) in the reals
• Equivalent formulations:
  • Convergence of Cauchy sequences: every Cauchy sequence of real numbers converges to a real number
  • Existence of limits: every convergent sequence of real numbers has a limit in the reals
  • Nested Intervals Property: if $I_1 \supset I_2 \supset I_3 \supset \ldots$ is a sequence of nested closed intervals with lengths tending to zero, then their intersection contains exactly one real number

Consequences and Applications of Completeness

• Continuous functions: a function $f: \mathbb{R} \to \mathbb{R}$ is continuous if and only if it preserves limits of convergent sequences
  • Intermediate Value Theorem: if $f$ is continuous on $[a, b]$ and $f(a) < y < f(b)$, then there exists a $c \in (a, b)$ such that $f(c) = y$
• Differentiation and integration: the completeness of the reals is necessary for the development of calculus and real analysis
  • Existence of derivatives: if $f$ is differentiable at $a$, then $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists and equals $f'(a)$
  • Riemann integrability: a bounded function $f$ on $[a, b]$ is Riemann integrable if and only if its set of discontinuities has measure zero