5.3 Ordering and Density of Real Numbers
3 min read•august 12, 2024
form a complete, ordered system that includes both rational and . This system allows for precise representation and comparison of quantities, extending beyond the limitations of alone.
The ordering and density properties of real numbers provide a foundation for mathematical analysis. These properties enable the development of calculus and other advanced mathematical concepts, making real numbers essential in various fields of study.
Ordering and the Number Line
Representing Real Numbers Visually
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- serves as a visual representation of real numbers
- Extends infinitely in both positive and negative directions
- Each point on the line corresponds to a unique real number
- Zero positioned at the center, dividing positive and negative numbers
- Integers placed at equal intervals along the line
- Rational numbers occupy positions between integers
- Irrational numbers fill gaps between rational numbers
Comparing and Ordering Real Numbers
- establishes the concept of "less than" and "greater than"
- For any two real numbers a and b, exactly one of the following holds true:
- a b (a is less than b)
- a = b (a is equal to b)
- a b (a is greater than b)
- applies to ordering (if a < b and b < c, then a < c)
- Real numbers can be arranged in ascending or descending order
- Inequalities used to express relationships between numbers (, )
Fundamental Properties of Real Numbers
- ensures no gaps exist in the real number line
- States that every nonempty set of real numbers with an upper bound has a least upper bound
- Guarantees the existence of limits and continuity in calculus
- demonstrates the relationship between natural and real numbers
- For any positive real number x, there exists a natural number n such that n > x
- Implies that the set of natural numbers is unbounded
- Allows for approximation of any real number using rational numbers
Density of Real Numbers
Density of Rational Numbers
- Between any two distinct rational numbers, there exists another rational number
- used to find a rational number between two given rationals
- For rationals a and b, the number lies between them
- Infinite number of rational numbers exists between any two distinct rationals
- of rationals either terminate or repeat (0.25, 0.333...)
Density of Irrational Numbers
- Between any two distinct real numbers, there exists an irrational number
- Irrational numbers have non-repeating, non-terminating decimal expansions (, )
- Constructive proof involves finding an irrational between two given reals
- Algebraic irrationals (, ) and transcendental irrationals (, e) exist
- Irrational numbers form a dense subset of the real numbers
Interplay Between Rational and Irrational Numbers
- Real number line consists of both rational and irrational numbers
- Rational and irrational numbers interspersed throughout the number line
- Any interval on the real number line contains infinitely many rationals and irrationals
- Rationals form a countable set, while irrationals form an uncountable set
- Real numbers can be approximated to any degree of accuracy using rational numbers
Bounds of Sets
Upper and Lower Bounds
- Upper bound of a set S represents a number greater than or equal to all elements in S
- Lower bound of a set S represents a number less than or equal to all elements in S
- A set may have multiple upper and lower bounds
- Bounded set has both upper and lower bounds (set of numbers between 0 and 1)
- Unbounded set lacks either upper or lower bound (set of all positive integers)
Supremum and Infimum
- Least upper bound () represents the smallest upper bound of a set
- Greatest lower bound () represents the largest lower bound of a set
- Supremum and infimum may or may not belong to the set itself
- For the set {1, 2, 3, 4, 5}, supremum is 5 and infimum is 1
- For the (0, 1), supremum is 1 and infimum is 0, neither belonging to the set
- Completeness axiom guarantees the existence of supremum and infimum for bounded sets of real numbers
Key Terms to Review (21)
<: The symbol '<' represents the mathematical concept of 'less than', indicating that one quantity is smaller than another. This relation is foundational in mathematics, especially in ordering numbers and comparing values, allowing us to understand the relative sizes of real numbers and their placement on the number line.
>: > is a symbol used in mathematics to represent the concept of 'greater than.' It establishes a relationship between two numbers, indicating that the number on the left is larger than the number on the right. This symbol is fundamental in ordering numbers and understanding their relative magnitudes, which is crucial for comparing values and making decisions based on numerical data.
≤: The symbol '≤' represents the mathematical concept of 'less than or equal to.' It is used to indicate that one value is either less than or equal to another value, establishing a relationship between two numbers. This concept is fundamental in understanding ordering and comparing real numbers, as it allows for a complete framework for expressing inequalities.
≥: The symbol '≥' represents the concept of 'greater than or equal to' in mathematics. It is used to compare two values, indicating that one value is either greater than or equal to another. This term plays a crucial role in understanding inequalities and helps in establishing the ordering and relationships between real numbers.
Archimedean Property: The Archimedean Property states that for any two positive real numbers, there exists a natural number such that when you multiply the smaller number by that natural number, it exceeds the larger number. This property is fundamental in understanding the ordering and density of real numbers, as it ensures that there are no 'gaps' in the real number system and that any real number can be approximated arbitrarily closely by rational numbers.
Arithmetic mean method: The arithmetic mean method is a statistical approach used to find the average of a set of numerical values by summing all the values and dividing by the count of those values. This concept is significant in understanding the ordering and density of real numbers as it provides a way to quantify central tendency within a dataset, illustrating how values relate to one another in a numeric continuum.
Betweenness: Betweenness is a concept that describes the relationship between three points on a number line or in a geometric context, where one point lies directly between two other points. This relationship is fundamental in understanding the ordering of real numbers, as it allows for the determination of relative positions and distances between numbers. It is also closely tied to the idea of density, highlighting how betweeness shows that there are infinitely many points between any two distinct real numbers.
Closed Interval: A closed interval is a set of real numbers that includes all the numbers between two endpoints, as well as the endpoints themselves. It is represented mathematically as $[a, b]$, where 'a' and 'b' are the endpoints. This concept is crucial when considering the ordering and density of real numbers, as it emphasizes the inclusion of boundary points, thus affecting how intervals are treated on the real number line. Additionally, the understanding of closed intervals is essential in defining distances and absolute values within a specified range.
Completeness axiom: The completeness axiom states that every non-empty set of real numbers that is bounded above has a least upper bound, or supremum. This concept is essential because it ensures that the real numbers are 'complete', meaning there are no gaps or missing points in the number line, allowing for the well-defined ordering and density of the real numbers.
Decimal expansions: Decimal expansions are a way of representing numbers using a base 10 system, where numbers are expressed as a whole number part followed by a decimal point and a fractional part. This representation allows for both rational numbers, which can be expressed as fractions, and irrational numbers, which cannot. Understanding decimal expansions is crucial for comparing sizes of real numbers and analyzing their density on the number line.
Density Property: The density property states that between any two real numbers, there exists another real number. This concept highlights the idea that real numbers are not isolated; instead, they fill the number line completely without any gaps. It applies to both rational and irrational numbers, reinforcing the idea that no matter how close two numbers are, one can always find another number in between them.
Inequality chains: Inequality chains are sequences of inequalities that connect multiple values in a logical manner, allowing for comparisons across different quantities. They enable the establishment of relationships among various numbers, helping to articulate complex ideas of order and density within the real numbers. This concept is essential in understanding how different numbers can be ordered and compared, leading to insights about their relative positions on the number line.
Infimum: The infimum of a set of real numbers is the greatest lower bound of that set, meaning it is the largest number that is less than or equal to every number in the set. This concept is essential in understanding how numbers are ordered and their density within the real number line. The infimum can be an element of the set, or it may not be; if it's not, it signifies that there is no smallest number in the set.
Irrational numbers: Irrational numbers are real numbers that cannot be expressed as a simple fraction or ratio of two integers. They have non-repeating, non-terminating decimal expansions, which means their decimal representation goes on forever without repeating any pattern. This characteristic sets them apart from rational numbers and connects them to various concepts such as decimal representations, the ordering of real numbers, and methods of proof.
Number Line: A number line is a visual representation of numbers arranged in a straight line, where each point corresponds to a specific number. It helps illustrate the relationships between numbers, such as their order and distance from one another. This tool is essential for understanding the properties of natural numbers and integers as well as the ordering and density of real numbers.
Open interval: An open interval is a set of real numbers that lies between two endpoints, where both endpoints are not included in the set. This concept highlights the idea that every number within the interval is a member, while the boundaries themselves are excluded. Open intervals are essential in understanding properties of real numbers, particularly regarding their ordering and the density of real numbers, as well as measuring distances and absolute values on the real line.
Ordering relation: An ordering relation is a mathematical concept that describes how elements in a set can be arranged or compared in a specific sequence based on certain properties. This concept is crucial for establishing notions of greater than, less than, or equal to among elements, thereby creating a structured framework for analysis and comparison. The relationship must be reflexive, antisymmetric, and transitive to qualify as an ordering relation, allowing for meaningful discussions about order and density among real numbers.
Rational Numbers: Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. This means that any number that can be written in the form $$\frac{a}{b}$$, where $$a$$ and $$b$$ are integers and $$b \neq 0$$, is considered a rational number. These numbers can be represented as both terminating and repeating decimals, making them a crucial part of understanding real numbers and their properties.
Real numbers: Real numbers are a set of values that include all the rational numbers (like integers and fractions) and all the irrational numbers (like the square root of 2 or pi). They can be represented on a number line and are used to measure quantities and represent continuous data. Real numbers also play a crucial role in defining operations and properties in various mathematical systems.
Supremum: The supremum of a set is the least upper bound of that set, meaning it is the smallest number that is greater than or equal to every element in the set. This concept is essential in understanding how numbers are ordered and how dense they are in the real number system, especially when dealing with bounded sets where a maximum may not exist.
Transitivity Property: The transitivity property states that if one element is related to a second element, and that second element is related to a third element, then the first element is also related to the third element. This concept is crucial in establishing ordered relationships among real numbers, which helps in understanding their density and structure.