Properties of Real Numbers

Interval Notation

Interval notation uses parentheses and brackets to describe sets of real numbers and their endpoints.

Learning Objectives

  • Use interval notation to represent sets of numbers

Key Takeaways

Key Points

  • A real interval is a set of real numbers with the property that any number that lies between two numbers included in the set is also included in the set.
  • The interval of numbers between [latex]a[/latex] and [latex]b[/latex], including [latex]a[/latex] and [latex]b[/latex], is denoted [latex][a,b][/latex]. The two numbers [latex]a[/latex] and [latex]b[/latex] are called the endpoints of the interval.
  • To indicate that an endpoint of a set is not included in the set, the square bracket enclosing the endpoint can be replaced with a parenthesis.
  • An open interval does not include its endpoints, and is enclosed in parentheses. A closed interval includes its endpoints, and is enclosed in square brackets.
  • An interval is considered bounded if both endpoints are real numbers. An interval is unbounded if both endpoints are not real numbers.
  • Replacing an endpoint with positive or negative infinity—e.g., [latex](- \infty, b][/latex]—indicates that a set is unbounded in one direction, or half-bounded.

Key Terms

  • interval: A distance in space.
  • bounded interval: A set for which both endpoints are real numbers.
  • open interval: A set of real numbers that does not include its endpoints.
  • endpoint: Either of the two points at the ends of a line segment.
  • half-bounded interval: A set for which one endpoint is a real number and the other is not.
  • closed interval: A set of real numbers that includes both of its endpoints.
  • unbounded interval: A set for which neither endpoint is a real number.

A “real interval” is a set of real numbers such that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers [latex]x[/latex] satisfying [latex]0 \leq x \leq 1[/latex] is an interval that contains 0 and 1, as well as all the numbers between them. Other examples of intervals include the set of all real numbers and the set of all negative real numbers.

The interval of numbers between [latex]a[/latex] and [latex]b[/latex], including [latex]a[/latex] and [latex]b[/latex], is often denoted [latex][a,b][/latex]. The two numbers are called the endpoints of the interval.

Open and Closed Intervals

An open interval does not include its endpoints and is indicated with parentheses. For example, [latex](0,1)[/latex] describes an interval greater than 0 and less than 1.

A closed interval includes its endpoints and is denoted with square brackets rather than parentheses. For example, [latex][0,1][/latex] describes an interval greater than or equal to 0 and less than or equal to 1.

To indicate that only one endpoint of an interval is included in that set, both symbols will be used. For example, the interval of numbers between 1 and 5, including 1 but excluding 5, is written as [latex][1,5)[/latex].

The image below illustrates open and closed intervals on a number line.

In the open interval (a, b), both endpoints are open circles and are not included in the interval. In the interval (a, b], the endpoint a is an open circle and the endpoint b is a shaded circle, indicating that b is included in the interval, but not a. In the interval [a, b), the endpoint a is a shaded circle and the endpoint b is an open circle. In the interval [a, b], both endpoints a and b are shaded to indicate that they are both included in the interval.

Intervals: Representations of open and closed intervals on the real number line.

Bounded and Unbounded Intervals

An interval is said to be bounded if both of its endpoints are real numbers. Bounded intervals are also commonly known as finite intervals. Conversely, if neither endpoint is a real number, the interval is said to be unbounded. For example, the interval [latex](1,10)[/latex] is considered bounded; the interval [latex](- \infty, + \infty)[/latex] is considered unbounded.

The set of all real numbers is the only interval that is unbounded at both ends; the empty set (the set containing no elements) is bounded.

An interval that has only one real-number endpoint is said to be half-bounded, or more descriptively, left-bounded or right-bounded. For example, the interval [latex](1, + \infty)[/latex] is half-bounded; specifically, it is left-bounded.

Absolute Value

Absolute value can be thought of as the distance of a real number from zero.

Learning Objectives

Define the absolute value of a number

Key Takeaways

Key Points

  • The absolute value of a real number may be thought of as its distance from zero along the real number line.
  • The absolute value of [latex]a[/latex] is denoted [latex]\left | a \right |[/latex].

Key Terms

  • absolute value: The distance of a real number from [latex]0[/latex] along the real number line.

In mathematics, the absolute value (sometimes called the modulus) of a real number [latex]a[/latex] is denoted [latex]\left | a \right |[/latex]. It refers to the distance of [latex]a[/latex] from zero. Therefore, [latex]\left | a \right |>0[/latex] for all numbers. For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5, because both numbers are the same distance from 0.

The absolute values of -5 and 5 are shown on a number line as the distance between -5 and 0 and the distance between 5 and 0. Both these distances are 5 units, so the absolute values are the same.

Absolute value: The absolute values of 5 and -5 shown on a number line.

When applied to the difference between real numbers, the absolute value represents the distance between the numbers on a number line.

History

Absolute value is closely related to the mathematical and physical concepts of magnitude, distance, and norm. The term “absolute value” has been used in this sense since at least 1806 in French and 1857 in English. The notation [latex]\left | a \right |[/latex] was introduced by Karl Weierstrass in 1841. Other names for absolute value include “numerical value,” “modulus,” and “magnitude.”

Examples

The following are some examples of equations involving absolute value:

  • [latex]\left | 7 \right |=7[/latex]
  • [latex]\left | -2 \right |=2[/latex]
  • [latex]-\left | 4 \right |=-4[/latex]
  • [latex]-\left | -3 \right |=-3[/latex]

Sets of Numbers

A set is a collection of unique numbers, often denoted with curly brackets: {}.

Learning Objectives

Use set notation to represent sets of numbers and describe the properties of commonly used sets of numbers

Key Takeaways

Key Points

  • A set is a collection of distinct objects and is considered an object in its own right. With numbers, a set is a collection of unique numbers, such as [latex]\left \{ 1, 2, 5, 8, 4 \right \}[/latex].
  • In sets, the order of numbers doesn’t matter; it is important only that no numbers are duplicated.
  • If every member of set A is also a member of set B, then A is said to be a subset of B, written [latex]A \subseteq B[/latex] (also pronounced “A is contained in B”). Conversely, [latex]B[/latex] can be considered a superset of [latex]A[/latex]. This is written [latex]B \supseteq A[/latex].
  • The common categories of number sets are natural numbers, real numbers, integers, rational numbers, imaginary numbers, and complex numbers.

Key Terms

  • superset: A set that contains another set.
  • set: A collection of unique objects, potentially infinite in size, that is not reliant on the order of the objects contained within it.
  • subset: A set that is also an element of another set.

Sets are one of the most fundamental concepts in mathematics. A set is a collection of distinct objects and is considered an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered together they form a single set of size three, written [latex]\left \{ 2,4,6 \right \}[/latex].

Defining a Set

There are two ways of describing, or specifying the members of, a set. One way is through intentional definition, using a rule or semantic description. For example: “[latex]A[/latex] is the set whose members are the first four positive integers.”

The second way of describing a set is through extension: listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets: [latex]C = \left \{ 4, 2, 1, 3 \right \}[/latex].

Every element of a set must be unique; no two members may be identical. All set operations preserve this property. The order in which the elements of a set are listed is irrelevant (unlike for a sequence). Therefore:

[latex]\left \{ 6, 11 \right \} = \left \{ 11, 6 \right \} = \left \{ 11, 6, 6, 11 \right \}[/latex]

because the extensional specification means merely that each of the elements listed is a member of the set.

For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive integers may be specified extensionally as:

[latex]\left \{ 1,2,3, \cdots, 1000 \right \}[/latex]

where the ellipsis ([latex]\cdots[/latex]) indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members. Thus the set of positive even numbers can be written as [latex]\left \{ 2,4,6,8, \cdots \right \}[/latex].

Subsets and Supersets

A subset is a set whose every element is also contained in another set. For example, if every member of set [latex]A[/latex] is also a member of set [latex]B[/latex], then [latex]A[/latex] is said to be a subset of [latex]B[/latex]. This is written [latex]A \subseteq B[/latex] (also pronounced “[latex]A[/latex] is contained in [latex]B[/latex]“). Equivalently, we can say that [latex]B[/latex] is a superset of [latex]A[/latex], which means that [latex]B[/latex] includes [latex]A[/latex], or [latex]B[/latex] contains [latex]A[/latex]. This is written [latex]B \supseteq A[/latex].

For example, [latex]\left \{ 1,3 \right \} \subseteq \left \{ 1,2,3,4 \right \}[/latex].

Common Sets

Some of the most commonly referenced sets of numbers are as follows.

The set of natural numbers, also known as “counting numbers,” includes all whole numbers starting at 1 and then increasing. The set of natural numbers is represented by the symbol [latex]\mathbb{N}[/latex] and can be denoted as [latex]\mathbb{N}=\left \{ 1,2,3,4, \cdots \right \}[/latex].

The set of real numbers includes every number, negative and decimal included, that exists on the number line. The set of real numbers is represented by the symbol [latex]\mathbb{R}[/latex].

The set of integers includes all whole numbers (positive and negative), including [latex]0[/latex]. The set of integers is represented by the symbol [latex]\mathbb{Z}[/latex]. (This may seem odd, but it stands for the German term “Zahlen,” which means “numbers.”)

The set of rational numbers, denoted by the symbol [latex]\mathbb{Q}[/latex], includes any number that is written as a fraction. The symbol [latex]\mathbb{Q}[/latex] is used because Q represents the word “quotient”.

The set of imaginary numbers, denoted by the symbol [latex]\mathbb{I}[/latex], includes all numbers that result in a negative number when squared.

The set of complex numbers, denoted by the symbol [latex]\mathbb{C}[/latex], includes a combination of real and imaginary numbers in the form of [latex]a+bi[/latex] where [latex]a[/latex] and [latex]b[/latex] are real numbers and [latex]i[/latex] is an imaginary number.

Factors

Any whole number greater than one can be factored, which means it can be broken down into smaller integers.

Learning Objectives

Calculate numbers’ factors and prime factors

Key Takeaways

Key Points

  • Factorization (or factoring ) is the process of breaking an object (such as a number or algebraic expression ) down into a product of other objects, or factors, which when multiplied together give the original number or expression.
  • Prime factorization is a particular type of factorization that breaks a number of interest into prime numbers that when multiplied back together produce the original number.
  • Every positive integer greater than 1 has a distinct prime factorization.
  • Factor trees can be used to find the prime factorization of a number.

Key Terms

  • prime factor: A factor that is also a prime number.
  • factor: Any of various objects multiplied together to form some whole.
  • factorization: The process of creating a list of items that, when multiplied together, will produce a desired quantity or expression.
  • prime number: A whole number greater than 1 that can be divided evenly by only the number 1 and itself.

In mathematics, factorization (or factoring) is the process of breaking an object (such as a number or algebraic expression) down into a product of other objects, or factors, which when multiplied together give the original number or expression. The aim of factoring is to reduce something to “basic building blocks.” This process has many real-life applications and can help us solve problems in mathematics.

In particular, factoring a number means to break it down into numbers that when multiplied back together produce the given number. For now, we will focus on factoring whole numbers.

For example, consider the number 24. To find the factors, consider the numbers that yield a product of 24. We know that [latex]6 \times 4 = 24[/latex], so both 6 and 4 are factors of 24. If we think about it, we can list all of the numbers that 24 is divisible by: 1, 2, 3, 4, 6, 8, 12, and 24. This is a complete list of the factors of 24.

The number 864 is broken into its factors 32 and 27. 32 is then broken into 4 and 8, which are respectively broken into 2 and 3 2's. 27 is broken into 9 and 3, and 9 is broken into 2 3's. The resulting prime factors are 5 2's and 3 3's.

Prime factorization example: This factor tree shows the factorization of 864. It shows that 864 is the product of five 2s and three 3s. A shorthand way of writing these resulting prime factors is [latex]2^5 \times 3^3[/latex].

Prime Factorization

Prime factorization is a particular type of factorization that breaks a number of interest into prime numbers that when multiplied back together produce the original number. Such prime numbers are called prime factors.

Example 1

For example, consider the number 6. We know that [latex]2 \times 3 = 6[/latex], so 2 and 3 are both factors of 6. Also note that 2 and 3 are prime numbers, because each is divisible by only 1 and itself. Therefore, 2 and 3 are prime factors of 6.

Example 2

Now, consider the number 12. We know that [latex]2 \times 6 = 12[/latex], so 2 and 6 are both factors of 12. However, 6 is not a prime factor. In this case, we must reduce 6 to its prime factors as well. Since we know from the previous example that the prime factors of 6 are 2 and 3 (because [latex]2 \times 3 = 6[/latex]), we can easily recognize that [latex]2 \times 2 \times 3 = 12[/latex]. We have now found factors for 12 that are all prime numbers. Therefore, the prime factorization for 12 is [latex]2 \times 2 \times 3[/latex].

Factor Trees and Prime Factorization

Every positive integer greater than 1 has a distinct prime factorization. To factor larger numbers, it can be helpful to draw a factor tree.

In a factor tree, the number of interest is written at the top. Then, two factors of that number are found and connected below that number with branches. This process repeats for each subsequent factor of the original number until all the factors at the bottoms of the branches are prime.