1.1.2: Sets of Numbers

Learning Objectives

  • Define natural numbers, whole numbers, integers, rational numbers, irrational numbers and real numbers in terms of sets.
  • Use interval notation to define sets of numbers
  • Use set-builder notation to define sets of numbers

KEY words

  • Natural numbers: [latex]\mathbb{N}[/latex] [latex]=\{1, 2, 3, …\}[/latex]
  • Whole numbers: [latex]\mathbb{W}[/latex] [latex]=\{0,1, 2, 3, …\}[/latex]
  • Integers:  [latex]\mathbb{Z}[/latex] [latex]=\{… -3, -2, -1, 0,1, 2, 3, …\}[/latex]
  • Rational numberst: [latex]\mathbb{Q}[/latex] [latex]=\,\left\{\dfrac{m}{n}\normalsize \;\large\vert\;\normalsize\,m\text{ and }{n}\text{ are integers and }{n}\ne{ 0 }\right\}[/latex]
  • Irrational numbers: the set of numbers that cannot be written as rational numbers
  • Real numbers: [latex]\mathbb{R}[/latex] = the union of the set of rational numbers and the set of irrational numbers
  • Interval notation: shows highest and lowest values in an interval inside brackets or parentheses
  • Set-builder notation: defines a set inside braces using variables, words, inequalities, etc.

Number Systems

The number system that we use today is called the Real Numbers. It is divided into subsets. We will define each subset and then further define the Real Numbers.

Natural numbers

The natural numbers (sometimes called counting numbers) are: [latex]1, 2, 3[/latex], and so on.  These are numbers we use for counting, or enumerating items.  Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The mathematical symbol for the set of all natural numbers is written as [latex]\mathbb{N}[/latex].  We describe them in set notation as [latex]\mathbb{N}[/latex] [latex]=\{1, 2, 3, …\}[/latex] where the ellipsis (…) indicates that the numbers continue following the same pattern to infinity.

Whole numbers

The set of whole numbers includes all natural numbers as well as [latex]0[/latex]. We describe them in set notation as [latex]\mathbb{W}[/latex] [latex]=\{0,1, 2, 3, …\}[/latex].

Integers

When the opposites of the natural numbers are combined with the set of whole numbers, the result is defined as the set of integers, [latex]\mathbb{Z}[/latex] [latex]=\{… -3, -2, -1, 0,1, 2, 3, …\}[/latex]. It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

[latex]\begin{array}{lll}{\text{negative integers}}\hfill & {\text{zero}}\hfill & {\text{positive integers}}\\{\dots ,-3,-2,-1,}\hfill & {0,}\hfill & {1,2,3,\dots }\end{array}[/latex]

Rational numbers

Numbers that can be written as one integer divided by another integer in the form [latex]\frac{p}{q}[/latex] are known as fractions. Fractions represent part of a whole. The bottom line (denominator) tells us how many equal parts the whole (1) is divided into, and the top line (numerator) tells us how many parts are being used. For example, the fraction [latex]\frac{3}{4}[/latex], read “three-fourths” or “three-quarters”, represents 3 out of 4 equal parts of a whole. When fractions are combined with the set of integers, the result is defined as the set of rational numbers, [latex]\mathbb{Q}[/latex]. A rational number is any number that can be written as a ratio of two integers. A ratio is just the comparison of two numbers, the numerator and denominator of the fraction.

Rational Numbers

A rational number is a number that can be written in the form [latex]{\frac{p}{q}}[/latex], where [latex]p[/latex] and [latex]q[/latex] are integers and [latex]q\ne 0[/latex].

A rational number, [latex]\mathbb{Q}[/latex], is a number that can be expressed as a fraction with integer numerator and denominator.  The set of rational numbers is written as  [latex]\mathbb{Q}[/latex] [latex]=\,\left\{\dfrac{p}{q}\normalsize \;\large\vert\;\normalsize\,p\text{ and }{q}\text{ are integers and }{q}\ne{ 0 }\right\}[/latex]. This is referred to as set builder notation, and is read, ” the set of fractions where the numerator and denominator are integers, and the denominator is never [latex]0[/latex]“. The vertical line [latex]\large\vert[/latex] is read “where” or “such that”. Recall that letters like [latex]p[/latex] and [latex]q[/latex] are called variables. In this context the variables are defined as integer values with [latex]p\neq 0[/latex].

Rational numbers are defined as fractions, so the fractions, [latex]\frac{4}{5} ,-\frac{7}{8} ,\frac{13}{4}[/latex] and [latex]-\frac{20}{3}[/latex] that all have numerators and each denominators that are integers are rational numbers.

The definition of rational numbers tells us that all fractions are rational, but what about natural numbers, whole numbers, and integers?

Are integers rational?

To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this is to write it as a fraction with denominator one.

[latex]3=\frac{3}{1}\normalsize ,\space-8=\frac{-8}{1} ,\space0=\frac{0}{1}[/latex]

Since any integer can be written as the ratio of two integers, all integers are rational numbers. And since natural numbers and whole numbers are subsets of the integers, they, too, are rational.

Can rational numbers be written as decimals?

Let’s look at the decimal form of the numbers we know are rational. We have seen that every integer is a rational number, since [latex]a=\frac{a}{1}[/latex] for any integer, [latex]a[/latex]. We can also change any integer to a decimal by adding a decimal point and a zero.

Integer [latex]-2,-1,0,1,2,3[/latex]

Decimal [latex]-2.0,-1.0,0.0,1.0,2.0,3.0[/latex]

These decimal numbers stop.

By definition, every fraction is a rational number. Look at the decimal form of some fractions.

Ratio of Integers [latex]\frac{4}{5} ,\frac{7}{8} ,\frac{13}{4} ,\frac{20}{3}[/latex]

Decimal Forms [latex]0.8,-0.875,3.25,-6.666\ldots,-6.\overline{66}[/latex]

These decimals either stop or repeat.

What do these examples tell us? Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats.

Because they are fractions, any rational number can also be expressed in decimal form by dividing the numerator of the fraction by the denominator. Any rational number can be represented as either:

  1. a terminating decimal: [latex]\dfrac{15}{8}\normalsize =1.875[/latex], or
  2. a repeating decimal: [latex]\dfrac{4}{11}\normalsize =0.36363636\dots =0.\overline{36}[/latex]

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Example

Write each of the following as a rational number.

  1. [latex]7[/latex]
  2. [latex]0[/latex]
  3. [latex]–8[/latex]

Example

Write each of the following rational numbers as either a terminating or repeating decimal.

  1. [latex]-\dfrac{5}{7}[/latex]
  2. [latex]\dfrac{15}{5}[/latex]
  3. [latex]\dfrac{13}{25}[/latex]

Are all decimals rational?

What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers. We’ve already seen that integers are rational numbers, so any decimal that is also an integer must also be a rational number. So, clearly, some decimals are rational.

A decimal number has an integer part before the decimal point, while numbers after the decimal point are parts of a whole. For example, the decimal [latex]12.645[/latex] consists of 12 wholes and 645 thousandths. We know it is thousandths because there are three places after the decimal point. Consequently, [latex]12.645[/latex] can be written as the mixed number [latex]12\frac{645}{1000}[/latex] or as the fraction [latex]\frac{12,645}{1000}[/latex]. So, [latex]12.645[/latex] is a rational number.

In general, any decimal that ends after a number of digits such as [latex]7.3[/latex] or [latex]-1.2684[/latex] is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction. Also, any decimal that never ends but repeats such as [latex]0.3333…[/latex] is a rational number. If you use your calculator to divide 1 by 3 you will find that  [latex]\frac{1}{3} = 0.3333…[/latex]. When a number repeats, we can use ellipses [latex](…)[/latex] or a bar over the repeating digit(s) [latex]0.\bar{3}[/latex] to show that the pattern repeats forever.

example

Write each number as the ratio of two integers:

1. [latex]-15[/latex]

2. [latex]6.81[/latex]

3. [latex]-3\frac{6}{7}[/latex]

Solution:

1.
[latex]-15[/latex]
Write the integer as a fraction with denominator 1. [latex]\frac{-15}{1}[/latex]
2.
[latex]6.81[/latex]
Write the decimal as a mixed number. [latex]6\frac{81}{100}[/latex]
Then convert it to an improper fraction. [latex]\frac{681}{100}[/latex]
3.
[latex]-3\frac{6}{7}[/latex]
Convert the mixed number to an improper fraction. [latex]-\frac{27}{7}[/latex]

 

try it

Irrational Numbers

Not all numbers are rational. Such numbers are said to be irrational because they cannot be written as fractions. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as [latex]\{\;x \;\;\large | \; \normalsize x \text{ is not a rational number}\}[/latex].

Examples of irrational numbers are [latex]pi[/latex], which is used with circles, and [latex]e[/latex], which is used in growth problems. Similarly, the decimal representations of square roots of numbers that are not perfect squares are irrational numbers. Approximations of these numbers can be found and used, but the numbers themselves are decimals that never repeat and never end.

For example,

[latex]\sqrt{5}=\text{2.236067978…..}[/latex]

A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number.

Irrational Number

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.

example

Identify each of the following as rational or irrational:
1. [latex]0.58\overline{3}[/latex]
2. [latex]0.475[/latex]
3. [latex]3.605551275\dots [/latex]

try it

example

Identify each of the following as rational or irrational:
1. [latex]\sqrt{36}[/latex]
2. [latex]\sqrt{44}[/latex]

try it

In the following video we show more examples of how to determine whether a number is irrational or rational.

Example

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

  1. [latex]\sqrt{25}[/latex]
  2. [latex]\dfrac{33}{9}[/latex]
  3. [latex]\sqrt{11}[/latex]
  4. [latex]\dfrac{17}{34}[/latex]
  5. [latex]0.3033033303333\dots[/latex]

Real numbers

Any number is either rational or irrational. It cannot be both. It can either be written as a fraction or it cannot. The sets of rational and irrational numbers together make up the set of real numbers, [latex]\mathbb{R}[/latex]. This means that the set of irrational numbers is the complement of the set of rational numbers in the set of real numbers.

Real Numbers

Real numbers are numbers that are either rational or irrational.

The real numbers include all the measuring numbers. The symbol for the real numbers is [latex]\mathbb{R}[/latex]. Real numbers are often represented using decimal numbers.  Like integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as [latex]0[/latex], with negative numbers to the left of [latex]0[/latex] and positive numbers to the right of [latex]0[/latex]. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of [latex]0[/latex]. Any real number corresponds to a unique position on the number line.  The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown below.

A number line that is marked from negative five to five

The real number line.

We have seen that all natural numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.  Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram.

Real Numbers Set

The set of irrational numbers is the complement of the set of rational numbers in the set of real numbers. The set of natural numbers is a subset of the set of whole numbers is a subset of the set of integers is a subset of the set of rational numbers is a subset of the set of real numbers.

Interval Notation

Another commonly used method for describing sets of numbers is called interval notation. With this convention, sets are built with parentheses or brackets, each having a distinct meaning. Interval notation is used to denote an interval of numbers. For example, the interval [latex](2,3)[/latex] represents the interval of numbers that are greater than two and less than three.

The main concept is that parentheses represent solutions greater than or less than the number, and brackets represent numbers that are greater than or equal to or less than or equal to the number. For example, the interval [latex][-2,3)[/latex] represents the interval greater than or equal to negative two to less than three.

Parentheses are used to represent infinity or negative infinity, as infinity is not a number in the usual sense of the word. For example [latex](-\infty, -3][/latex] is all numbers less than and including -3. However, the interval [latex](-\infty, -3)[/latex] is all numbers less than -3, not including -3 itself.

Example

Use interval notation to indicate all real numbers greater than or equal to -2.

Representing an Interval on a Number Line

Intervals can be graphed on a number line. Graphs of number lines and intervals can be very helpful in visualizing the interval. For example the interval [latex][-3, 4)[/latex] can be represented by the following graph:

Note that the closed circle is used to represent the inclusion of that point in the set, and the open point is used to demonstrate that the point is not included in the set.

Example

Use a real number line to describe the interval [latex](-2,\,6][/latex].

try it

Use a real number line to describe the interval [latex][3,\,5)[/latex].

Set-Builder Notation

Another way to represent an interval of real numbers is to use set-builder notation. An example of set-builder notation is the set of real numbers that are greater than 5: [latex]\left\{x\in\mathbb{R}\;\large\vert\;\normalsize\,x\gt\,5\right\}[/latex]. This is read, ” the set of all real numbers, [latex]x[/latex], where [latex]x[/latex] is greater than 5″. The vertical line [latex]\large\vert[/latex] is read “where” or “such that”.

Example

Use set builder notation to describe the real numbers that lie in the interval [latex](-2,\,6][/latex].

try it

Use set builder notation to describe the real numbers that lie in the interval [latex][3,\,5)[/latex].