The classes of numbers we will explore include:

Natural numbers

The most familiar numbers are the natural numbers (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. The mathematical symbol for the set of all natural numbers is written as [latex]\mathbb{N}[/latex], and sometimes  [latex]\mathbb{N_0}[/latex]  or  [latex]\mathbb{N_1}[/latex] when it is necessary to indicate whether the set should start with 0 or 1, respectively.

Integers

When the set of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set of integers, [latex]\mathbb{Z}[/latex]

Rational numbers

Real numbers

The numbers we use for counting, or enumerating items, are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe them in set notation as [latex]\{1, 2, 3, …\}[/latex] where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers. 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 set of whole numbers is the set of natural numbers plus zero: [latex]\{0, 1, 2, 3,…\}[/latex].

The set of integers adds the opposites of the natural numbers to the set of whole numbers: [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.

The set of rational numbers is written as [latex]\left\{\frac{m}{n}|m\text{ and }{n}\text{ are integers and }{n}\ne{ 0 }\right\}[/latex]. Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

1. a terminating decimal: [latex]\frac{15}{8}=1.875[/latex], or
2. a repeating decimal: [latex]\frac{4}{11}=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.

Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even [latex]\frac{3}{2}[/latex], but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. 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 shown.

 [latex]\mathbb{Q'}=[/latex]{h | h is not a rational number}

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]\frac{33}{9}[/latex]
3. [latex]\sqrt{11}[/latex]
4. [latex]\frac{17}{34}[/latex]
5. [latex]0.3033033303333\dots[/latex]

Show Solution

1. [latex]\sqrt{25}:[/latex] This can be simplified as [latex]\sqrt{25}=5[/latex]. Therefore, [latex]\sqrt{25}[/latex] is rational.
2. [latex]\frac{33}{9}:[/latex] Because it is a fraction, [latex]\frac{33}{9}[/latex] is a rational number. Next, simplify and divide.
   [latex]\frac{33}{9}=\frac{11\cdot3}{3\cdot3}=\frac{11}{3}=3.\overline{6}[/latex]

   So, [latex]\frac{33}{9}[/latex] is rational and a repeating decimal.

3. [latex]\sqrt{11}:[/latex] This cannot be simplified any further. Therefore, [latex]\sqrt{11}[/latex] is an irrational number.
4. [latex]\frac{17}{34}:[/latex] Because it is a fraction, [latex]\frac{17}{34}[/latex] is a rational number. Simplify and divide.
   [latex]\frac{17}{34}=\frac{{1}{\overline{)17}}}{\underset{2}{\overline{)34}}}=\frac{1}{2}=0.5[/latex]

   So, [latex]\frac{17}{34}[/latex] is rational and a terminating decimal.

5. 0.3033033303333… is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.

Real Numbers

Given any number n, we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with 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 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. 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 in Figure 1.

Figure 1. The real number line

 

Example

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

1. [latex]-\frac{10}{3}[/latex]
2. [latex]\sqrt{5}[/latex]
3. [latex]-\sqrt{289}[/latex]
4. [latex]-6\pi[/latex]
5. [latex]0.615384615384\dots[/latex]

Show Solution

1. [latex]-\frac{10}{3}[/latex] is negative and rational. It lies to the left of 0 on the number line.
2. [latex]\sqrt{5}[/latex] is positive and irrational. It lies to the right of 0.
3. [latex]-\sqrt{289}=-\sqrt{{17}^{2}}=-17[/latex] is negative and rational. It lies to the left of 0.
4. [latex]-6\pi [/latex] is negative and irrational. It lies to the left of 0.
5. [latex]0.615384615384\dots [/latex] is a repeating decimal so it is rational and positive. It lies to the right of 0.

Sets of Numbers as Subsets

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.

 

Figure 2. Sets of numbers.   N: the set of natural numbers   W: the set of whole numbers   Z: the set of integers   Q: the set of rational numbers   Q´: the set of irrational numbers

A General Note: Sets of Numbers

The set of natural numbers, [latex]\mathbb{N}[/latex], includes the numbers used for counting: [latex]\{1,2,3,\dots\}[/latex].

The set of whole numbers, [latex]\mathbb{W}[/latex], is the set of natural numbers plus zero: [latex]\{0,1,2,3,\dots\}[/latex].

The set of integers, [latex]\mathbb{Z}[/latex] adds the negative natural numbers to the set of whole numbers: [latex]\{\dots,-3,-2,-1,0,1,2,3,\dots\}[/latex].

The set of rational numbers, [latex]\mathbb{Q}[/latex] includes fractions written as [latex]\{\frac{m}{n}|m\text{ and }n\text{ are integers and }n\ne 0\}[/latex].

The set of irrational numbers, [latex]\mathbb{Q'}[/latex] is the set of numbers that are not rational, are nonrepeating, and are nonterminating: [latex]\{h|h\text{ is not a rational number}\}[/latex].

Example

Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q’).

1. [latex]\sqrt{36}[/latex]
2. [latex]\frac{8}{3}[/latex]
3. [latex]\sqrt{73}[/latex]
4. [latex]-6[/latex]
5. [latex]3.2121121112\dots [/latex]

Show Solution

	N	W	Z	Q	Q’
1. [latex]\sqrt{36}=6[/latex]	X	X	X	X
2. [latex]\frac{8}{3}=2.\overline{6}[/latex]				X
3. [latex]\sqrt{73}[/latex]					X
4. [latex]–6[/latex]			X	X
5. [latex]3.2121121112\dots[/latex]					X

Use Properties of Real Numbers

For some activities we perform, the order of certain processes does not matter, but the order of others do. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for addition and multiplication.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

We can better see this relationship when using real numbers.

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

Again, consider an example with real numbers.

Caution! It is important to note that neither subtraction nor division is commutative. For example, [latex]17 - 5[/latex] is not the same as [latex]5 - 17[/latex]. Similarly, [latex]20\div 5\ne 5\div 20[/latex].

Associative Properties – Grouping

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

Consider this example.

This property can be especially helpful when dealing with negative integers. Consider this example.

Are subtraction and division associative? Review these examples.

Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

To be more precise when describing this property, we say that multiplication distributes over addition.

The reverse is not true, as we can see in this example.

[latex]\begin{array}{ccc}\hfill 6+\left(3\cdot 5\right)& \stackrel{?}{=}& \left(6+3\right)\cdot \left(6+5\right) \\ \hfill 6+\left(15\right)& \stackrel{?}{=}& \left(9\right)\cdot \left(11\right)\hfill \\ \hfill 21& \ne & \text{ }99\hfill \end{array}[/latex]

A special case of the distributive property occurs when a sum of terms is subtracted.

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms.

Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

Inverse Properties

The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted−a, that, when added to the original number, results in the additive identity, 0.

For example, if [latex]a=-8[/latex], the additive inverse is 8, since [latex]\left(-8\right)+8=0[/latex].

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\frac{1}{a}[/latex], that, when multiplied by the original number, results in the multiplicative identity, 1.

Evaluate Algebraic Expressions

In mathematics, we may see expressions such as [latex]x+5,\frac{4}{3}\pi {r}^{3}[/latex], or [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex]. In the expression [latex]x+5[/latex], 5 is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.
In the following example, we will practice identifying constants and variables in mathematical expressions.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before. In the next example we show how to substitute various types of numbers into a mathematical expression.

Now we will show more examples of evaluating a variety of mathematical expressions for various values.

In the following video we present more examples of evaluating a variety of expressions for given values.