1.1 Real Numbers: Algebra Essentials

Classifying a Real Number

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 $\left\{1,2,3,\mathrm{...}\right\}$ 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: $\left\{0,1,2,3,\mathrm{...}\right\}.$

The set of integers adds the opposites of the natural numbers to the set of whole numbers: $\left\{\mathrm{...},\mathrm{-3},\mathrm{-2},\mathrm{-1},0,1,2,3,\mathrm{...}\right\}.$ 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 $\left\{\frac{m}{n}|m\text{and }n\text{are integers and }n\ne 0\right\}.$ 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 as a terminating or repeating decimal. Any rational number can be represented as either:

1. ⓐa terminating decimal: $\frac{15}{8}=1.875,$ or
2. ⓑa repeating decimal: $\frac{4}{11}=0.36363636\dots =0.\overline{36}$

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

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 4

Classifying Real Numbers

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. ⓐ $-\frac{10}{3}$
2. ⓑ $\sqrt{5}$
3. ⓒ $-\sqrt{289}$
4. ⓓ $\mathrm{-6}\pi$
5. ⓔ $0.615384615384\dots$

Solution

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

Try It #4

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. ⓐ $\sqrt{73}$
2. ⓑ $\mathrm{-11.411411411}\dots$
3. ⓒ $\frac{47}{19}$
4. ⓓ $-\frac{\sqrt{5}}{2}$
5. ⓔ $6.210735$

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, such as Figure 2.

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

Sets of Numbers

The set of natural numbers includes the numbers used for counting: $\left\{1,2,3,\mathrm{...}\right\}.$

The set of whole numbers is the set of natural numbers plus zero: $\left\{0,1,2,3,\mathrm{...}\right\}.$

The set of integers adds the negative natural numbers to the set of whole numbers: $\left\{\mathrm{...},\mathrm{-3},\mathrm{-2},\mathrm{-1},0,1,2,3,\mathrm{...}\right\}.$

The set of rational numbers includes fractions written as $\left\{\frac{m}{n}|m\text{and }n\text{are integers and }n\ne 0\right\}.$

The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: $\left\{h|h\text{is not a rational number}\right\}.$

Example 5

Differentiating the Sets of Numbers

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

1. ⓐ $\sqrt{36}$
2. ⓑ $\frac{8}{3}$
3. ⓒ $\sqrt{73}$
4. ⓓ $\mathrm{-6}$
5. ⓔ $3.2121121112\dots$

Solution

	N	W	I	Q	Q′
a. $\sqrt{36}=6$	X	X	X	X
b. $\frac{8}{3}=2.\overline{6}$				X
c. $\sqrt{73}$					X
d. –6			X	X
e. 3.2121121112...					X

Try It #5

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

1. ⓐ $-\frac{35}{7}$
2. ⓑ $0$
3. ⓒ $\sqrt{169}$
4. ⓓ $\sqrt{24}$
5. ⓔ $4.763763763\dots$

Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example, $4^2=4\cdot 4=16.$ We can raise any number to any power. In general, the exponential notation $a^n$ means that the number or variable $a$ is used as a factor $n$ times.

In this notation, $a^n$ is read as the nth power of $a,$or $a$ to the $n$ where $a$ is called the base and $n$ is called the exponent. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, $24+6\cdot \frac{2}{3}-4^2$ is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify $4^2$ as 16.

Next, perform multiplication or division, left to right.

Lastly, perform addition or subtraction, left to right.

Therefore, $24+6\cdot \frac{2}{3}-4^2=12.$

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. 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 operations in mathematics.

Commutative Properties

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

$$a+b=b+a$$

We can better see this relationship when using real numbers.

$$\begin{array}{lllll}\left(\mathrm{-2}\right)+7=5\hfill & \hfill & \text{and}\hfill & \hfill & 7+\left(\mathrm{-2}\right)=5\hfill \end{array}$$

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

$$a\cdot b=b\cdot a$$

Again, consider an example with real numbers.

$$\begin{array}{ccccc}\left(\mathrm{-11}\right)\cdot \left(\mathrm{-4}\right)=44& & \text{and}& & \left(\mathrm{-4}\right)\cdot \left(\mathrm{-11}\right)=44\end{array}$$

It is important to note that neither subtraction nor division is commutative. For example, $17-5$ is not the same as $5-17.$ Similarly, $20÷5\ne 5÷20.$

Associative Properties

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.

$$a\left(bc\right)=\left(ab\right)c$$

Consider this example.

$$\begin{array}{ccccc}\left(3\cdot 4\right)\cdot 5=60& & \text{and}& & 3\cdot \left(4\cdot 5\right)=60\end{array}$$

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

$$a+\left(b+c\right)=\left(a+b\right)+c$$

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

$$\begin{array}{ccccc}\left[15+\left(\mathrm{-9}\right)\right]+23=29& & \text{and}& & 15+\left[\left(\mathrm{-9}\right)+23\right]=29\end{array}$$

Are subtraction and division associative? Review these examples.

$$\begin{array}{cccccc}\hfill 8-(3-15)& \stackrel{?}{=}& (8-3)-15\hfill & \hfill 64÷(8÷4)& \stackrel{?}{=}& (64÷8)÷4\hfill \\ \hfill 8-(-12)& =& 5-15\hfill & \hfill 64÷2& \stackrel{?}{=}& 8÷4\hfill \\ \hfill 20& \ne & -10\hfill & \hfill 32& \ne & 2\hfill \end{array}$$

As we can see, neither subtraction nor division is associative.

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.

$$a\cdot \left(b+c\right)=a\cdot b+a\cdot c$$

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

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.

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.

$$\begin{array}{ccc}\hfill 6+(3\cdot 5)& \stackrel{?}{=}& (6+3)\cdot (6+5)\hfill \\ \hfill 6+(15)& \stackrel{?}{=}& (9)\cdot (11)\hfill \\ \hfill 21& \ne & 99\hfill \end{array}$$

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

$$a-b=a+\left(-b\right)$$

For example, consider the difference $12-\left(5+3\right).$ We can rewrite the difference of the two terms 12 and $\left(5+3\right)$ by turning the subtraction expression into addition of the opposite. So instead of subtracting $\left(5+3\right),$ we add the opposite.

$$12+\left(\mathrm{-1}\right)\cdot \left(5+3\right)$$

Now, distribute $\mathrm{-1}$ and simplify the result.

$$\begin{array}{ccc}\hfill 12-(5+3)& =\hfill & 12+(\mathrm{-1})\cdot (5+3)\hfill \\ & =\hfill & 12+[(\mathrm{-1})\cdot 5+(\mathrm{-1})\cdot 3]\hfill \\ & =\hfill & 12+(\mathrm{-8})\hfill \\ & =\hfill & 4\hfill \end{array}$$

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. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

$$\begin{array}{ccc}\hfill 12-(5+3)& =& 12+(\mathrm{-5}-3)\\ & =& 12+(\mathrm{-8})\hfill \\ & =& 4\hfill \end{array}$$

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.

$$a+0=a$$

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.

$$a\cdot 1=a$$

For example, we have $\left(\mathrm{-6}\right)+0=\mathrm{-6}$ and $23\cdot 1=23.$ There are no exceptions for these properties; they work for every real number, including 0 and 1.

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 by (−a), that, when added to the original number, results in the additive identity, 0.

$$a+\left(-a\right)=0$$

For example, if $a=\mathrm{-8},$ the additive inverse is 8, since $\left(\mathrm{-8}\right)+8=0.$

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 $\frac{1}{a},$ that, when multiplied by the original number, results in the multiplicative identity, 1.

$$a\cdot \frac{1}{a}=1$$

For example, if $a=-\frac{2}{3},$ the reciprocal, denoted $\frac{1}{a},$ is $-\frac{3}{2}$ because

$$a\cdot \frac{1}{a}=\left(-\frac{2}{3}\right)\cdot \left(-\frac{3}{2}\right)=1$$

Properties of Real Numbers

The following properties hold for real numbers a, b, and c.

	Addition	Multiplication
Commutative Property	$a+b=b+a$	$a\cdot b=b\cdot a$
Associative Property	$a+\left(b+c\right)=\left(a+b\right)+c$	$a\left(bc\right)=\left(ab\right)c$
Distributive Property	$a\cdot \left(b+c\right)=a\cdot b+a\cdot c$
Identity Property	There exists a unique real number called the additive identity, 0, such that, for any real number a $$a+0=a$$	There exists a unique real number called the multiplicative identity, 1, such that, for any real number a $$a\cdot 1=a$$
Inverse Property	Every real number a has an additive inverse, or opposite, denoted –a, such that $$a+\left(-a\right)=0$$	Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted $\frac{1}{a},$ such that $$a\cdot \left(\frac{1}{a}\right)=1$$

Example 7

Using Properties of Real Numbers

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

1. ⓐ $3\cdot 6+3\cdot 4$
2. ⓑ $\left(5+8\right)+\left(\mathrm{-8}\right)$
3. ⓒ $6-\left(15+9\right)$
4. ⓓ $\frac{4}{7}\cdot \left(\frac{2}{3}\cdot \frac{7}{4}\right)$
5. ⓔ $100\cdot \left[0.75+\left(\mathrm{-2.38}\right)\right]$

Solution

1. ⓐ
   $\begin{array}{cccc}\hfill 3\cdot 6+3\cdot 4& =& 3\cdot (6+4)\hfill & \text{Distributive property.}\hfill \\ & =& 3\cdot 10\hfill & \text{Simplify.}\hfill \\ & =& 30\hfill & \text{Simplify.}\hfill \end{array}$
2. ⓑ
   $\begin{array}{cccc}\hfill (5+8)+(\mathrm{-8})& =& 5+[8+(\mathrm{-8})]\hfill & \text{Associative property of addition.}\\ & =& 5+0\hfill & \text{Inverse property of addition.}\hfill \\ & =& 5\hfill & \text{Identity property of addition.}\hfill \end{array}$
3. ⓒ
   $\begin{array}{cccc}\hfill 6-(15+9)\hfill & =& 6+[(\mathrm{-15})+(\mathrm{-9})]\hfill & \text{Distributive property.}\hfill \\ & =& 6+(\mathrm{-24})\hfill & \text{Simplify.}\hfill \\ & =& \mathrm{-18}\hfill & \text{Simplify.}\hfill \end{array}$
4. ⓓ
   $\begin{array}{cccc}\hfill \frac{4}{7}\cdot \left(\frac{2}{3}\cdot \frac{7}{4}\right)& =& \frac{4}{7}\cdot \left(\frac{7}{4}\cdot \frac{2}{3}\right)\hfill & \text{Commutative property of multiplication.}\hfill \\ & =& \left(\frac{4}{7}\cdot \frac{7}{4}\right)\cdot \frac{2}{3}\hfill & \text{Associative property of multiplication.}\hfill \\ & =& 1\cdot \frac{2}{3}\hfill & \text{Inverse property of multiplication.}\hfill \\ & =& \frac{2}{3}\hfill & \text{Identity property of multiplication.}\hfill \end{array}$
5. ⓔ
   $\begin{array}{cccc}\hfill 100\cdot [0.75+(-2.38)]& =& 100\cdot 0.75+100\cdot (\mathrm{-2.38})\hfill & \text{Distributive property.}\\ & =& 75+(\mathrm{-238})\hfill & \text{Simplify.}\hfill \\ & =& \mathrm{-163}\hfill & \text{Simplify.}\hfill \end{array}$

Try It #7

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

1. ⓐ $\left(-\frac{23}{5}\right)\cdot \left[11\cdot \left(-\frac{5}{23}\right)\right]$
2. ⓑ $5\cdot \left(6.2+0.4\right)$
3. ⓒ $18-\left(7\mathrm{-15}\right)$
4. ⓓ $\frac{17}{18}+\left[\frac{4}{9}+\left(-\frac{17}{18}\right)\right]$
5. ⓔ $6\cdot \left(\mathrm{-3}\right)+6\cdot 3$