1.3.2: Rational and Irrational Numbers

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Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia
Rio Hondo College

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1.3.2 Learning Objectives

Order real numbers using inequality symbols
Add, subtract, multiply and divide fractions
Name and write decimals
Add, subtract, multiply, and divide decimals
Add, subtract, multiply, and divide certain irrational numbers

Rational Numbers

Rational numbers are a subset of the real numbers.

Rational numbers are sometimes called fractions. They are numbers that can be written as the quotient of two integers. They have decimal representations that either terminate or do not terminate but contain a repeating block of digits. Some examples are

$\dfrac{3}{4}=0.75$ or $8\dfrac{11}{7}=8.407407407...$

Some rational numbers are graphed below.

$A number line containing hash marks for numbers -3 through 4. There are dots for negative three and one-eighths, negative one-half, two-fifths, two divided by one, and three and one-half.$

Notice that every integer is a rational number.

Ordering Real Numbers

Our number system is called an ordered number system because the numbers in the system can be placed in order from smaller to larger. This is easily seen on the number line.

$A number line showing marks for 0 through 10. An arrow points to the left, labeled smaller. Another arrow points to the right, labeled larger.$

On the number line, a number that appears to the right of another number is larger than that other number. For example, 5 is greater than 2 because 5 is located to the right of 2 on the number line. We may also say that 2 is less than 5.

To make the inequality phrases "greater than" and "less than" more brief, mathematicians represent them with the symbols > and <, respectively.

Symbols for Greater Than > and Less Than <
> represents the phrase "greater than."
< represents the phrase "less than."

5 > 2 represents "5 is greater than 2."

2 < 5 represents "2 is less than 5."

Ordering Real Numbers
A real number $b$ is said to be greater than a real number $a$, denoted $b > a$, if $b$ is to the right of $a$ on the number line. Thus, as we would expect, $5 > 2$ since 5 is to the right of 2 on the number line. Also, $-2 > -5$ since -2 is to the right of -5 on the number line.

$A number line containing hash marks for numbers -5 through 5. There are dots on the hash marks for -5, -2, 2, and 5. Above the left side of the number line is the expression -2 > -5, and on the left side is 5 > 2.$

If we let $a$ and $b$ represent two numbers, then $a$ and $b$ are related in exactly one of three ways: Either

Equality Symbol
$a = b$ $a$ and $b$ are equal (8 = 8)

Inequality Symbols
$\begin{cases} a > b & a \text{ is greater than }b & (8 > 5) \\ a < b & a \text{ is less than } b & (5 < 8) \\ \ & \text{Some variations of these symbols are} & \ \\ a \ne b & a \text{ is not equal to } b & (8 \ne 5) \\ a \ge b & a \text{ is greater than or equal to } b & (a \ge 8) \\ a \le b & a \text{ is less than or equal to } b & (a \le 8) \end{cases}$

Comparing Fractions

Recall that the fraction $\dfrac{4}{5}$ indicates that we have 4 of 5 parts of some whole quantity, and the fraction $\dfrac{3}{5}$ indicates that we have 3 of 5 parts. Since 4 of 5 parts is more than 3 of 5 parts, $\dfrac{4}{5}$ is greater than $\dfrac{3}{5}$; that is,

$\dfrac{4}{5} > \dfrac{3}{5}$

We have just observed that when two fractions have the same denominator, we can determine which is larger by comparing the numerators.

Comparing Fractions
If two fractions have the same denominators, the fraction with the larger numerator is the larger fraction.

Thus, to compare the sizes of two or more fractions, we need only convert each of them to equivalent fractions that have a common denominator. We then compare the numerators. It is convenient if the common denominator is the LCD. The fraction with the larger numerator is the larger fraction.

Equivalent Fractions

Fundamental Principle of Rational Numbers (Fractions)

If a fraction is multiplied by the same value to the numerator and the denominator, the fraction that results is called and equivalent fraction as long as the mulitplied number is not zero.

Example 1

Produce an equivalent fraction for $\dfrac{2}{3}$.

Solution

Choose any value except zero. If we choose 3, then we multiply both the numerator and denominator by that number.

$\dfrac{2\times 3}{3\times 3}=\dfrac{6}{9}$.

Therefore $\dfrac{6}{9}$ is an equivalent fraction to $\dfrac{2}{ 3}$.

Least Common Multiple (LCM) and Least Common Denominator (LCD)
The least common multiple (LCM) of a collection of numbers is the smallest value that all the numbers in a collection can produce with multiplication. If these numbers are used as denominators of fractions, we call the least common multiple, the least common denominator (LCD).

Example 2

Compare $\dfrac{8}{9}$ and $\dfrac{14}{15}$. Find the LCD. Convert each fraction to an equivalent fraction with the LCD as the denominator.

Solution

$\left \{ \begin{array} {rcl} {9} & = & {3^2} \\ {15} & = & {3 \cdot 5} \end{array} \right \} \text{ The LCD } = 3^2 \cdot 5 = 9 \cdot 5 = 45$

$\dfrac{8}{9} = \dfrac{8 \cdot 5}{45} = \dfrac{40}{45}$

$\dfrac{14}{15} = \dfrac{14 \cdot 3}{45} = \dfrac{42}{45}$

Since $40 < 42$.

$\dfrac{40}{45} < \dfrac{42}{45}$

Thus $\dfrac{8}{9} < \dfrac{14}{15}$.

Example 3

Write $\dfrac{5}{6}$, $\dfrac{7}{10}$, and $\dfrac{13}{15}$ in order from smallest to largest.

Find the LCD of the three fractions. Convert each fraction to an equivalent fraction with the LCD as the denominator.

Solution

$\left \{ \begin{array} {rcl} {6} & = & {2 \cdot 3} \\ {10} & = & {2 \cdot 5} \\ {15} & = & {3 \cdot 5} \end{array} \right \} \text{ The LCD } = 2 \cdot 3 \cdot 5 = 30$

$\dfrac{5}{6} = \dfrac{5 \cdot 5}{30} = \dfrac{25}{30}$

$\dfrac{7}{10} = \dfrac{7 \cdot 3}{30} = \dfrac{21}{30}$

$\dfrac{13}{15} = \dfrac{13 \cdot 2}{30} = \dfrac{26}{30}$

Since $21 < 25 < 26$.

$\dfrac{21}{30} < \dfrac{25}{30} < \dfrac{26}{30}$

$\dfrac{7}{10} < \dfrac{5}{6} < \dfrac{13}{15}$

Writing these numbers in order from smallest to largest, we get $\dfrac{7}{10}$, $\dfrac{5}{6}$, $\dfrac{13}{15}$.

Example 4

Compare $8 \dfrac{6}{7}$ and $6 \dfrac{3}{4}$.

Solution

To compare mixed numbers that have different whole number parts, we need only compare whole number parts. Since 6 < 8, $6 \dfrac{3}{4} < 8 \dfrac{6}{7}$

Example 5

Compare $4 \dfrac{5}{8}$ and $4 \dfrac{7}{12}$

Solution

To compare mixed numbers that have the same whole number parts, we need only compare fractional parts.

$\left \{ \begin{array} {rcl} {8} & = & {2 \cdot 3} \\ {12} & = & {2^2 \cdot 3} \end{array} \right \} \text{ The LCD } = 2^3 \cdot 3 = 8 \cdot 3 = 24$

$\dfrac{5}{8} = \dfrac{5 \cdot 3}{24} = \dfrac{15}{24}$

$\dfrac{7}{12} = \dfrac{7 \cdot 2}{24} = \dfrac{14}{24}$

Since $14 < 15$.

$\dfrac{14}{24} < \dfrac{15}{24}$

$\dfrac{7}{12} < \dfrac{5}{8}$

Hence, $4 \dfrac{7}{12} < 4 \dfrac{5}{8}$

Addition and Subtraction of Fractions With Like Denominators

Let's examine the following diagram.

$A rectangle divided in five parts. Each part is labeled one-fifth. Two of the parts are shaded, and labeled two-fifths. A third part is shaded, and is labeled one-fifth.$

2 one-fifths and 1 one fifth is shaded.

It is shown in the shaded regions of the diagram that

(2 one-fifths) + (1 one-fifth) = (3 one-fifths)

That is,

$\dfrac{2}{5} + \dfrac{1}{5} = \dfrac{3}{5}$

From this observation, we can suggest the following rule.

Method of Adding Fractions Having Like Denominators
To add two or more fractions that have the same denominators, add the numerators and place the resulting sum over the common denominator. Simplify, if necessary.

Example 6

Find the following sums.

$\dfrac{3}{7} + \dfrac{2}{7}$. The denominators are the same. Add the numerators and place that sum over 7.

Solution

$\dfrac{3}{7} + \dfrac{2}{7} = \dfrac{3 + 2}{7} = \dfrac{5}{7}$

Example 7

$\dfrac{1}{8} + \dfrac{3}{8}$. The denominators are the same. Add the numerators and place the sum over 8. Simplify.

Solution

$\dfrac{1}{8} + \dfrac{3}{8} = \dfrac{1 + 3}{8} = \dfrac{4}{8} = \dfrac{1}{2}$

Example 8

To see what happens if we mistakenly add the denominators as well as the numerators, let's add

$\dfrac{1}{2} + \dfrac{1}{2}$

Adding the numerators and mistakenly adding the denominators produces

$\dfrac{1}{2} + \dfrac{1}{2} = \dfrac{1 + 1}{2 + 2} = \dfrac{2}{4} = \dfrac{1}{2}$

This means that two $\dfrac{1}{2}$'s is the same as one $\dfrac{1}{2}$. Preposterous! We do not add denominators.

Subtraction of Fractions With Like Denominators

We can picture the concept of subtraction of fractions in much the same way we pictured addition.

$A visualization of a subtraction problem. There are three rows displayed, and each row has an element that corresponds with it. In the first row are three rectangles, each divided into five parts. Each part in each fraction is labeled one-fifth. The first rectangle has three shaded parts. Next to this is the statement, take away. Next to this is the second rectangle, with one part shaded. Next to this is an equals sign. Finally, the third rectangle has two shaded parts. The second row reads three-fifths minus one-fifth equals two-fifths. The third row shows the same equation written in words.$

From this observation, we can suggest the following rule for subtracting fractions having like denominators:

Subtraction of Fractions with Like Denominators
To subtract two fractions that have like denominators, subtract the numerators and place the resulting difference over the common denominator. Simplify, if possible.

Example 9

Find the following differences.

$\dfrac{3}{5} - \dfrac{1}{5}$. The denominators are the same. Subtract the numerators. Place the difference over 5.

Solution

$\dfrac{3}{5} - \dfrac{1}{5} = \dfrac{3 - 1}{5} = \dfrac{2}{5}$

Example 10

$\dfrac{8}{6} - \dfrac{2}{6}$. The denominators are the same. Subtract the numerators. Place the difference over 6.

Solution

$\dfrac{8}{6} - \dfrac{2}{6} = \dfrac{8 - 2}{6} = \dfrac{6}{6} = 1$

Example 11

To see what happens if we mistakenly subtract the denominators, let's consider

$\dfrac{7}{15} - \dfrac{4}{15} = \dfrac{7 - 4}{15 - 15} = \dfrac{3}{0}$

We get division by zero, which is undefined. We do not subtract denominators.

A Basic Rule

There is a basic rule that must be followed when adding or subtracting fractions.

A Basic Rule
Fractions can only be added or subtracted conveniently if they have like denominators.

To see why this rule makes sense, let's consider the problem of adding a quarter and a dime.

$\text{1 quarter + 1 dime = 35 cents}$

Now,

$\left \{ \begin{array} {r} {\text{1 quarter } = \dfrac{25}{100}} \\ {\text{1 dime } = \dfrac{10}{100}} \end{array} \right \} \text{ same denominations}$

$35, \cancel{c} = \dfrac{35}{100}$

$\dfrac{25}{100} + \dfrac{10}{100} = \dfrac{25 + 10}{100} = \dfrac{35}{100}$

In order to combine a quarter and a dime to produce 35¢, we convert them to quantities of the same denomination.

Same denomination $\rightarrow$ same denominator

Addition and Subtraction of Fractions with Unlike Denominators

Method of Adding or Subtracting Fractions with Unlike Denominators
To add or subtract fractions having unlike denominators, convert each fraction to an equivalent fraction having as a denominator the least common denominator ( LCD) of the original denominators.

Example 12

Find the following sums and differences.

$\dfrac{1}{6} + \dfrac{3}{4}$. The denominators are not the same. Find the LCD of 6 and 4.

$\left\{ \begin{array} {c} {6 = 2 \cdot 3} \\ {4 = 2^2} \end{array} \right \} \text{ The LCD = } 2^2 \cdot 3 = 4 \cdot 3 = 12$

Write each of the original fractions as a new, equivalent fraction having the common denominator 12.

$\dfrac{1}{6} + \dfrac{3}{4} = \dfrac{}{12} + \dfrac{}{12}$

To find a new numerator, we divide the original denominator into the LCD. Since the original denominator is being multiplied by this quotient, we must multiply the original numerator by this quotient.

$12 \div 6 = 2$

$Multiply 1 by 2: 1 times 2 equals 2. The 1 is the original numerator, and the 2, the product of the multiplication is the new numerator.$

$12 \div 4 = 3$

$Multiply 3 by 3: 3 times 3 equals 9. 3 is the original numerator. 9 is the new numerator.$

$\begin{array} {rcll} {\dfrac{1}{6} + \dfrac{3}{4}} & = & {\dfrac{1 \cdot 2}{12} + \dfrac{3 \cdot 3}{12}} & {} \\ {} & = & {\dfrac{2}{12} + \dfrac{9}{12}} & {\text{ Now the denominators are the same.}} \\ {} & = & {\dfrac{2 + 9}{12}} & {\text{ Add the numerators and place the sum over the common denominator.}} \\ {} & = & {\dfrac{11}{12}} & {} \end{array}$

Example 13

$\dfrac{1}{2} + \dfrac{2}{3}$. The denominators are not the same. Find the LCD of 2 and 3.

$\text{LCD} = 2 \cdot 3 = 6$

Write each of the original fractions as a new, equivalent fraction having the common denominator 6.

$\dfrac{1}{2} + \dfrac{2}{3} = \dfrac{}{6} + \dfrac{}{6}$

$6 \div 2 = 3$ Multiply the numerator 1 by 3.
$6 \div 2 = 3$ Multiply the numerator 2 by 2.

$\begin{array} {rcl} {\dfrac{1}{2} + \dfrac{2}{3}} & = & {\dfrac{1 \cdot 3}{6} + \dfrac{2 \cdot 3}{6}} \\ {} & = & {\dfrac{3}{6} + \dfrac{4}{6}} \\ {} & = & {\dfrac{3 + 4}{6}} \\ {} & = & {\dfrac{7}{6} \text{ or } 1 \dfrac{1}{6}} \end{array}$

Example 14

$\dfrac{5}{9} - \dfrac{5}{12}$. The denominators are not the same. Find the LCD of 9 and 12.

$\left \{ \begin{array} {l} {9 = 3 \cdot 3 = 3^2} \\ {12 = 2 \cdot 6 = 2 \cdot 2 \cdot 3 = 2^2 \cdot 3} \end{array} \right \} \text{ LCD} = 2^2 \cdot 3^2 = 4 \cdot 9 = 36$

$\dfrac{5}{9} - \dfrac{5}{12} = \dfrac{}{36} - \dfrac{}{36}$

$36 \div 9 = 4$ Multiply the numerator 5 by 4.
$36 \div 12 = 3$ Multiply the numerator 5 by 3.

$\begin{array} {rcl} {\dfrac{5}{9} - \dfrac{5}{12}} & = & {\dfrac{5 \cdot 4}{36} - \dfrac{5 \cdot 3}{36}} \\ {} & = & {\dfrac{20}{36} - \dfrac{15}{36}} \\ {} & = & {\dfrac{20 - 15}{6}} \\ {} & = & {\dfrac{5}{36}} \end{array}$

Multiplication of Fractions

When multiplying numbers in fraction form, the numerator and denominator are calculated separately. The numbers in the numerator are multiplied across and the numbers of the denominator are multiplied across. Again, the final answer should be reduced to the simplest form.

Example 15

Multiply the following rational numbers in fraction form.

$\dfrac{2}{9}\times \dfrac{5}{9}$
$\dfrac{7}{12}\times \dfrac{15}{16}$

Solution

1. $\dfrac{2}{9}\times \dfrac{5}{9}$

$\dfrac{2}{9}\times \dfrac{5}{9}=\dfrac{2\times 5}{9\times 9}=\dfrac{10}{81}$

$\dfrac{10}{81}$ cannot be simplified, so this is the final answer.

2. $\dfrac{7}{12}\times \dfrac{15}{16}$

In multiplication, simplification can also be done before multiplying. Notice in this example, 12 is in the denominator of the first fraction and 15 is in the numerator of the second fraction. Both of these numbers are divisible by 3. Since one is in the numerator and the other is in the denominator, this is the same as dividing by 3 in both places in the final step of the process above. Reduce those numbers then multiply.

$\dfrac{7}{12}\times \dfrac{15}{16}=\dfrac{7}{12\div 3}\times \dfrac{15\div 3}{16}=\dfrac{7}{4}\times \dfrac{5}{16}=\dfrac{7 \times 5}{4\times 16}=\dfrac{35}{64}$

$\dfrac{35}{64}$ cannot be simplified, so this is the final answer.

Division of Fractions

Fractions divided by fractions are typically called complex fractions. The first step to division will be to convert the operation to multiplication. In order to convert the operation to multiplication, the reciprocal of the fraction in the denominator must be used. The numerator fraction will remain the same, switch the operation to multiplication, and then write the reciprocal of the denominator. Once the operation is converted use the above steps for multiplication to determine the final answer. Remember that whole numbers can be written as fractions and be careful not to simplify anything until after the operation is converted.

Example 16

Divide the following rational numbers.

1. $\dfrac{\dfrac{5}{14}}{\dfrac{7}{6}}$

2. $\dfrac{\dfrac{2}{9}}{ \dfrac{5}{9}}$

3. $\dfrac{\dfrac{2}{9}}{11}$

Solution

1. $\dfrac{\dfrac{5}{14}}{\dfrac{7}{6}}$

Keep, Switch, Flip first.

$\dfrac{\dfrac{5}{14}}{\dfrac{7}{6}}=\dfrac{5}{14}\times \dfrac{6}{7}$

Notice the numerator and denominator have numbers that can be simplified. Both the 14 and 6 are divisible by 2.

$\dfrac{5}{14}\times \dfrac{6}{7}=\dfrac{5}{14\div 2}\times \dfrac{6\div 2}{7}=\dfrac{5}{7}\times \dfrac{3}{7}=\dfrac{5\times 3}{7 \times 7}=\dfrac{15}{49}$

$\dfrac{15}{49}$ cannot be simplified, so this is the final answer.

2. $\dfrac{\dfrac{2}{9}}{ \dfrac{5}{9}}$

Keep, Switch, Flip first.

$\dfrac{\dfrac{2}{9}}{ \dfrac{5}{9}}=\dfrac{2}{9}\times \dfrac{9}{5}$

Notice the numerator and denominator have a common factor of nine nines that can be reduced.

$\dfrac{2}{9}\times \dfrac{9}{5}=\dfrac{2}{9\div 9}\times \dfrac{9\div 9}{5}=\dfrac{2}{1}\times \dfrac{1}{5}=\dfrac{2\times 1}{1\times 5}=\dfrac{2}{5}$

$\dfrac{2}{5}$ cannot be simplified, so this is the final answer.

3. $\dfrac{\dfrac{2}{9}}{11}$

Keep, Switch, Flip first. Since 11 can be written as $\dfrac{11}{1}$, the reciprocal would be $\dfrac{1}{11}$.

$\dfrac{\dfrac{2}{9}}{11}=\dfrac{2}{9}\times \dfrac{1}{11}=\dfrac{2\times 1}{9\times 11}=\dfrac{2}{99}$

$\dfrac{2}{99}$ cannot be simplified, so this is the final answer.

Try It Now 2

Multiply or Divide the following.

$\dfrac{5}{14} \times \dfrac{7}{16}$
$\dfrac{\dfrac{2}{5}}{\dfrac{3}{4}}$

Answer

$\dfrac{5}{14}\times \dfrac{7}{16}$

$\dfrac{5}{14}\times \dfrac{7}{16}=\dfrac{5\times 7}{14 \times 16}=\dfrac {35}{224}$

$\dfrac {35}{224}$ can be simplified since 35 and 224 are both divisible by 7.

$\dfrac {35}{224}=\dfrac {35\div 7}{224 \div 7}=\dfrac{5}{32}$

$\dfrac{5}{32}$ cannot be simplified, so this is the final answer.
$\dfrac{\dfrac{2}{5}}{\dfrac{3}{4}}$

Remember keep the numerator, switch the operation, flip the denominator before doing anything else. "Keep, Switch, Flip."

$\dfrac{\dfrac{2}{5}}{\dfrac{3}{4}}=\dfrac{2}{5}\times \dfrac{4}{3}=\dfrac{2\times 4}{5\times 3}=\dfrac{8}{15}$

$\dfrac{8}{15}$ cannot be simplified, so this is the final answer.

Improper Fractions and Mixed Numbers

At times, a rational number is written with a numerator that is larger than the denominator. This is called an improper fraction. A fraction represents a part of a whole. When there is an improper fraction it represents a combination of whole numbers with extra pieces called mixed numbers.

An improper fraction can be converted to a mixed number to more clearly display the whole part with the fraction part.

Example 17

Convert the improper fractions to mixed numbers.

$\dfrac{3}{2}$
$\dfrac{7}{3}$
$\dfrac{26}{7}$

Solution

$\dfrac{3}{2}$ can be rewritten as $\dfrac{2}{2}$+$\dfrac{1}{2}$=1+$\dfrac{1}{2}$=1$\dfrac{1}{2}$
$\dfrac{7}{3}$ can also be viewed as division. $7 \div 3$. Using division we see that 3 goes into 7 two times with a remainder of 1. So the mixed number would be $2\dfrac{1}{3}$
$\dfrac{26}{7}$. Viewing it as several fractions or as division, we find that $\dfrac{26}{7}=3\dfrac{5}{7}$

A mixed number can also be converted to an improper fraction to make mathematical calculations easier than working with mixed numbers. The whole number part of the mixed number is changed to the same denominator as the fraction part, and the two parts are added.

Example 18

Convert to improper fractions.

4 $\dfrac{2}{5}$
12 $\dfrac{3}{8}$

Solution

4 $\dfrac{2}{5}= 4 + \dfrac{2}{5}=4 \cdot\dfrac{5}{5}+\dfrac{2}{5}=\dfrac{20}{5}+\dfrac{2}{5}=\dfrac{22}{5}$
12 $\dfrac{3}{8}=12 + \dfrac{3}{8}=12 \cdot \dfrac{8}{8}+\dfrac{3}{8}=\dfrac{96}{8}+\dfrac{3}{8}=\dfrac{99}{8}$

A shortcut method is to keep the denominator the same and to find the new numerator. To find the new numerator, multiply the whole number to the denominator and then add the original numerator.

4 $\dfrac{2}{5}= \dfrac{4 \cdot 5+2}{2}=\dfrac{22}{5}$
12 $\dfrac{3}{8}=\dfrac{12 \cdot 8+3}{8}=\dfrac{99}{8}$

Fractions in Decimal Form

Fractions can also be written as decimals that are terminating or have a repeating pattern.

The place value system extends to the right of the ones place, so 0.2 has 2 in the tenths place and is equal to the fraction $\dfrac{2}{10}$. 0.07 has 7 in the hundredths place, so it is the same as $\dfrac{7}{100}$. A decimal point is used to identify which place is the ones place.

Reading decimal numbers

Decimal numbers like 0.27 mean $\dfrac{2}{10} +\dfrac{7}{100}$. Using equivalent fractions, $\dfrac{2}{10} = \dfrac{20}{100}$, so $0.27 = \dfrac{20}{100} + \dfrac{7}{100}$ or $\dfrac{27}{100}$. To read 0.27, we look at the digits to the right of the decimal point and read them as if they were a whole number: twenty seven. Then we identify the place value of the right-hand digit: hundredths. 0.27 is read "twenty-seven hundredths."

Mixed numbers are also written in decimal form such as 16.352. Just as with mixed fractions, we say the whole number part, followed by "and," followed by the fraction part. We read 16.352 as "sixteen AND three hundred fifty-two thousandths."

Example 19

Write these decimal numbers in words

487.59
83.9403

Solution

four hundred eighty-seven and fifty-nine hundredths
eighty-three and nine thousand four hundred three ten-thousandths

Example 20

Write these decimal numbers using digits

twenty and forty-one hundredths
five and eighty-nine ten-thousandths
three hundred and thirty-six thousandths

Solution

20.41
5.0089
300.036

Add and Subtract Decimals

To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.

Steps for adding or subtracting decimals.

Write the numbers so the decimal points line up vertically.
Use zeros as place holders, as needed.
Add or subtract the numbers as if they were whole numbers. Then place the decimal point in the answer under the decimal points in the given numbers.

Example 21

Add: $23.5+41.38$.

Solution

$\text{Write the numbers so that the decimal points line up vertically.} \quad \begin{array} {r} { 23.50 } \\ { + 41.38 } \\ \hline \end{array}$
$\text{Put 0 as a placeholder after the 5 in 23.5. Remember, } \frac{5}{10} = \frac{50}{100}, \text{ so } 0.5 = 0.50 \quad \begin{array} {r} { 23.50 } \\ { + 41.38 } \\ \hline \end{array}$
$\text{Add the numbers as if they were whole numbers . Then place the decimal point in the sum.} \quad \begin{array} {r} { 23.50 } \\ { + 41.38 } \\ \hline 64.88 \end{array}$

Try It Now 3

Add: $4.8+11.69$.

Answer: $16.49$

Multiply and Divide Decimals

Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first convert them to fractions and then multiply.

So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side. Look for a pattern!

Table $\PageIndex{1}$
	$two multiplication problems. The first one has 0.3 with a bracket under showing 1 decimal place multiplied by 0.7 with a bracket under showing 1 decimal place. The second one has 0.2 with a bracket under showing 1 decimal place multiplied by 0.46 with a bracket under showing 2 decimal places.$
Convert to fractions.	$\dfrac{3}{10} \cdot \dfrac{7}{10}$	$\dfrac{2}{10} \cdot \dfrac{46}{100}$
Multiply.	$\dfrac{21}{100}$	$\dfrac{92}{1000} $
Convert to decimals.	$The solutions for both columns. The first solution is 0.21. This has a bracket under showing 2 decimal places. The second solution is 0.092 which has a bracket under showing 3 decimal places.$

Notice, in the first example, we multiplied two numbers that each had one digit after the decimal point and the product had two decimal places. In the second example, we multiplied a number with one decimal place by a number with two decimal places and the product had three decimal places.

We multiply the numbers just as we do whole numbers, temporarily ignoring the decimal point. We then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product.

The rules for multiplying positive and negative numbers apply to decimals, too, of course!

When multiplying two numbers,

if their signs are the same the product is positive.
if their signs are different the product is negative.

When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. Finally, we write the product with the appropriate sign.

Steps for multiplying decimals.

Determine the sign of the product.
Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.
Write the product with the appropriate sign.

Example 22

Multiply: $(−3.9)(4.075)$.

Solution

	$(−3.9)(4.075)$
The signs are different. The product will be negative.
Write in vertical format, lining up the numbers on the right.	$Using vertical format to multiply two numbers. 4.075 is on top with the multiplication symbol and 3.9 on the next line. The 5 and 9 are lined up. There is a line under 3.9 to show that the solution belongs under that line.$
Multiply.	$4.075 time 3.9 with the solution bar is repeated from the image above. Under the solution bar is the solution from multiplying 9 time 4.075, 36675. Below that solution is the solution from multipling 3 times 4.075, 12225 with a blank space at the end. The sum of those two numbers are listed below another solution bar. The sum is 158925.$
Add the number of decimal places in the factors $(1 + 3)$. $.$ Place the decimal point 4 places from the right.	$The solution from above is shown again, this time there is a decimal after the 15: 15.8925. A bracket is below the decimal part stating 4 decimal places.$
The signs are different, so the product is negative.	$(−3.9)(4.075) = −15.8925$

Try It Now 4

Multiply: $−4.5(6.107)$.

Answer: $−27.4815$

Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.

To divide decimals, determine what power of 10 to multiply the denominator by to make it a whole number. Then multiply the numerator by that same power of 10. Because of the equivalent fractions property, we haven’t changed the value of the fraction! The effect is to move the decimal points in the numerator and denominator the same number of places to the right. For example:

$\begin{array} { c } { \frac { 0.8 } { 0.4 } } \\ { \frac { 0.8 ( 10 ) } { 0.4 ( 10 ) } } \\ { \frac { 8 } { 4 } } \end{array}$

We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.

We review the notation and vocabulary for division:

$\begin{array} {ll} {} &{\underset{\text{quotient}}{c}} \\ {\underset{\text{dividend}}{a} \div \underset{\text{divisor}}{b} = \underset{\text{quotient}}{c}} & {\underset{\text{divisor}}{b})\overline{\underset{\text{dividend}}{a}}} \end{array}$

We’ll write the steps to take when dividing decimals, for easy reference.

Steps for dividing decimals.

Determine the sign of the quotient.
Make the divisor a whole number by “moving” the decimal point all the way to the right. “Move” the decimal point in the dividend the same number of places—adding zeros as needed.
Divide. Place the decimal point in the quotient above the decimal point in the dividend.
Write the quotient with the appropriate sign.

Example 23

Divide: $−25.65\div (−0.06)$.

Solution

Remember, you can “move” the decimals in the divisor and dividend because of the Equivalent Fractions Property.

	$−25.65\div (−0.06)$
The signs are the same.	The quotient is positive.
Make the divisor a whole number by “moving” the decimal point all the way to the right.
“Move” the decimal point in the dividend the same number of places.	$0.06 outside the dividing bracket with an arrow pointing to the right below the .06. 2.65 is under the dividing bracket with an arrow pointing to the right below the .65$
Divide. Place the decimal point in the quotient above the decimal point in the dividend.
Write the quotient with the appropriate sign.	$−25.65\div (−0.06) = 427.5$

Try It Now 5

Divide: $−23.492\div (−0.04)$.

Answer: 687.3

Irrational Numbers

Irrational numbers are also a subset of the real numbers.

Irrational numbers are numbers with decimal representations that do not terminate or contain a repeating block of digits. Some examples are:

$\pi, e, \sqrt{2}$.

If you put these in your calculator, it will display the first several digits of the decimal approximation until it runs out of space. The decimal approximation of these numbers continues beyond what your calculator can display.

Operations with Irrational Numbers

Most often when working with irrational numbers, an exact number is more desirable than a decimal approximation. An exact number will contain the irrational number in its nondecimal form such as $\pi$ or $\sqrt{2}$. It is important to pay attention to the coefficient of the irrational number to determine how to calculate the final answer.

Multiplication and Division

When an irrational number is the same type, for example both are square roots of a number, they will remain the same type while the multiplication or division is performed within the type.

Example 24

Multiply or divide the following irrational numbers. Write the answer in exact form.

$\pi \times \pi$
$\sqrt{5}\times\sqrt{5}$
$\sqrt{2}\times \sqrt{3}$
$\sqrt{2}\times \sqrt{8}$
$7\sqrt{2}\times \pi$
$\sqrt{2}\div \sqrt{8}$
$15\pi \div 3\pi$
$\sqrt{6}\div 2\pi$

Solution

1. $\pi \times \pi$

$\pi \times \pi=\pi^{2}$

2. $\sqrt{5}\times\sqrt{5}$

Since these are both square roots, do the multiplication or division within the square root. first Then simplify, if possible.

$\sqrt{5}\times\sqrt{5}=\sqrt{5^{2}}=5$

3. $\sqrt{2}\times \sqrt{3}$

$\sqrt{2}\times \sqrt{3}=\sqrt{2\times 3}=\sqrt{6}$

4. $\sqrt{2}\times \sqrt{8}$

$\sqrt{2}\times \sqrt{8}=\sqrt{2\times 8 }=\sqrt{16}=4$

5. $7\sqrt{2}\times \pi$

Since these are different types of irrational numbers they cannot be combined in exact form.

$7\sqrt{2}\times \pi=7\sqrt{2}\pi$

6. $\sqrt{2}\div \sqrt{8}$

$\sqrt{2}\div \sqrt{8}=\sqrt{2\div 8}=\sqrt{\dfrac{1}{4}}=\dfrac{\sqrt{1}}{\sqrt{4}}=\dfrac{1}{2}$

7. $15\pi \div 3\pi$

Since both the numerator and denominator are divisible by $\pi$, the $\pi$ will divide out.

$15\pi \div 3\pi=15 \div 3=5$

8. $\sqrt{6}\div 2\pi$

Since these are different types of irrational numbers they cannot be combined in exact form. It can be rewritten as a fraction.

$\dfrac{\sqrt{6}}{ 2\pi}$

Addition and Subtraction

When adding and subtracting the irrational number is treated like a variable. Only "like terms" can be added or subtracted, meaning the terms must already have or be able to be converted to the same irrational number. Like term coefficients are what are added or subtracted, while the irrational number remains a part of the final answer.

Example 25

Add the following irrational numbers in exact form, if possible.

$2\sqrt{3}+5\sqrt{3}$
$7\pi +2\sqrt{5}$
$9\sqrt{2}-4\sqrt{2}$
$\sqrt{5}+7\sqrt{20}$

Solution

1. $2\sqrt{3}+5\sqrt{3}$

Since the irrational number $\sqrt{3}$ is in both terms, these can be added.

$2\sqrt{3}+5\sqrt{3}=\left(2+5\right)\sqrt{3}=7\sqrt{3}$

2. $7\pi +2\sqrt{5}$

These do not have the same irrational number, so they cannot be added in exact form.

3. $9\sqrt{2}-4\sqrt{2}$

Since the irrational number $\sqrt{2}$ is in both terms, these can be subtracted.

$9\sqrt{2}-4\sqrt{2}=\left(9-4\right)\sqrt{2}=5\sqrt{2}$

4. $\sqrt{5}+7\sqrt{20}$

At first glance, these look like they cannot be added, but 20 is divisible by 5 leaving a perfect square factor of 4. Also, when there is no visible coefficient, assume that the coefficient is 1. Rewrite the sum and then try again.

$\sqrt{5}+7\sqrt{20}=1\sqrt{5}+7\sqrt{4\times 5}=1\sqrt{5}+7\sqrt{4}\sqrt{5}=1\sqrt{5}+\left(7\times 2\right)\sqrt{5}=1\sqrt{5}+14\sqrt{5}=\left(1+14\right)\sqrt{5}=15\sqrt{5}$

	$two multiplication problems. The first one has 0.3 with a bracket under showing 1 decimal place multiplied by 0.7 with a bracket under showing 1 decimal place. The second one has 0.2 with a bracket under showing 1 decimal place multiplied by 0.46 with a bracket under showing 2 decimal places.$
Convert to fractions.	\(\dfrac{3}{10} \cdot \dfrac{7}{10}\)	\(\dfrac{2}{10} \cdot \dfrac{46}{100}\)
Multiply.	\(\dfrac{21}{100}\)	\(\dfrac{92}{1000} \)
Convert to decimals.	$The solutions for both columns. The first solution is 0.21. This has a bracket under showing 2 decimal places. The second solution is 0.092 which has a bracket under showing 3 decimal places.$

	\((−3.9)(4.075)\)
The signs are different. The product will be negative.
Write in vertical format, lining up the numbers on the right.	$Using vertical format to multiply two numbers. 4.075 is on top with the multiplication symbol and 3.9 on the next line. The 5 and 9 are lined up. There is a line under 3.9 to show that the solution belongs under that line.$
Multiply.	$4.075 time 3.9 with the solution bar is repeated from the image above. Under the solution bar is the solution from multiplying 9 time 4.075, 36675. Below that solution is the solution from multipling 3 times 4.075, 12225 with a blank space at the end. The sum of those two numbers are listed below another solution bar. The sum is 158925.$
Add the number of decimal places in the factors \((1 + 3)\). $.$ Place the decimal point 4 places from the right.	$The solution from above is shown again, this time there is a decimal after the 15: 15.8925. A bracket is below the decimal part stating 4 decimal places.$
The signs are different, so the product is negative.	\((−3.9)(4.075) = −15.8925\)

	\(−25.65\div (−0.06)\)
The signs are the same.	The quotient is positive.
Make the divisor a whole number by “moving” the decimal point all the way to the right.
“Move” the decimal point in the dividend the same number of places.	$0.06 outside the dividing bracket with an arrow pointing to the right below the .06. 2.65 is under the dividing bracket with an arrow pointing to the right below the .65$
Divide. Place the decimal point in the quotient above the decimal point in the dividend.
Write the quotient with the appropriate sign.	\(−25.65\div (−0.06) = 427.5\)