Decimals

Decimals are another way of writing fractions whose denominators are powers of ten.

Just as in whole numbers, each digit of a decimal corresponds to the place value based on the powers of ten. shows the names of the place values to the left and right of the decimal point.

When we work with decimals, it is often necessary to round the number to the nearest required place value. We summarize the steps for rounding a decimal here.

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.

When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. We multiply the numbers temporarily ignoring the decimal point and then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product. Finally, we write the product with the appropriate sign.

Often, especially in the sciences, you will multiply decimals by powers of 10 (10, 100, 1000, etc). If you multiply a few products on paper, you may notice a pattern relating the number of zeros in the power of 10 to number of decimal places we move the decimal point to the right to get the product.

We review the notation and vocabulary for division:

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

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

In our work, it is often necessary to change the form of a number. We may have to change fractions to decimals or decimals to percent.

We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal \(0.03.\) the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.

The steps to take to convert a decimal to a fraction are summarized in the procedure box.

A percent is a ratio whose denominator is 100. Percent means per hundred. We use the percent symbol, %, to show percent. Since a percent is a ratio, it can easily be expressed as a fraction. Percent means per 100, so the denominator of the fraction is 100. We then change the fraction to a decimal by dividing the numerator by the denominator. After doing this many times, you may see the pattern.

To convert a percent number to a decimal number, we move the decimal point two places to the left.

To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is 100, it is easy to change that fraction to a percent. After many conversions, you may recognize the pattern.

To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Remember that when a number \(n\) is multiplied by itself, we write \({n}^{2}\) and read it “\(n\) squared.” The result is called the square of a number n. For example, \({8}^{2}\) is read “8 squared” and 64 is called the square of 8. Similarly, 121 is the square of 11 because \({11}^{2}\) is 121. It will be helpful to learn to recognize the perfect square numbers.

What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.

Because \({10}^{2}=100,\) we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is m is called a square root of a number m.

Notice \({(-10)}^{2}=100\) also, so \(-10\) is also a square root of 100. Therefore, both 10 and \(-10\) are square roots of 100. So, every positive number has two square roots—one positive and one negative. The radical sign, \(\sqrt{m}\), denotes the positive square root. The positive square root is called the principal square root. When we use the radical sign that always means we want the principal square root.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write \(\sqrt{100}=10.\) If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, \(\text{-}\sqrt{100}=-10.\) We read \(\text{-}\sqrt{100}\) as “the opposite of the principal square root of 100.”

We have already described numbers as counting numbers, whole numbers, and integers. What is the difference between these types of numbers? Difference could be confused with subtraction. How about asking how we distinguish between these types of numbers?

What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

In general, any decimal that ends after a number of digits (such as 7.3 or \(-1.2684\)) is a rational number. Simply write the decimal as a mixed number. The decimal for \(\frac{1}{3}\) is the number \(0.\overset{-}{3}.\) The bar over the 3 indicates that the number 3 repeats infinitely. Continuously has an important meaning in calculus. The number(s) under the bar is called the repeating block and it repeats continuously.

Since all integers can be written as a fraction whose denominator is 1, the integers (and so also the counting and whole numbers. are rational numbers.

Every rational number can be written both as a ratio of integers \(\frac{p}{q},\) where p and q are integers and \(q\ne 0,\) and as a decimal that stops or repeats.

Are there any decimals that do not stop or repeat? Yes! The number \(\pi\) (the Greek letter pi, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat. We use three dots (…) to indicate the decimal does not stop or repeat.

The square root of a number that is not a perfect square is a decimal that does not stop or repeat.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

We now want to include fractions and decimals on the number line. Let’s start with fractions and locate \(\frac{1}{5},-\frac{4}{5},3,\frac{7}{4},-\frac{9}{2},-5\) and \(\frac{8}{3}\) on the number line.

We’ll start with the whole numbers 3 and \(-5\) because they are the easiest to plot. See .

The proper fractions listed are \(\frac{1}{5}\) and \(-\frac{4}{5}.\) We know the proper fraction \(\frac{1}{5}\) has value less than one and so would be located between 0 and 1. The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts \(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}.\) We plot \(\frac{1}{5}.\)

Similarly, \(-\frac{4}{5}\) is between 0 and \(-1.\) After dividing the unit into 5 equal parts we plot \(-\frac{4}{5}.\)

Finally, look at the improper fractions \(\frac{7}{4},\frac{9}{2},\frac{8}{3}.\) Locating these points may be easier if you change each of them to a mixed number.

shows the number line with all the points plotted.

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

• How to round decimals.
  1. Locate the given place value and mark it with an arrow.
  2. Underline the digit to the right of the place value.
  3. Is the underlined digit greater than or equal to \(5?\)
     • Yes: add 1 to the digit in the given place value.
     • No: do not change the digit in the given place value
  4. Rewrite the number, deleting all digits to the right of the rounding digit.
• How to add or subtract decimals.
  1. Determine the sign of the sum or difference.
  2. Write the numbers so the decimal points line up vertically.
  3. Use zeros as placeholders, as needed.
  4. 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.
  5. Write the sum or difference with the appropriate sign
• How to multiply decimals.
  1. Determine the sign of the product.
  2. 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.
  3. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.
  4. Write the product with the appropriate sign.
• How to multiply a decimal by a power of ten.
  1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
  2. Add zeros at the end of the number as needed.
• How to divide decimals.
  1. Determine the sign of the quotient.
  2. 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.
  3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
  4. Write the quotient with the appropriate sign.
• How to convert a decimal to a proper fraction and a fraction to a decimal.
  
  1. To convert a decimal to a proper fraction, determine the place value of the final digit.
  2. Write the fraction.
     • numerator—the “numbers” to the right of the decimal point
     • denominator—the place value corresponding to the final digit
  3. To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.
• How to convert a percent to a decimal and a decimal to a percent.
  1. To convert a percent to a decimal, move the decimal point two places to the left after removing the percent sign.
  2. To convert a decimal to a percent, move the decimal point two places to the right and then add the percent sign.
• Square Root Notation
  \(\sqrt{m}\) is read “the square root of m.”
  If \(m={n}^{2},\) then \(\sqrt{m}=n,\) for \(n\ge 0.\)
  The square root of m, \(\sqrt{m},\) is the positive number whose square is m.
• Rational or Irrational
  If the decimal form of a number
  
  • repeats or stops, the number is a rational number.
  • does not repeat and does not stop, the number is an irrational number.
• Real Numbers

Round Decimals

In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.

Add and Subtract Decimals

In the following exercises, add or subtract.

Multiply and Divide Decimals

In the following exercises, multiply.

In the following exercises, divide. Round money monetary answers to the nearest cent.

Convert Decimals, Fractions and Percents

In the following exercises, write each decimal as a fraction.

In the following exercises, convert each fraction to a decimal.

In the following exercises, convert each percent to a decimal.

In the following exercises, convert each decimal to a percent.

Simplify Expressions with Square Roots

In the following exercises, simplify.

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.

Locate Fractions and Decimals on the Number Line

In the following exercises, locate the numbers on a number line.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Decimals are another way of writing fractions whose denominators are powers of 10.

Notice that “ten thousand” is a number larger than one, but “one ten-thousandth” is a number smaller than one. The “th” at the end of the name tells you that the number is smaller than one.

When we name a whole number, the name corresponds to the place value based on the powers of ten. We read 10,000 as “ten thousand” and 10,000,000 as “ten million.” Likewise, the names of the decimal places correspond to their fraction values. shows the names of the place values to the left and right of the decimal point.

We summarize the steps needed to name a decimal below.

When we write a check we write both the numerals and the name of the number. Let’s see how to write the decimal from the name.

We summarize the steps to writing a decimal.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Rounding decimals is very much like rounding whole numbers. We will round decimals with a method based on the one we used to round whole numbers.

We summarize the steps for rounding a decimal here.

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.

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!

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.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal 0.03 the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.

Notice, when the number to the left of the decimal is zero, we get a fraction whose numerator is less than its denominator. Fractions like this are called proper fractions.

The steps to take to convert a decimal to a fraction are summarized in the procedure box.

We’ve learned to convert decimals to fractions. Now we will do the reverse—convert fractions to decimals. Remember that the fraction bar means division. So \(\frac{4}{5}\) can be written \(4\div 5\) or \(54.\) This leads to the following method for converting a fraction to a decimal.

When we divide, we will not always get a zero remainder. Sometimes the quotient ends up with a decimal that repeats. A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly. A bar is placed over the repeating block of digits to indicate it repeats.

A bar is placed over the repeating block of digits to indicate it repeats.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

• Name a Decimal
  1. Name the number to the left of the decimal point.
  2. Write ”and” for the decimal point.
  3. Name the “number” part to the right of the decimal point as if it were a whole number.
  4. Name the decimal place of the last digit.
• Write a Decimal
  1. Look for the word ‘and’—it locates the decimal point. Place a decimal point under the word ‘and.’ Translate the words before ‘and’ into the whole number and place it to the left of the decimal point. If there is no “and,” write a “0” with a decimal point to its right.
  2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
  3. Translate the words after ‘and’ into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
  4. Fill in zeros for place holders as needed.
• Round a Decimal
  1. Locate the given place value and mark it with an arrow.
  2. Underline the digit to the right of the place value.
  3. Is this digit greater than or equal to 5? Yes—add 1 to the digit in the given place value. No—do not change the digit in the given place value.
  4. Rewrite the number, deleting all digits to the right of the rounding digit.
• Add or Subtract Decimals
  1. Write the numbers so the decimal points line up vertically.
  2. Use zeros as place holders, as needed.
  3. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.
• Multiply Decimals
  1. Determine the sign of the product.
  2. 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.
  3. Place the decimal point. The number of decimal places in the product is the sum of the decimal places in the factors.
  4. Write the product with the appropriate sign.
• Multiply a Decimal by a Power of Ten
  1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
  2. Add zeros at the end of the number as needed.
• Divide Decimals
  1. Determine the sign of the quotient.
  2. 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.
  3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
  4. Write the quotient with the appropriate sign.
• Convert a Decimal to a Proper Fraction
  1. Determine the place value of the final digit.
  2. Write the fraction: numerator—the ‘numbers’ to the right of the decimal point; denominator—the place value corresponding to the final digit.
• Convert a Fraction to a Decimal Divide the numerator of the fraction by the denominator.

Name and Write Decimals

In the following exercises, write as a decimal.

In the following exercises, name each decimal.

Round Decimals

In the following exercises, round each number to the nearest tenth.

In the following exercises, round each number to the nearest hundredth.

In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.

Add and Subtract Decimals

In the following exercises, add or subtract.

Multiply and Divide Decimals

In the following exercises, multiply.

In the following exercises, divide.

Convert Decimals, Fractions and Percents

In the following exercises, write each decimal as a fraction.

In the following exercises, convert each fraction to a decimal.

In the following exercises, convert each percent to a decimal.

In the following exercises, convert each decimal to a percent.

Condensed — the full section is in OpenStax Elementary Algebra 2e.