5: Decimals
- Page ID
- 114906
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Gasoline price changes all the time. They might go down for a period of time, but then they usually rise again. One thing that stays the same is that the price is not usually a whole number. Instead, it is shown using a decimal point to describe the cost in dollars and cents. We use decimal numbers all the time, especially when dealing with money. In this chapter, we will explore decimal numbers and how to perform operations using them.
- 5.1: Introduction to Decimals
- Decimals are another way of writing fractions whose denominators are powers of ten. To convert a decimal number to a fraction or mixed number, look at the number to the left of the decimal. If it is zero, the decimal converts to a proper fraction. If not, the decimal converts to a mixed number. The numbers to right of the decimal point become the numerator while the place value corresponding to the final digit represent to the denominator. Finally, simplify the fraction if possible.
- 5.2: Decimals
- Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line. To round a decimal, locate the given place value and mark it with an arrow. Underline the digit to the right of the place value and determine if it is greater than or equal to 5. If it is, add one to the digit in the given place value. If not, don't change the digit. Finally, rewrite the number, removing all digits to the right of the given place value.
- 5.3: Decimal Operations
- To add or subtract decimals, write the numbers vertically so the decimal points line up. Use zeros for place holders, as needed. Then, add or subtract the numbers as if they were whole numbers. Lastly, place the decimal in the answer under the decimal points in the given numbers. Multiplying decimals is like multiplying whole numbers—we just have to determine where to place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.
- 5.4: Decimals and Fractions
- To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction. To add a fraction and a decimal, we would need to either convert the fraction to a decimal or the decimal to a fraction. To compare a decimal to a fraction, we will first convert the fraction to a decimal and then compare the decimals. A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly.
- 5.5: Solve Equations with Decimals
- Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family’s budget, or planning for the future of your business, you’ll be solving equations with decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal.
- 5.6: Averages and Probability
- The mean is often called the arithmetic average. It is computed by dividing the sum of the values by the number of values. The median of a set of data values is the middle value. So, half of the data values are less than or equal to the median while the other half of the data values are greater than or equal to the median. The mode of a set of numbers is the number with the highest frequency. The frequency is the number of times a number occurs.
- 5.7: Ratios and Rate
- A ratio compares two numbers or two quantities that are measured with the same unit. When a ratio is written in fraction form, the fraction should be simplified. To find the ratio of two measurements, we must make sure the quantities have been measured with the same unit. If the measurements are not in the same units, we must first convert them to the same units. A rate compares two quantities of different units. A rate is usually written as a fraction.
- 5.8: Simplify and Use Square Roots
- If m is the square of n, then n is the square root of m and m is the product of a number n multiplied by itself. If n is a whole number, then m is a perfect square. Any positive number squared is positive, and any negative number squared is also positive. When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. The radical sign stands for the positive square root, which is also called the principal square root.
Figure 5.1 - The price of a gallon of gasoline is written as a decimal number. (credit: Mark Turnauckus, Flickr)
Contributors and Attributions
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."