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Intermediate Algebra

1.4 Decimals

Intermediate Algebra1.4 Decimals
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Round decimals
  • Add and subtract decimals
  • Multiply and divide decimals
  • Convert decimals, fractions, and percents
  • Simplify expressions with square roots
  • Identify integers, rational numbers, irrational numbers, and real numbers
  • Locate fractions and decimals on the number line
Be Prepared 1.4

A more thorough introduction to the topics covered in this section can be found in the Elementary Algebra chapter, Foundations.

Round Decimals

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

0.1=110is “one tenth” 0.01=1100is “one hundredth” 0.001=11000is “one thousandth” 0.0001=110,000is “one ten-thousandth”0.1=110is “one tenth” 0.01=1100is “one hundredth” 0.001=11000is “one thousandth” 0.0001=110,000is “one ten-thousandth”

Just as in whole numbers, each digit of a decimal corresponds to the place value based on the powers of ten. Figure 1.6 shows the names of the place values to the left and right of the decimal point.

This table is labeled place value and has 12 columns. The seventh column is blank. Starting from here and going left the columns are labeled: ones, tens, hundreds, thousands, ten thousands, hundred thousands. Starting from the blank column and going right the columns are labeled: tenths, hundredths, thousandths, ten thousandths hundred thousandths. There is a dot under the blank column.
Figure 1.6

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.

How To

Round decimals.

  1. Step 1. Locate the given place value and mark it with an arrow.
  2. Step 2. Underline the digit to the right of the place value.
  3. Step 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. Step 4. Rewrite the number, deleting all digits to the right of the rounding digit.

Example 1.34

Round 18.379 to the nearest hundredth tenth whole number.

Try It 1.67

Round 6.5826.582 to the nearest hundredth tenth whole number.

Try It 1.68

Round 15.217515.2175 to the nearest thousandth hundredth tenth.

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.

How To

Add or subtract decimals.

  1. Step 1. Determine the sign of the sum or difference.
  2. Step 2. Write the numbers so the decimal points line up vertically.
  3. Step 3. Use zeros as placeholders, as needed.
  4. Step 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. Step 5. Write the sum or difference with the appropriate sign.

Example 1.35

Add or subtract: −23.541.38−23.541.38 14.6520.14.6520.

Try It 1.69

Add or subtract: −4.811.69−4.811.69 9.5810.9.5810.

Try It 1.70

Add or subtract: −5.12318.47−5.12318.47 37.4250.37.4250.

Multiply and Divide Decimals

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.

How To

Multiply decimals.

  1. Step 1. Determine the sign of the product.
  2. Step 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. Step 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. Step 4. Write the product with the appropriate sign.

Example 1.36

Multiply: (−3.9)(4.075).(−3.9)(4.075).

Try It 1.71

Multiply: −4.5(6.107).−4.5(6.107).

Try It 1.72

Multiply: −10.79(8.12).−10.79(8.12).

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.

How To

Multiply a decimal by a power of ten.

  1. Step 1. Move the decimal point to the right the same number of places as the
    number of zeros in the power of 10.
  2. Step 2. Add zeros at the end of the number as needed.

Example 1.37

Multiply: 5.63 by 10 100 1000.

Try It 1.73

Multiply 2.58 by 10 100 1000.

Try It 1.74

Multiply 14.2 by 10 100 1000.

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

In the expression a divided by b equals c, a is the dividend, b is the divisor and c is the quotient. This can be written as b right parentheses a overbar, with c on top of the bar. In this case too, a is the dividend, b is the divisor and c is the quotient.

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

How To

Divide decimals.

  1. Step 1. Determine the sign of the quotient.
  2. Step 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. Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
  4. Step 4. Write the quotient with the appropriate sign.

Example 1.38

Divide: −25.65÷(−0.06).−25.65÷(−0.06).

Try It 1.75

Divide: −23.492÷(−0.04).−23.492÷(−0.04).

Try It 1.76

Divide: −4.11÷(−0.12).−4.11÷(−0.12).

Convert Decimals, Fractions, and Percents

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 0.03.0.03. the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.

0.03=31000.03=3100

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

How To

Convert a decimal to a proper fraction and a fraction to a decimal.

  1. Step 1. To convert a decimal to a proper fraction, determine the place value of the final digit.
  2. Step 2. Write the fraction.
    • numerator—the “numbers” to the right of the decimal point
    • denominator—the place value corresponding to the final digit
  3. Step 3. To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.

Example 1.39

Write: 0.3740.374 as a fraction 5858 as a decimal.

Try It 1.77

Write: 0.2340.234 as a fraction 7878 as a decimal.

Try It 1.78

Write: 0.0240.024 as a fraction 3838 as a decimal.

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.

Figure shows the value 6 percent. An arrow indicates that the decimal is moved two places to the left. Hence the value is equal to 0.06. Similarly, 78 percent is 0.78, 2.7 percent is 0. 027 and 135 percent is 1.35.

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.

Figure shows value 0.05. An arrow indicates that the decimal is moved two places to the right. Hence the value becomes 5 percent. Similarly, 0.83 is 83 percent, 1.05 is 105 percent, 0.075 is 7.5 percent and 0.3 is 30 percent.

How To

Convert a percent to a decimal and a decimal to a percent.

  1. Step 1. To convert a percent to a decimal, move the decimal point two places to the left after removing the percent sign.
  2. Step 2. To convert a decimal to a percent, move the decimal point two places to the right and then add the percent sign.

Example 1.40

Convert each:

percent to a decimal: 62%, 135%, and 13.7%.

decimal to a percent: 0.51, 1.25, and 0.093.

Try It 1.79

Convert each:

percent to a decimal: 9%, 87%, and 3.9%.

decimal to a percent: 0.17, 1.75, and 0.0825.

Try It 1.80

Convert each:

percent to a decimal: 3%, 91%, and 8.3%.

decimal to a percent: 0.41, 2.25, and 0.0925.

Simplify Expressions with Square Roots

Remember that when a number nn is multiplied by itself, we write n2n2 and read it “nn squared.” The result is called the square of a number n. For example, 8282 is read “8 squared” and 64 is called the square of 8. Similarly, 121 is the square of 11 because 112112 is 121. It will be helpful to learn to recognize the perfect square numbers.

Square of a number

If n2=m,n2=m, then m is the square of n.

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.

(−3)2=9(−8)2=64(−11)2=121(−15)2=225(−3)2=9(−8)2=64(−11)2=121(−15)2=225

Because 102=100,102=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.

Square Root of a Number

If n2=m,n2=m, then n is a square root of m.

Notice (−10)2=100(−10)2=100 also, so −10−10 is also a square root of 100. Therefore, both 10 and −10−10 are square roots of 100. So, every positive number has two square roots—one positive and one negative. The radical sign, mm, 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.

Square Root Notation

mm is read “the square root of mm.”

Figure shows the expression square root of m. The square root sign is labeled radical sign and m is labeled radicand.

If m=n2,m=n2, then m=n,m=n, for n0.n0.

The square root of m, m,m, is the positive number whose square is m.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write 100=10.100=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, 100=−10.100=−10. We read 100100 as “the opposite of the principal square root of 10.”

Example 1.41

Simplify: 2525 121121 144.144.

Try It 1.81

Simplify: 3636 169169 225.225.

Try It 1.82

Simplify: 1616 196196 100.100.

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

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?

Counting numbers1,2,3,4,.. Whole numbers0,1,2,3,4,. Integers.3,−2,−1,0,1,2,3,. Counting numbers1,2,3,4,.. Whole numbers0,1,2,3,4,. Integers.3,−2,−1,0,1,2,3,.

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 −1.2684−1.2684) is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction. The decimal for 1313 is the number 0.3.0.3. 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 pq,pq, where p and q are integers and q0,q0, and as a decimal that stops or repeats.

Rational Number

A rational number is a number of the form pq,pq, where p and q are integers and q0.q0.

Its decimal form stops or repeats.

Are there any decimals that do not stop or repeat? Yes! The number ππ (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.

π=3.141592654...π=3.141592654...

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

A numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call this an irrational number.

Irrational Number

An irrational number is a number that cannot be written as the ratio of two integers.

Its decimal form does not stop and does not repeat.

Let’s summarize a method we can use to determine whether a number is rational or irrational.

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.

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.

Real Number

A real number is a number that is either rational or irrational.

Later in this course we will introduce numbers beyond the real numbers. Figure 1.7 illustrates how the number sets we’ve used so far fit together.

A chart shows that counting numbers 1, 2, 3 are a part of whole numbers 0, 1, 2, 3. Whole numbers are a part of integers minus 2, minus 1, 0, 1, 2. Integers are a part of rational numbers. Rational numbers along with irrational numbers form the set of real numbers.
Figure 1.7 This chart shows the number sets that make up the set of real numbers.

Does the term “real numbers” seem strange to you? Are there any numbers that are not “real,” and, if so, what could they be? Can we simplify −25?−25? Is there a number whose square is −25?−25?

()2=−25?()2=−25?

None of the numbers that we have dealt with so far has a square that is −25.−25. Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to −25.−25. The square root of a negative number is not a real number.

Example 1.42

Given the numbers −7,145,8,5,5.9,64,−7,145,8,5,5.9,64, list the whole numbers integers rational numbers irrational numbers real numbers.

Try It 1.83

Given the numbers −3,2,0.3,95,4,49,−3,2,0.3,95,4,49, list the whole numbers integers rational numbers
irrational numbers real numbers.

Try It 1.84

Given numbers 25,38,−1,6,121,2.041975...,25,38,−1,6,121,2.041975..., list the whole numbers integers rational numbers irrational numbers real numbers.

Locate Fractions and Decimals on the Number Line

We now want to include fractions and decimals on the number line. Let’s start with fractions and locate 15,45,3,74,92,−515,45,3,74,92,−5 and 8383 on the number line.

We’ll start with the whole numbers 3 and −5−5 because they are the easiest to plot. See Figure 1.8.

The proper fractions listed are 1515 and 45.45. We know the proper fraction 1515 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 15,25,35,45.15,25,35,45. We plot 15.15.

Similarly, 4545 is between 0 and −1.−1. After dividing the unit into 5 equal parts we plot 45.45.

Finally, look at the improper fractions 74,92,83.74,92,83. Locating these points may be easier if you change each of them to a mixed number.

74=13492=−41283=22374=13492=−41283=223

Figure 1.8 shows the number line with all the points plotted.

Figure shows a number line with numbers ranging from minus 6 to 6. Various points on the line are highlighted. From left to right, these are: minus 5, minus 9 by 2, minus 4 by 5, 1 by 5, 4 by 5, 8 by 3 and 3.
Figure 1.8

Example 1.43

Locate and label the following on a number line: 4,34,14,−3,65,52,4,34,14,−3,65,52, and 73.73.

Try It 1.85

Locate and label the following on a number line: −1,13,65,74,92,5,83.−1,13,65,74,92,5,83.

Try It 1.86

Locate and label the following on a number line: −2,23,75,74,72,3,73.−2,23,75,74,72,3,73.

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

Example 1.44

Locate on the number line: 0.4 −0.74.−0.74.

Try It 1.87

Locate on the number line: 0.60.6 −0.25.−0.25.

Try It 1.88

Locate on the number line: 0.90.9 −0.75.−0.75.

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with decimals.

Section 1.4 Exercises

Practice Makes Perfect

Round Decimals

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

239.

5.781

240.

1.638

241.

0.299

242.

0.697

243.

63.479

244.

84.281

Add and Subtract Decimals

In the following exercises, add or subtract.

245.

−16.5324.38−16.5324.38

246.

−19.4732.58−19.4732.58

247.

−38.69+31.47−38.69+31.47

248.

−29.83+19.76−29.83+19.76

249.

72.510072.5100

250.

86.210086.2100

251.

91.75(−10.462)91.75(−10.462)

252.

94.69(−12.678)94.69(−12.678)

253.

55.013.755.013.7

254.

59.084.659.084.6

255.

2.517.42.517.4

256.

3.846.13.846.1

Multiply and Divide Decimals

In the following exercises, multiply.

257.

94.69(−12.678)94.69(−12.678)

258.

(−8.5)(1.69)(−8.5)(1.69)

259.

(−5.18)(−65.23)(−5.18)(−65.23)

260.

(−9.16)(−68.34)(−9.16)(−68.34)

261.

(0.06)(21.75)(0.06)(21.75)

262.

(0.08)(52.45)(0.08)(52.45)

263.

(9.24)(10)(9.24)(10)

264.

(6.531)(10)(6.531)(10)

265.

(0.025)(100)(0.025)(100)

266.

(0.037)(100)(0.037)(100)

267.

(55.2)(1000)(55.2)(1000)

268.

(99.4)(1000)(99.4)(1000)

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

269.

$117.25÷48$117.25÷48

270.

$109.24÷36$109.24÷36

271.

1.44÷(−0.3)1.44÷(−0.3)

272.

−1.15÷(−0.05)−1.15÷(−0.05)

273.

5.2÷2.55.2÷2.5

274.

14÷0.3514÷0.35

Convert Decimals, Fractions and Percents

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

275.

0.040.04

276.

1.464

277.

0.0950.095

278.

−0.375−0.375

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

279.

17201720

280.

174174

281.

3102531025

282.

18111811

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

283.

71%71%

284.

150%150%

285.

39.3%39.3%

286.

7.8%7.8%

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

287.

1.561.56

288.

3

289.

0.06250.0625

290.

2.2542.254

Simplify Expressions with Square Roots

In the following exercises, simplify.

291.

6464

292.

169169

293.

144144

294.

44

295.

100100

296.

121121

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.

297.

−8,0,1.95286...,125,36,9−8,0,1.95286...,125,36,9

298.

−9,−349,9,0.409,116,7−9,−349,9,0.409,116,7

299.

100,−7,83,−1,0.77,314100,−7,83,−1,0.77,314

300.

−6,52,0,0.714285———,215,14−6,52,0,0.714285———,215,14

Locate Fractions and Decimals on the Number Line

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

301.

310,72,116,4310,72,116,4

302.

710,52,138,3710,52,138,3

303.

34,34,123,−123,52,5234,34,123,−123,52,52

304.

25,25,134,−134,83,8325,25,134,−134,83,83

305.

0.80.8 −1.25−1.25

306.

−0.9−0.9 −2.75−2.75

307.

−1.6−1.6 3.253.25

308.

3.13.1 −3.65−3.65

Writing Exercises

309.

How does knowing about U.S. money help you learn about decimals?

310.

When the Szetos sold their home, the selling price was 500% of what they had paid for the house 30 years ago. Explain what 500% means in this context.

311.

In your own words, explain the difference between a rational number and an irrational number.

312.

Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.

Self Check

Use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 columns, 6 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don’t get it. The statements in the first column are: round decimals, add and subtract decimals, multiply and divide decimals, convert decimals, fractions and percents, simplify expressions with square roots, identify integers, rational numbers, irrational numbers and real numbers, locate fractions and decimals on the number line. The remaining columns are blank.

On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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