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Chapter 2: Properties of Real Numbers

Difficulty Level: Basic Created by: CK-12

Introduction

Integers and Rational Numbers

Integers and rational numbers are important in daily life. The price per square yard of carpet is a rational number. The number of frogs in a pond is expressed using an integer. The organization of real numbers can be drawn as a hierarchy. Look at the hierarchy below.

Real numbers are all around us. The majority of numbers calculated are considered real numbers. This chapter defines a real number and explains important properties and rules that apply to real numbers. The most generic number is the real number; it can be a combination of negative, positive, decimal, fractional, or non-repeating decimal values. Real numbers have two major categories: rational numbers and irrational numbers. Irrational numbers are non-repeating, non-terminating decimals such as \begin{align*}\pi\end{align*} or \begin{align*}\sqrt{2}\end{align*}. In this chapter, you will learn how to perform operations and solve problems involving rational numbers and integers, which are a special kind of rational number.

Chapter Outline

Summary

This chapter talks about the properties of rational numbers and how to add, subtract, multiply, and divide them. It also shows how to add integers and covers additive inverses, absolute values, reciprocals, and the distributive property. Square roots and irrational numbers are discussed as well, and how to classify and order real numbers is explained in detail. Finally, many real-world problems are given, and the chapter concludes by going over the guess and check and working backward problem-solving strategies.

Properties of Real Numbers Review

Compare the real numbers.

1. 7 and –11
2. \begin{align*}\frac{4}{5}\end{align*} and \begin{align*}\frac{11}{16}\end{align*}
3. \begin{align*}\frac{10}{15}\end{align*} and \begin{align*}\frac{2}{3}\end{align*}
4. 0.985 and \begin{align*}\frac{31}{32}\end{align*}
5. –16.12 and \begin{align*}\frac{-300}{9}\end{align*}

Order the real numbers from least to greatest.

1. \begin{align*}\frac{8}{11}, \frac{7}{10}, \frac{5}{9}\end{align*}
2. \begin{align*}\frac{2}{7}, \frac{1}{11}, \frac{8}{13}, \frac{4}{7}, \frac{8}{9}\end{align*}

Graph these values on the same number line.

1. \begin{align*}3\frac{1}{3}\end{align*}
2. –1.875
3. \begin{align*}\frac{7}{8}\end{align*}
4. \begin{align*}0.1\bar{6}\end{align*}
5. \begin{align*}\frac{-55}{5}\end{align*}

Simplify by applying the Distributive Property.

1. \begin{align*}6n(-2+5n)-n(-3n-8)\end{align*}
2. \begin{align*}7x+2(-6x+2)\end{align*}
3. \begin{align*}-7x(x+5)+3(4x-8)\end{align*}
4. \begin{align*}-3(-6r-5)-2r(1+6r)\end{align*}
5. \begin{align*}1+3(p+8)\end{align*}
6. \begin{align*}3(1-5k)-1\end{align*}

Approximate the square root to the nearest hundredth.

1. \begin{align*}\sqrt{26}\end{align*}
2. \begin{align*}\sqrt{330}\end{align*}
3. \begin{align*}\sqrt{625}\end{align*}
4. \begin{align*}\sqrt{121}\end{align*}
5. \begin{align*}\sqrt{225}\end{align*}
6. \begin{align*}\sqrt{11}\end{align*}
7. \begin{align*}\sqrt{8}\end{align*}

Rewrite the square root without using a calculator.

1. \begin{align*}\sqrt{50}\end{align*}
2. \begin{align*}\sqrt{8}\end{align*}
3. \begin{align*}\sqrt{80}\end{align*}
4. \begin{align*}\sqrt{32}\end{align*}

Simplify by combining like terms.

1. \begin{align*}8+b+1-7b\end{align*}
2. \begin{align*}9n+9n+17\end{align*}
3. \begin{align*}7h-3+3\end{align*}
4. \begin{align*}9x+11-x-3+5x+2\end{align*}

Evaluate.

1. \begin{align*}\frac{8}{5}-\frac{4}{3}\end{align*}
2. \begin{align*}\frac{4}{3}-\frac{1}{2}\end{align*}
3. \begin{align*}\frac{1}{6}+ 1 \frac{5}{6}\end{align*}
4. \begin{align*}\frac{-5}{4}\times \frac{1}{3}\end{align*}
5. \begin{align*}\frac{4}{9} \times \frac{7}{4}\end{align*}
6. \begin{align*}-1\frac{5}{7} \times -2\frac{1}{2}\end{align*}
7. \begin{align*}\frac{1}{9} \div -1\frac{1}{3}\end{align*}
8. \begin{align*}\frac{-3}{2} \div \frac{-10}{7}\end{align*}
9. \begin{align*}-3\frac{7}{10} \div 2\frac{1}{4}\end{align*}
10. \begin{align*}1\frac{1}{5}-\left (-3\frac{3}{4}\right )\end{align*}
11. \begin{align*}4 \frac{2}{3}+3\frac{2}{3}\end{align*}
12. \begin{align*}5.4+(-9.7)\end{align*}
13. \begin{align*}(-7.1)+(-0.4)\end{align*}
14. \begin{align*}(-4.79)+(-3.63)\end{align*}
15. \begin{align*}(-8.1)-(-8.9)\end{align*}
16. \begin{align*}1.58-(-13.6)\end{align*}
17. \begin{align*}(-13.6)+12-(-15.5)\end{align*}
18. \begin{align*}(-5.6)-(-12.6)+(-6.6)\end{align*}
19. \begin{align*}19.4+24.2\end{align*}
20. \begin{align*}8.7+3.8+12.3\end{align*}
21. \begin{align*}9.8-9.4\end{align*}
22. \begin{align*}2.2-7.3\end{align*}

List all the categories that apply to the following numbers.

1. 10.9
2. \begin{align*}\frac{-9}{10}\end{align*}
3. \begin{align*}3\pi\end{align*}
4. \begin{align*}\frac{\pi}{2}-\frac{\pi}{2}\end{align*}
5. –21
6. 8

Which property has been applied?

1. \begin{align*}6.78+(-6.78)=0\end{align*}
2. \begin{align*}9.8+11.2+1.2=9.8+1.2+11.2\end{align*}
3. \begin{align*}3a+(4a+8)=(3a+4a)+8\end{align*}
4. \begin{align*}\frac{4}{3}-\left (-\frac{5}{6}\right )=\frac{4}{3}+\frac{5}{6}\end{align*}
5. \begin{align*}(1)j=j\end{align*}
6. \begin{align*}8(11)\left (\frac{1}{8}\right )=8\left (\frac{1}{8}\right )(11)\end{align*}

Solve the real-world situation.

1. Carol has 18 feet of fencing and purchased an additional 132 inches. How much fencing does Carol have?
2. Ulrich is making cookies for a fundraiser. Each cookie requires \begin{align*}\frac{3}{8}\end{align*}-pound of dough. He has 12 pounds of cookie dough. How many cookies can Ulrich make?
3. Herrick bought 11 DVDs at 19.99 each. Use the Distributive Property to show how Herrick can mentally calculate the amount of money he will need. 4. Bagger 288 is a trench digger, which moves at \begin{align*}\frac{3}{8} \ miles/hour\end{align*}. How long will it take to dig a trench 14 miles long? 5. Georgia started with a given amount of money, \begin{align*}a\end{align*}. She spent4.80 on a large latte, $1.20 on an English muffin,$68.48 on a new shirt, and $32.45 for a present. She now has$0.16. How much money, \begin{align*}a\end{align*}, did Georgia have in the beginning?
6. The formula for the area of a square is \begin{align*}A=s^{2}\end{align*}. A square garden has an area of 145 \begin{align*}\text{meters}^{2}\end{align*}. Find the exact length of the garden.

Properties of Real Numbers Test

Simplify by using the Distributive Property.

1. \begin{align*}-3+7(3a-2)\end{align*}
2. \begin{align*}8(3+2q)+5(q+3)\end{align*}

Simplify.

1. \begin{align*}8p-5p\end{align*}
2. \begin{align*}9z+33-2z-15\end{align*}
3. \begin{align*}-\frac{9}{5}\div 2\end{align*}
4. \begin{align*}1\frac{6}{7}\times 5\frac{3}{4}\end{align*}
5. \begin{align*}\frac{1}{2}-3\frac{2}{3}\end{align*}
6. \begin{align*}\frac{3}{14}+\frac{15}{8}\end{align*}
7. \begin{align*}3.5-5-10.4\end{align*}
8. \begin{align*}\frac{1}{6}-(-6.5)\times \frac{6}{5}\end{align*}

Find the exact value of the square root without a calculator.

1. \begin{align*}\sqrt{125}\end{align*}
2. \begin{align*}\sqrt{18}\end{align*}
3. How is the multiplicative inverse different from the additive inverse?
4. A square plot of land has an area of \begin{align*}168 \ miles^2\end{align*}. To the nearest tenth, what is the length of the land?
5. Troy plans to equally divide 228 candies among 16 people. Can this be done? Explain your answer.
6. Laura withdrew $15 from the ATM, wrote a check for$46.78, and deposited her paycheck of $678.12. After her deposit she had$1123.45 in her account. How much money did Laura begin with?
7. Will the area of a circle always be an irrational number? Explain your reasoning.
8. When would you use the Commutative Property of Multiplication? Give an example to help illustrate your explanation.

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Difficulty Level:
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