Properties. Comparing and Ordering Rational Numbers Using a Number Line
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1 Chapter 5 Summary Key Terms natural numbers (counting numbers) (5.1) whole numbers (5.1) integers (5.1) closed (5.1) rational numbers (5.1) irrational number (5.2) terminating decimal (5.2) repeating decimal (5.2) bar notation (5.2) real number (5.3) Venn diagram (5.3) closure (5.3) Properties Additive Identity (5.3) Multiplicative Identity (5.3) Additive Inverse (5.3) Multiplicative Inverse (5.3) Commutative Property of Addition (5.3) Commutative Property of Multiplication (5.3) Associative Property of Addition (5.3) Associative Property of Multiplication (5.3) Reflexive Property of Symmetric Property of Transitive Property of Comparing and Ordering Rational Numbers Using a Number Line Fractions and decimals can be compared and ordered by converting fractions to decimals and plotting on a number line. The number line is used to show The fraction 5 5 is equal to Chapter 5 Summary 305
2 Performing Operations with Rational Numbers A rational number is a number that can be written in the form a, where a and b are both b integers and b is not equal to 0. You can add, subtract, multiply, and divide rational numbers in much the same way that you do using integers Identifying Terminating and Repeating Decimals A terminating decimal is a decimal that has a last digit. A repeating decimal is a decimal with digits that repeat in sets of one or more. Two different notations are used to represent repeating decimals. One notation is to write the decimal, including one set of digits that repeat, and place a bar over the repeating digits. Another notation is to write the decimal, including two sets of the digits that repeat, and using dots to indicate repetition. s 8 is a terminating decimal: and 4 is a repeating decimal: ) ) Chapter 5 The Real Number System
3 Writing Repeating Decimals as Fractions Some repeating decimals represent common fractions, such as , and are 3 used often enough that we recognize the fraction by its decimal representation. However, there are decimals in which it is difficult to determine the fractional equivalent. To determine the fraction for the decimal, first, write an equation by setting the decimal equal to a variable that will represent the fraction. Next, write another equation by multiplying each side of the equation by a power of 10. The exponent on the power of 10 is equal to the number of decimal places until the decimal begins to repeat. Then, subtract the first equation from the second equation. Finally, solve the equation. The repeating decimal is equal to the fraction w w w w w 5 15 w Identifying Irrational Numbers Decimals that do not repeat and do not terminate are said to be irrational numbers. An irrational number is a number that cannot be written in the form a, where a and b are both b integers and b fi 0. An example of an irrational number is 11 because it is a square root that is not a perfect square and therefore has no repeating patterns of digits. Chapter 5 Summary 307
4 Classifying Numbers in the Real Number System Combining the set of rational numbers and the set of irrational numbers produces the set of real numbers. Within the set of rational numbers, a number can be or not be an integer, whole number, natural number, or some combination. s π is an irrational number. 28 is a rational number and an integer. 23 is a natural number, whole number, integer, and rational number. 1 is a rational number. 4 Understanding the Properties of Real Numbers The real numbers, together with their operations and properties, form the real number system. The properties of real numbers include: Closure: A set of numbers is said to be closed under an operation if the result of the operation on two numbers in the set is another member of the set. Additive Identity: An additive identity is a number such that when you add it to a second number, the sum is equal to the second number. Multiplicative Identity: A multiplicative identity is a number such that when you multiply it by a second number, the product is equal to the second number. Additive Inverse: Two numbers are additive inverses if their sum is the additive identity. Multiplicative Inverse: Two numbers are multiplicative inverses if their product is the multiplicative identity. Commutative Property of Addition: Changing the order of two or more addends in an addition problem does not change the sum. Commutative Property of Multiplication: Changing the order of two or more factors in a multiplication problem does not change the product. Associative Property of Addition: Changing the grouping of the addends in an addition problem does not change the sum. 308 Chapter 5 The Real Number System
5 Associative Property of Multiplication: Changing the grouping of the factors in a multiplication problem does not change the product. Reflexive Property of Equality: For any real number a, a 5 a. Symmetric Property of Equality: For any real numbers a and b, if a 5 b, then b 5 a. Transitive Property of Equality: For any real numbers a, b, and c, if a 5 b and b 5 c, then a 5 c. s (2128) 5 0 shows the additive inverse (27) shows the commutative property of multiplication shows the multiplicative identity. (31 3 x) 1 y (x 1 y) shows the associative property of addition. If x y and 7 1 y 5 21, then x 5 21 shows the transitive property of equality. Chapter 5 Summary 30
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