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Transcript
MTH 232 Section 7.1 Decimals and Real Numbers Objectives 1. Define decimal numbers and represent them using manipulatives; 2. Write decimals in expanded form (with and without exponents) 3. Express terminating and repeating decimals as fractions. Definition • A decimal is a base-ten positional numeral, either positive or negative, in which there are finitely many digits to a left of a point (called the decimal point) that represent units (ones), tens, hundreds, and so on, and a finite or infinite sequence of digits to the right of the decimal point that represent tenths, hundredths, thousandths, and so on. The Big Idea • Place values to the left of the decimal point represent increasingly large powers of 10: The Big Idea (Continued) • Place values to the right of the decimal point represent divisions of 1 into increasingly large powers of 10: Another Representation • Decimals can also be represented, in a somewhat limited way, by using dollar coins, dimes, and pennies: • 10 pennies = 1 dime • 10 dimes = 1 dollar coin • Unfortunately, in this overly simplified representation, nickels and quarters have no place (value). Expanded Notation • When working with whole numbers, we use place value to expand into increasing detailed notations: 674 600 70 4 (6 100) (7 10) (4 1) 6 10 7 10 4 10 2 1 0 Continued • The same strategy can be applied to a decimal number: 38.12 30 8 0.1 0.02 3 10 8 1 1 0.1 2 0.01 1 1 3 10 8 1 1 2 10 100 1 1 1 0 3 10 8 10 1 1 2 2 10 10 3 10 8 10 110 2 10 1 0 1 2 Types of Decimal Numbers 1. Decimal numbers that terminate, or end. 2. Decimal numbers that do not terminate and have a digit or series of digits that repeat forever. 3. Decimal numbers that do not terminate but do not have a digit or series of digits that repeat forever. Terminating Decimals • Terminating decimals can be written as fractions by adding the fractions associated with each place value: 0.79 0.7 0.09 7 0.1 9 0.01 1 1 7 9 10 100 7 9 10 100 70 9 100 100 79 100 Repeating Decimals • Repeating decimals can be written as fractions by algebraic manipulation of the repeating digit or digits. • Recall that multiplying by 10 will effectively move the decimal point in a number one place to the right: 10 0.5555.... 5.5555.... Continued • Let x = 0.5555….. • Then 10x = 5.5555…. 10 x x 5.5555..... 0.5555..... 9x 5 9x 5 9 9 5 x 9 Non-terminating, Non-repeating Decimals • Decimals that do not terminate but also do not repeat cannot be written as fractions. • These decimal numbers are called irrational numbers. • The most commonly-referenced irrational number is pi: Pi, to 224 Decimal Places