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Definition of a Rational Number A number is a rational number if and only if it can be a a represented by a pair of integers, , where b ≠ 0 and b b represents the quotient a ÷ b. Rational Number Ideas and Symbols Section 6.1 Mathematics for Elementary School Teachers - 4th Edition O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK Linda Roper 1 Fractions ! A fraction is a symbol, 2 Definition of Equivalent Fractions where a and b are numbers and b ≠ 0. ! Two fractions, ! is the numerator of the fraction and b is the denominator of the fraction. !a , are equivalent fractions if and only if ad = bc. ! Example: 2 4 and , are equivalent fractions since 2· 8 = 4· 4 4 8 !Proper Fraction: the numerator of a fraction is less than the denominator. !Improper Fraction: the numerator of a fraction is greater than or equal to the denominator. 3 4 Fractions (continued) Example Fundamental Law of Fractions Given a fraction a/b and a number c ≠ 0, a/b = ac/bc. Find two fractions equivalent to Include its simplest form, and verify your choices. Solution: Description of the Simplest Form of a Fraction A fraction representing a rational number is in simplest form when the numerator and denominator are both integers that are relatively prime and the denominator is greater than zero. 20 5 10 and are equivalent to . 12 3 6 The fraction 5 is in simplest form. 3 5 Description of a Decimal Expanded Notation ! A decimal is a symbol that uses a base-ten place-value Example: Write the number 423.85 in expanded notation. system with tenths and powers of tenths to represent a number. Thousands 1000 1 Hundreds 100 4 Tens 10 6 Ones 1 . Tenths Hundredths Thousandths 0 . 4 5 8 6 5 . 2 5 3 . 8 7 9 ( ) ( ) ( ) ( ) ( 423.85 = 4 !10 2 + 2 !101 + 3 !10 0 + 8 !10 "1 + 5 !10 "2 7 ) 8 Terminating and Repeating Decimals Changing a fraction to decimal form ! Every rational number can be expressed as a terminating or repeating decimal. ! We can translate any rational number in fraction form into decimal form by dividing. ! Terminating Decimals: A decimal that has a fixed number of places. The remainder eventually becomes 0. The denominator of a fraction converted from a terminating decimal will be a multiple of 2 and/or 5. Examples .5 or .25 ! Repeating Decimals: A decimal that has a digit or group of digits that repeat. Examples .333 or .3636 9 The End Section 6.1 Linda Roper 11 10