Section 5'3: The Rational Numbers

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Section 5.3 The Rational Numbers

• Dr. Fred Butler
• Math 121 Fall 2004

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Other Numbers on the Number Line

• The numbers that fall between the integers on the
  number line are either rational or irrational
  numbers.
• rational irrational
• -1 ½ v2
• -4 -3 -2 -1 0 1 2
  3 4

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The Rational Numbers

• The set of rational numbers, denoted by Q, is the
  set of all numbers that can be written in the
  form p/q, where p and q are integers and q?0.
• The following are examples of rational numbers
• 1/3 -3/4 12/7.
• Numbers like these are called fractions. The
  number to the left of (above) the fraction line
  is the numerator, and the number to the right of
  (below) the fraction line is the denominator.

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Every Integer is a Rational Number

• The integers -2 and 0 are also rational numbers
  because -2 can be written as -2/1 and 0 can be
  written as 0/1.
• In fact, every integer can be expressed as a
  rational number.
• We express the integer n as n/1, as above in the
  case of -2 and 0.

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Class Question 3.5

• Which of the following is true? (Only one)
• 1. 2. 3.

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Reducing Fractions

• Sometimes the numerator and the denominator of a
  fraction have a common divisor.
• When this is the case, we can reduce the fraction
  to its lowest terms.
• A fraction is said to be in lowest terms when the
  numerator and denominator are relatively prime
  (have no common factors other than 1).

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Reducing Fractions contd.

• For example the GCD of 6 and 10 is 2, so
• 6/10 (62)/(102)3/5.
• In general we find the GCD of the numerator and
  the denominator, and then divide them both by
  this GCD, to reduce the fraction to lowest terms.
• Last time we found the GCD of 54 and 90 is 18, so
  to reduce 54/90 to lowest terms
• 54/90 (5418)/(9018) 3/5.

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Mixed Numbers

• A mixed number is a number that consists of an
  integer part and a fraction part.
• For example, 2 3/4 is a mixed number. The
  integer part is 2, and the fraction part is 3/4.
• When we write 2 3/4, what we mean is
• 2 3/4.
• The mixed number -4 1/4 means
• (4 1/4).

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Improper Fractions

• An improper fraction is a fraction whose
  numerator is bigger (forgetting or signs)
  than its denominator.
• For example, 8/5 is an improper fraction.
• So is -9/7.

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Mixed Numbers and Improper Fractions on the
Number Line

• Below we show where both mixed numbers and
  improper fractions fall on the number line.
• -4 ¼ - 1 2/7 1 3/5 2
  3/4
• -4 -3 -2 -1 0 1 2
  3 4
• -17/4 -9/7 8/5
  11/4

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Converting a Mixed Number to an Improper Fraction

• If the number is negative, omit the sign (for
  now).
• Multiply the denominator of the fraction part by
  the integer part.
• Add the product obtained in step 2 to the
  numerator of the fraction part.
• The numerator is the number obtained in step 3,
  and the denominator is the denominator of the
  fraction part.
• If the original number was negative, put the
  negative sign back (if it wasnt, dont do
  anything).

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Mixed Number to Improper Fraction Example

• Lets use this method to convert 3 5/6 to an
  improper fraction
• 3 5/6 (6x35)/6 (185)/6 23/6.
• For a negative mixed number -2 1/3, we convert 2
  1/3 first
• 2 1/3 (3x21)/3 (61)/3 7/3,
• then we add back the negative. Our final answer
  is -7/3.

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Converting an Improper Fraction to a Mixed Number

• If the number is negative, omit the sign (for
  now).
• Divide the numerator by the denominator.
  Identify the quotient and the remainder.
• The quotient obtained in step 2 is the integer
  part.
• The numerator of the fraction part is the
  remainder obtained in step 2, and the denominator
  is the denominator of the original improper
  fraction.
• If the original number was negative, put the
  negative sign back (if it wasnt, dont do
  anything).

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Improper Fraction to Mixed Number Example

• Lets use this method to convert 21/5 to a mixed
  number
• 215 4 remainder 1, so
• 21/5 4 1/5.
• For a negative improper fraction -13/7, we
  convert 13/7 first
• 137 1 remainder 6, so
• 13/7 1 6/7,
• then we add back the negative. Our final answer
  is -1 6/7.

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Fractions as Decimals

• Every rational number, when expressed as a
  decimal number, will be either a terminating or a
  repeating decimal number.
• Examples of terminating decimal numbers are
• 0.5 0.75 0.9875213 4.125.
• Examples of repeating decimal numbers are
• 0.3333 0.232323 8.13456456456 .
• We represent repeating digits with a bar over
  top, so 0.333 is written and 8.13456456
  is written

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The Places in a Decimal (Put on Board)

• Consider the decimal number below, with the each
  decimal place named.
• 0 . 1 2 1 4 6 8
• units hundredths ten-thousandths
  millionths
• tenths thousandths
  hundred-thousandths

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Converting a Terminating Decimal to a Fraction

• We use the place names from the previous slide to
  determine the denominator of the fractional form
  of a terminating decimal number.
• 0.4 ends in the tenths place, so as a fraction
  0.4 is 4/10.
• 0.062 ends in the thousandths place, so it is
  written as a fraction as 62/1000.

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Converting a Repeating Decimal to a Fraction

• Lets convert to a fraction.
• Note that
• Lets suppose n is the fraction equal to
• so 100xn
• Then

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Converting a Repeating Decimal to a Fraction
contd.

• From the previous slide we have that the fraction
  n which is equal to the repeating decimal
  satisfies the equation
• 99xn35.
• So if we divide both sides of the equation by 99
  we get our final answer,
• n35/99.
• In general, if there is one repeating digit we
  multiply our repeating decimal by 10, two
  repeating digits by 100, three by 1000, etc., and
  then repeat the procedure we just did.

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Multiplication of Fractions

• We multiply two fractions by the following
  formula (note that b?0 and d?0)
• (a/b)x(c/d)(axc)/(bxd)ac/bd.
• For example,
• (-2/3)x(4/7)(-2)x4/(3x7)-8/21.
• To multiply mixed numbers, first convert to
  improper fractions.
• For example, 1 7/8 15/8 and 2 ¼ 9/4, so
• (1 7/8)x(2 ¼) (15/8)x(9/4)(15x9)/(8x4)135/32
  4 7/32.

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Reciprocals

• The reciprocal of any number is 1 divided by the
  number.
• The product of a number and its reciprocal must
  equal 1.
• The reciprocal of a/b is always b/a (since
  (a/b)x(b/a)ab/ab1).

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Division of Fractions

• To find the quotient of two fractions, multiply
  the first fraction by the reciprocal of the
  second fraction.
• That is (note again that b?0 and d?0),
• (a/b) (c/d) (a/b)x(d/c) ad/bc.
• For example
• (-3/5)(5/7) (-3/5)x(7/5) -21/25.

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Lowest Common Denominator

• Before we can add or subtract fractions, the
  fractions must have a common denominator.
• A common denominator is another name for a common
  multiple of the denominators.
• The lowest common denominator is the least common
  multiple of the denominators.

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Addition and Subtraction of Fractions with a
Common Denominator

• To add two fractions with a common denominator c
  ?0, we use the formula
• (a/c) (b/c) (ab)/c,
• and to subtract we use the formula
• (a/c)-(b/c) (a-b)/c.
• So for example
• (3/8) (2/8) (32)/8 5/8
• and
• (15/16) (7/16) (15-7)/16 8/16 1/2.

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Fundamental Law of Rational Numbers

• If a, b, and c are integers with b?0 and c?0,
  then
• a/b (a/b)x(c/c) ac/bc.
• So for example
• 5/7 (5x3)/(7x3) 15/21.
• Two such fractions a/b and (axc)/(bxc) are called
  equivalent fractions.

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Adding and Subtracting Fractions with Unlike
Denominators

• When we add two fractions with unlike
  denominators, recall we said we have to have a
  common denominator, preferably the lowest common
  denominator.
• To add two fractions once we find the LCD, we
  need to find fractions that are equivalent to
  each of the fractions we want to add with
  denominator equal to the LCD.

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An Example of Adding and Subtracting Fractions
with Unlike Denominators

• Say we want to add (1/54)(1/90).
• The LCD is 270 in this case, and we see that
  27054x5 and 27090x3.
• Thus
• (1/54)(1/90)(1x5)/(54x5)(1x3)/(90x3)
• (5/270)(3/270)8/2704/135.
• Similarly,
• (5/12) (3/10) (5x5)/(12x5) (3x6)/(10x6)
• (25/60)-(18/60) 7/60.

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Lecture Summary

• The set of rational numbers Q is the set of all
  numbers that can be written in the form p/q,
  where p and q are integers and q?0.
• Every rational number, when expressed as a
  decimal number, will be either a terminating or a
  repeating decimal number.
• We can perform addition, subtraction,
  multiplication, and division with rational
  numbers in order to add or subtract, we must
  have a common denominator.

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Homework

• Do problems from Section 5.3 of text.
• Print lecture notes.