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Number Theory Chapter 7: Fractions (pt 1)

Number Theory Chapter 7: Fractions (pt 1). Kaitlyn Haase. Historical Perspective. Egyptians focused on unit fractions included 2/3 (this was the only anomaly). Greeks - used Egyptian fractions, including 2/3 Greek papyri included problems with common fractions & sexagesimal fractions

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Number Theory Chapter 7: Fractions (pt 1)

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  1. Number TheoryChapter 7: Fractions (pt 1) KaitlynHaase

  2. Historical Perspective Egyptians • focused on unit fractions • included 2/3 (this was the only anomaly)

  3. Greeks - used Egyptian fractions, including 2/3 • Greek papyri included problems with common fractions & sexagesimal fractions India - fractional notation was the same as today, excluding the bar

  4. Developmental Perspective Treating fractions as whole-number-partitioning problems Recognizing fractions as portions that take into account the size of individual fractions

  5. Fraction Arithmetic Define a fraction as the solution x to an equation of the form: where a and b are integers and b = # of parts in the whole a = # of parts selected x = the fraction “a out of b” parts

  6. Hershey’s chocolate bar The whole bar is split into 12 equal parts, b = 12 If we select 3 pieces, a = 3 The fraction x, is a/b = 3/12

  7. Unit Fractions (when a = 1) When a = 1 x is the unit fraction 1/b This picture represents when a = 1, b = 5 So the unit fraction is x = 1/5

  8. Equivalent Fractions The quantity ½ is the solution to the equation However, it is also the solution to the equation In fact, if k is a non-zero integer, the ½ is the solution to any equation of the form

  9. Determining When 2 Fractions are Equivalent We have fractions x & y b• x = a c • y = d ** We want to determine when x = y (i.e. x-y=0) x = a/b y = c/d

  10. Multiply 1st equation by d: d•b•x=d•a Multiply 2nd equation by b: b•d•y=b•c Subtract the yields: d•b•x-b•d•y=d•a-b•c d•b(x-y)=d•a-b•c Divide both sides by d•b: x-y=(a/b)-(c/d) x-y=0 iff (a/b)=(c/d)

  11. If x is not equivalent to y: For example:

  12. Jim is given 14/15 of a pie. Mary is given 13/14 of a pie of the same size. “That’s not fair,” Jim says, “Mary’s got more pie than me because since her pieces are bigger!” Mary replies, “That’s not true. Jim’s got more pie than me since he has more pieces!” Who has more pie, Mary or Jim, and why?

  13. Who has more pie? Let x = 14/15 Let y = 13/14 Since x = a/b a= 14 b= 15 Since y=c/d c= 13 d= 14 d•a = (14)(14) = 196 b•c = (15)(13) = 195 d•a > b•c so x > y Jim has more pie than Mary. Jim is given 14/15 of a pie. Mary is given 13/14 of a pie of the same size. “That’s not fair,” Jim says, “Mary’s got more pie than me because since her pieces are bigger!” Mary replies, “That’s not true. Jim’s got more pie than me since he has more pieces!” Who has more pie, Mary or Jim, and why?

  14. Reducing Simplifying Fractions Given a fraction a/b we want to find the “minimum equivalent” Strategy: Find the largest positive integer k such that k divides a & k divides b [k is the greatest common divisor of a & b]

  15. Euclid’s Algorithm

  16. EXAMPLES Simplify 63020/76084 Find a partner: Decide who will represent the numerator, and who will represent the denominator. Numerator: Multiply the number of sheets of toilet paper you have by 16 & add 5 Denominator: Multiply the number of sheets of toilet paper you have by 25 & add 17 Simplify your fraction using the Euclidean Algorithm.

  17. Adding & Subtracting Fractions The notion of a common denominator is similar to the idea of units. Example: 5 yards + 4 feet We need to convert to the same units So the problem becomes: 5(36 in)+4(12 in) = 228 in or 5(3 feet)+4 feet = 19 feet

  18. d•b is a common denominator

  19. d•b may not be the least common denominator Example:

  20. To calculate the least common denominator *proof by contradiction in book pg 110

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