Number TheoryChapter 7: Fractions (pt 1)

Kaitlyn Haase

Historical PerspectiveEgyptians- focused on unit fractions

- included 2/3 (this was the only anomaly)

€

78

=12

+14

+18

€

78

=23

+18

+1

12

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

€

311

€

311

Developmental PerspectiveTreating fractions as whole-number-

partitioning problems

Recognizing fractions as portions that take into account the size of individual fractions

Fraction ArithmeticDefine a fraction as the solution x to an equation of

the form:

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b • x = awhere a and b are integers and

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b ≠ 0

b = # of parts in the wholea = # of parts selectedx = the fraction “a out of b” parts

Hershey’s chocolate bar

The whole bar is split into 12 equal parts, b = 12

If we select 3 pieces, a = 3The fraction x, is a/b =

3/12€

b • x = a

Unit Fractions (when a = 1)When a = 1x is the unit fraction 1/b

This picture represents when a = 1, b = 5So the unit fraction is x = 1/5

Equivalent FractionsThe quantity ½ is the solution to the equation

€

2 • x =1

However, it is also the solution to the equation

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4 • x = 2In fact, if k is a non-zero integer, the ½ is the solution to any equation of the form

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2k • x = k

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x =k

2k

Determining When 2 Fractions are EquivalentWe have fractions x & yb • x = ac • y = d

** We want to determine when x = y (i.e. x-y=0)

x = a/by = c/d

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)

If x is not equivalent to y:

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x > y ⇔ab

>cd

⇔ d • b(x − y) > 0 ⇔ d • a −b • c > 0 ⇔ d • a > b • c

For example:

€

12

>25

since 5 • 1 > 2 • 2

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x < y ⇔ab

<cd

⇔ d • b(x − y) < 0 ⇔ d • a −b • c < 0 ⇔ d • a < b • c

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?

Who has more pie?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?

Let x = 14/15Let y = 13/14

Since x = a/ba= 14b= 15

Since y=c/dc= 13d= 14

d•a = (14)(14) = 196b•c = (15)(13) = 195

d•a > b•c so x > y

Jim has more pie than Mary.

Reducing Simplifying FractionsGiven 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]

Euclid’s Algorithm

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Without loss of generality, b ≥ aDivide by a with respect to the least positive integerb = q1 • a+ r1 0 ≤ r1 < aIf r1 ≠ 0, divide a by r1a = q1 • r1 + r2 0 ≤ r2 < r1If r1 = 0, then the algorithm is terminated.Continue this process until rn +1 = 0

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b = q1 • a+ r1a = q2 • r1 + r2r1 = q3 • r2 + r3

rn−2 = qn • rn−1 + rnrn−1 = qn +1 • rn + 0

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Now rn is a common divisor of a and b.Every divisor of a and b must divide rn .Let c be a divisor of a and b.We can show in the :1st step that c divides r12nd step that c divides r2

... c divides rn

Every divisor of a and b divides rn

∴ rn is the greatest common divisor of a and b

EXAMPLESSimplify 63020/76084Find 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.

Adding & Subtracting FractionsThe 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

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Add the fractions x and y that satisfy the equationsb • x = ac • y = d

b • x +d • y = a +cHowever, this only represents x + y iff b = d(The two fractions already share a common denominator)

Then we can say ab

+cd

=a+ cb

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When b ≠ d we must convert to equivalent units.Multiply expression for x by d :d • b • x = d • aMultiply expression for y by b :b • d • y = b • cAdd the yields together :ab

+cd

=d • a +b • cd • b

d•b is a common denominator

d•b may not be the least common denominator

Example:

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Add 16

and 14

A common denominator is (6)(4) = 24

So we have 16

+14

=4

24+

624

=1024

However, 1024

=5

12where 12 is the least common denominator

To calculate the least common denominator

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[b,d] =b • dD

[b,d] = least common denominator of ab

+cd

D = greatest common divisor of b and d

*proof by contradiction in book pg 110