Download Fun with Infinity What is a Fraction?

Fun with Infinity
led by Thomas Clark
November 10, 2014
What is a Fraction?
1. What exactly is a rational number
a
, where a and b are positive integers?
b
2. What exactly is a decimal number 0.abcd..., where a, b, c, d, ... are all digits among 0, 1, 2, 3, ..., 9?
3. Show that 1/4 = 0.25 and 31/250 = 0.124?
4. What is the relationship between rational numbers and decimal numbers? Which decimal
numbers are rational?
5. What is the decimal representation of 3/7? Does this method work to turn any fraction into
a decimal?
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What is a Decimal?
1. We want to find a way to convert a decimal number to a fraction. What is different about
doing this with 0.123 and with 0.123123123.
2. Now let’s exploit the repetition of the decimal(let’s call it R) R = 0.123123123 to find the
rational form of R. Multiply R by 1000. What is 1000R? What is 1000R − R? Can you use
that to solve for R?
3. Use a similar strategy to find the rational form of 0.272727.
4. What is the rational form of 0.012345679012345679? (Note the 8 digit is missing.)
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Making the Infinite Finite
Each of the following problems can be solved by exploiting the symmetry of the expression in a
way similar to what we’ve done above. Try to evaluate each of the following problems. They can
be solved using a similar technique. Namely, assume the solution to the problem is the number x.
Then, use the repeating pattern to create an equation involving x and finally solve for x.
1. The value of 1 +
1 1 1
1
+ + + ....
2 4 8 16
s
r
2+
2. The value of
3. The value of 1 +
4. The value of 1 +
5. The value of
q
√
2 + 2 + 2 + ....
√
1
1+
1+
√
2
2
. (Do you recognize this number?)
1
1
1+ ...
2
√
2
1
1+
2
.
1+ 2 2
1+ ...
√ ...
2
.
6. Use the answer to 1. above to find the value of 1 +
3
2 3 4
5
6
+ + +
+ ....
2 4 8 16 32
Classroom Connections
1. Are there any connections between the math we’ve done tonight and what you teach in your
classroom?
2. Is there something we did tonight you would incorporate into one of your lessons?
3. How would you have improved the activity we did tonight?
4. What standards from the CCSS connect to what we did tonight? (I thought about making a
list but so many standards talk about rational numbers and decimals, they wouldn’t all fit!)
Which ones from your grade level apply?
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