Decimals

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Problems about decimal numbers.

On the number line below, a point is marked $A$ . Select all options which could be candidates for the value of $A$ .

$3.62678$ $3.6267783$ $3.68$ $3.626788983$ $3.627$

Select all fractions below that have terminating decimal representation.

$\frac {1}{10}$ $\frac {1}{30}$ $\frac {1}{80}$ $\frac {1}{64}$ $\frac {1}{125}$ $\frac {1}{250}$ $\frac {1}{385}$ $\frac {1}{2048}$ $\frac {1}{4228}$ $\frac {1}{2^{19} \times 5^{47}}$

A harder version of the previous problem: select all fractions below that have terminating decimal representation.

$\frac {14}{10}$ $\frac {6}{30}$ $\frac {4}{60}$ $\frac {7}{98}$ $\frac {11}{125}$ $\frac {3}{150}$ $\frac {11}{385}$ $\frac {2}{2049}$ $\frac {1057}{4228}$ $\frac {3^4 \times 7^{11} \times 19}{3^2 \times 5^{22} \times 19}$

Don’t forget to reduce the fractions to lowest terms!

Give an example of an irrational number. For a challenge, don’t pick $\pi$ , $e$ , or $\sqrt {p}$ where $p$ is prime.

One of my favorites is $0.01001000100001000001\dots$ . This number’s decimal representation is neither terminating nor repeating, though it does have a pattern!

Without doing the long division, after how many places would you expect $\frac {1}{47}$ to repeat?

We expect the repetition to occur after at most $\answer [given]{46}$ places.

Without doing the long division, after how many places would you expect $\frac {3}{104}$ to repeat?

We expect the repetition to occur after at most $\answer [given]{103}$ places.

Write each of the following decimals as a fraction using the patterns we observed in class.

(a): $0.\overline {4}$ = $\frac {\answer {4}}{\answer {9}}$
(b): $0.\overline {42}$ = $\frac {\answer {42}}{\answer {99}}$
(c): $0.\overline {215}$ = $\frac {\answer {215}}{\answer {999}}$
(d): $0.\overline {234584}$ = $\frac {\answer {234584}}{\answer {999999}}$

It is true that $0.\overline {9} = 1$ . What do you expect the following to be equal to?

(a): $1.\overline {9} = \answer {2}$
(b): $0.5\overline {9} = \answer {0.6}$
(c): $2.34\overline {9} = \answer {2.35}$

Given a prime number $p$ , we will explore a relationship between the number of decimal places in which $\frac {1}{p}$ repeats, and the smallest value of $n$ where $p$ divides $10^n - 1$ .

Consider the case of $p=3$ . We know that $\frac {1}{3} = 0.\overline {\answer [given]{3}}$ , or $\frac {1}{3}$ repeats after $\answer [given]{1}$ decimal place. What is the smallest value of $n$ so that $3 \vert 10^n - 1$ ?

Choose potential values for $n$ in an organized fashion. What is the prime factorization of $10^n-1$ ?

For $p=3$ , we have $n=\answer {1}$ .

Consider the case of $p=7$ . We know that $\frac {1}{7} = 0.\overline {\answer [given]{142857}}$ , or $\frac {1}{7}$ repeats after $\answer [given]{6}$ places. What is the smallest value of $n$ so that $7 \vert 10^n -1$ ?

For $p=7$ , we have $n=\answer {6}$ .

Consider the case of $p=11$ . We know that $\frac {1}{11} = 0.\overline {\answer [given]{09}}$ , or $\frac {1}{11}$ repeats after $\answer [given]{2}$ places. What is the smallest value of $n$ so that $11 \vert 10^n -1$ ?

For $p=11$ , we have $n=\answer {2}$ .

Consider the case of $p=13$ . We know that $\frac {1}{13} = 0.\overline {\answer [given]{076923}}$ , or $\frac {1}{13}$ repeats after $\answer [given]{6}$ places. What is the smallest value of $n$ so that $13 \vert 10^n -1$ ?

For $p=13$ , we have $n=\answer {6}$ .

Consider the case of $p=37$ . We know that $\frac {1}{37} = 0.\overline {\answer [given]{027}}$ , or $\frac {1}{37}$ repeats after $\answer [given]{3}$ places. What is the smallest value of $n$ so that $37 \vert 10^n -1$ ?

For $p=37$ , we have $n=\answer {3}$ .

What pattern are you observing?

The number of places in the decimal’s repeat is the same as the value of $n$ .