Exploding Dots
2.2 Division and Repeating Decimals
Lesson materials located below the video overview.
A fraction, in reality, is an answer to a division problem. For example, the fraction \(\frac{1}{8}\) is the result of dividing \(1\) by \(8\).
Let’s actually compute \(1 \div 8\) in a \(1 \leftarrow 10\) machine, making use of decimals. We seek groups of eight in the following picture:
Clearly none are to be found, so let’s unexplode:
(Is it okay if I don’t draw the ten actual dots?)
Now there is one group of \(8\), leaving two behind:
Let’s unexplode again, twice:
This gives two groups of \(8\) leaving four behind:
Unexploding again:
And here we have five groups of \(8\) leaving no remainders:
We now see that as a decimal, \(\frac{1}{8}\) turns out to be \(0.125\).
Comment: And as a check we see: \(0.125 = \frac{125}{1000} = \frac{25}{200} = \frac{5}{40} = \frac{1}{8}\).
Question 64:
a) Perform the division in a \(1 \leftarrow 10\) machine to show that \(\frac{1}{4}\), as a decimal, is \(0.25\) .
b) Perform the division in a \(1 \leftarrow 10\) machine to show that \(\frac{1}{2}\), as a decimal, is \(0.5\).
c) Perform the division in a \(1 \leftarrow 10\) machine to show that \(\frac{3}{5}\), as a decimal, is \(0.6\).
d) CHALLENGE: Perform the division in a \(1 \leftarrow 10\) machine to show that \(\frac{3}{16}\), as a decimal, is \(0.1875\). 
Question 65: In simplest terms, what fraction is represented by each of these decimals?
a) \(0.75\) b) \(0.625\) c) \(0.16\) d) \(0.85\) e) \(0.0625\)

Not all fractions lead to simple decimal representations. For example, consider the fraction \(\frac{1}{3}\). We seek groups of three in the following picture:
Unexploding requires us to look for groups of 3 instead in:
Here there are three groups of \(3\) leaving one behind:
Unexploding gives:
in which we find another three groups of \(3\) leaving one behind:
Unexploding gives:
and we seem to be caught in an infinitely repeating cycle.
We are now in a philosophically interesting position. As human beings, we cannot conduct this, or any, activity for an infinite amount of time. But it seems very tempting to write:
\(\dfrac{1}{3} = 0.33333\cdots\).
with the ellipsis “\(\cdots\)” representing the instruction “keep going forever with this pattern.” In our minds it seems we can almost imagine what this means, but as a practical human being it is beyond our abilities: one cannot actually write down those infinitely many \(3\)s represented by \(\cdots\).
Nonetheless, many people choose not to contemplate what an infinite statement means and like to carry on and say: “Some decimals are infinitely long” and simply not be worried by it. In which case, the fraction \(\frac{1}{3}\) is one of those fractions whose decimal expansion goes on forever!
COMMENT: Many people make use of a vinculum (horizontal bar) to represent infinitely long repeating decimals. For example, \(0.\bar{3}\) means “repeat the 3 forever”:
\(0.\bar{3} = 0.33333\cdots\)
and \(0.\overline{412}\) means “repeat 412 forever”:
\(0.\overline{412} = 0.412412412412\cdots\).
As another (complicated) example, here is the work that converts the fraction \(\dfrac{6}{7}\) to an infinitely long repeating decimal. Make sure to understand the steps one line to the next.
Do you see, with this \(6\) in the final rightmost box that we have returned to the very beginning of the problem? This means that we shall simply repeat the work we have done and obtain the same sequence \(857142\) of answers, and then again, and then again.
We have:
\(\dfrac{6}{7} = 0.857142857142857142857142\cdots\).
Question 66:
a) Compute \(\dfrac{4}{7}\) as an infinitely long repeating decimal.
b) Compute \(\dfrac{1}{11}\) as an infinitely long repeating decimal.

Question 67: Which of the following fractions give infinitely long decimal expansions?
\(\dfrac{1}{2}\) \(\dfrac{1}{3}\) \(\dfrac{1}{4}\) \(\dfrac{1}{5}\) \(\dfrac{1}{6}\) \(\dfrac{1}{7}\) \(\dfrac{1}{8}\) \(\dfrac{1}{9}\) \(\dfrac{1}{10}\) 
Question 68:
a) Use a \(1 \leftarrow 10\) machine to compute \( 133 \div 6\), writing the answer as a decimal.
b) Use a \(1 \leftarrow 10\) machine to compute \(255 \div 11\) , writing the answer as a decimal. 
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