## 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 right-most 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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