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Decimalia
Decimalia
Decimalia

Station  8D
Converting Fractions into Decimals 

A fraction is a number that is an answer to a division problem. For example, the fraction 18\dfrac {1} {8} is the result of dividing 11 by 88. And we can compute 1÷81 \div 8 in a 1101 \leftarrow 10 machine by making use of decimals. The method is exactly the same as for division without decimals.

Question 1

For 1÷81 \div 8 we seek groups of eight in the following picture.
I8S8D - Image209

None are to be found right away, so let’s unexplode.
I8S8D - Image210

We have one group of 88, leaving two behind.
I8S8D - Image211

Two more unexplosions.
I8S8D - Image212

This gives two more groups of 88 leaving four behind.
I8S8D - Image213

Unexploding again
I8S8D - Image214

reveals five more groups of 88 leaving no remainders.
I8S8D - Image215

We see that, as a decimal, 18\dfrac {1}{8} turns out to be 0.1250.125.
And as a check we have 0.125=1251000=25200=540=180.125 = \dfrac {125}{1000} = \dfrac {25}{200} = \dfrac {5}{40} = \dfrac {1}{8}.

Perform the division in a 1101 \leftarrow 10 machine to show that 14\dfrac {1}{4}, as a decimal, is 0.250.25.

Question 2

Perform the division in a 1101 \leftarrow 10 machine to show that 12\dfrac {1}{2}, as a decimal, is 0.50.5.

Question 3

Perform the division in a 1101 \leftarrow 10 machine to show that 35\dfrac {3}{5}, as a decimal, is 0.60.6.

Question 4

Perform the division in a 1101 \leftarrow 10 machine to show that 316\dfrac {3}{16}, as a decimal, is 0.18750.1875.

Question 5

In simplest terms, what fraction is represented by each of these decimals?

0.750.75

0.6250.625

0.160.16

0.850.85

0.06250.0625

Question 6

Not all fractions lead to simple decimal representations. For example, consider the fraction 13\dfrac {1}{3}. To compute it, we seek groups of three in the following picture.
I8S8D - Image234

Let’s unexplode.
I8S8D - Image235

We see three groups of 33 leaving one behind.
I8S8D - Image236

Unexploding gives another ten dots to examine.
I8S8D - Image237

We find another three groups of 3{3} leaving one behind.
I8S8D - Image238

And so on. We are caught in an infinitely repeating cycle.
I8S8D - Image239

This puts us 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 13=0.33333...\dfrac{1}{3}=0.33333... with the ellipsis representing the instruction “keep going with this pattern forever.” In our minds 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 33s represented by the ellipses.

Nonetheless, many people choose not to contemplate what an infinite statement like this means and just carry on to say that some decimals are infinitely long and not be worried by it. In which case, the fraction 13\dfrac{1}{3} is one of those fractions whose decimal expansion goes on forever.

Notation: Many people make use of a vinculum (a horizontal bar) to represent infinitely long repeating decimals. For example, 0.30.\overline {3} means “repeat the 33 forever”

0.3=0.33333...0.\overline {3}=0.33333...

and 0.381420.38\overline {142} means “repeat the group 142142 forever” after the beginning “3838” hiccup:

0.38142=0.38142142142142...0.38\overline {142}=0.38142142142142....

As another (complicated) example, here is the work that converts the fraction 67\dfrac {6}{7} to an infinitely long repeating decimal. Make sure to understand the steps one line to the next.
I8S8D - Image247
I8S8D - Image248
I8S8D - Image249

Do you see, with this 66 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 857142857142 of answers, and then again, and then again. We have

67=0.857142857142857142857142...\dfrac{6}{7}=0.857142857142857142857142....

Compute 47\dfrac{4}{7} as an infinitely long repeating decimal.

Question 7

Compute 111\dfrac{1}{11} as an infinitely long repeating decimal.

Question 8

Which of the following fractions give infinitely long decimal expansions?

12,13,14,15,16,17,18,19,110\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 9

Use a 1101 \leftarrow 10 machine to compute 133÷6133 \div 6, writing the answer as a decimal.

Question 10

Use a 1101 \leftarrow 10 machine to compute 255÷11255 \div 11, writing the answer as a decimal.

You can either play with some of the optional stations below or go to the next island!

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