A fraction is a number that is an answer to a division problem. For example, the fraction is the result of dividing by . And we can compute in a machine by making use of decimals. The method is exactly the same as for division without decimals.
For we seek groups of eight in the following picture.
None are to be found right away, so let’s unexplode.
We have one group of , leaving two behind.
Two more unexplosions.
This gives two more groups of leaving four behind.
Unexploding again
reveals five more groups of leaving no remainders.
We see that, as a decimal, turns out to be .
And as a check we have .
Perform the division in a machine to show that , as a decimal, is .
Perform the division in a machine to show that , as a decimal, is .
Perform the division in a machine to show that , as a decimal, is .
Perform the division in a machine to show that , as a decimal, is .
In simplest terms, what fraction is represented by each of these decimals?
Not all fractions lead to simple decimal representations. For example, consider the fraction . To compute it, we seek groups of three in the following picture.
Let’s unexplode.
We see three groups of leaving one behind.
Unexploding gives another ten dots to examine.
We find another three groups of leaving one behind.
And so on. We are caught in an infinitely repeating cycle.
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 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 s 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 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, means “repeat the forever”
and means “repeat the group forever” after the beginning “” hiccup:
.
As another (complicated) example, here is the work that converts the fraction to an infinitely long repeating decimal. Make sure to understand the steps one line to the next.
Do you see, with this 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 of answers, and then again, and then again. We have
.
Compute as an infinitely long repeating decimal.
Compute as an infinitely long repeating decimal.
Which of the following fractions give infinitely long decimal expansions?
Use a machine to compute , writing the answer as a decimal.
Use a machine to compute , writing the answer as a decimal.
You can either play with some of the optional stations below or go to the next island!
