Exploding Dots

2.4 Some Irrational Thoughts

Lesson materials located below the video overview.

We’ve seen that fractions can possess finitely long decimal expansions (for example, \(\frac{1}{8} = 0.125\) or \(\frac{1}{2} = 0.5\)) and can possess infinitely long decimal expansion (for example,  \(\frac{1}{3} = 0.3333\cdots\) and \(\frac{6}{7} = 0.857142857142857241\cdots\).)


Have you noticed that for all the examples we’ve seen thus far of fractions with infinitely long decimal expansions those expansions fell into a repeating pattern?


We can even say that our finite examples fall into a repeating pattern too, a repeating pattern of zeros (after an initial non-repeating start):


\(\dfrac{1}{8}  = 0.1250000000\cdots = 0.125\bar{0}\)

\(\dfrac{1}{2} = 0.500000\cdots = 0.5\bar{0}\)

\(\dfrac{1}{3} = 0.\bar{3}\)

\(\dfrac{6}{7} = 0.\overline{854271}\)


Does every fraction have a decimal representation that eventually repeats?


The answer to this question is YES and our method of division explains why!


Let’s go through the division process again, slowly, first with a familiar example. Let’s compute the decimal expansion of \(\frac{1}{3}\) again in a \(1 \leftarrow 10\) machine. We need to find groups of three within the following diagram:


Unexplode the single dot to make ten dots in the tenths position. There we find three groups of three leaving a remainder of \(1\) in that box.


Now we can unexploded the single dot in the tenths box and write ten dots in the hundredths box. There we find three more groups of three, leaving a single dot behind:


And so on. We are caught in a cycle of having the same remainder of one dot from cell to cell, meaning that the same pattern repeats. Thus we conclude:

\(\dfrac{1}{3} = 0.3333\cdots\).


A more complicated example: Let’s compute the decimal expansion of \(\frac{4}{7}\) in the \(1 \leftarrow 10\) machine.


We start by unexploding the four dots and writing “\(40\)” in the tenths cell. There we find \(5\) groups of seven, leaving five dots left over.


Now unexplode those five dots to make \(50\) dots in the hundreds position. There we find \(7\) groups of seven, leaving one dot over.


Unexplode this single dot. This yields one group of seven leaving three remainder.


Unexplode these three dots. This finds us \(4\) groups of seven with \(2\) remainder.


Unexplode the two dots. This finds us \(2\) groups of seven with \(6\) remainder:


Unexplode the six dots. This finds us \(8\) groups of seven with \(4\) remainder:


But this is predicament we started with: FOUR dots in a box. We are now going to repeat the pattern and produce a cycle in the decimal representation.


We have: \(\dfrac{4}{7} = 0.571428  571428  571428  \cdots\).


Stepping back from the specifics of this problem, it is clear now that one must be forced into a repeating pattern. In dividing a quantity by seven, there are only seven possible numbers for a remainder number of dots in a cell – 0, 1, 2, 3, 4, 5, or 6 – and there is no option but to eventually repeat a remainder and so enter a cycle.


In the same way, the decimal expansion of \(\dfrac{18}{37}\) must also cycle. In doing the division, there are only thirty-seven possible remainders for dots in a cell (0, 1, 2, …, 36). As we complete the division computation, we must eventually repeat a remainder and again cycle.


This establishes:



EXERCISE:  Conduct the division procedure for the fraction \(\frac{1}{8}\). Make sure to understand where the cycle of repeated remainders commences.


This now opens up a curious idea:


A quantity given by a decimal expansion that does not repeat cannot be a fraction!


For example, the quantity:


\(0.10 1100 111000 111100000 11111000000 \cdots\)


designed not to repeat (though there is a pattern to this decimal expansion) represents a number that is not a fraction.


Definition: A number that is the ratio of two whole numbers (i.e. a fraction) is called a rational number. A number that cannot be represented this way is called an irrational number.


It looks like we have established the existence of irrational numbers.


In fact, we can now invent all sorts of numbers that can’t be fractions! For example:






are irrational numbers.







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