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Station  8E
Irrational Numbers 

Question 1

We have seen that fractions can possess finitely long decimal expansions.

For example, 18=0.125\dfrac{1}{8} = 0.125 and 12=0.5\dfrac{1}{2} = 0.5.

And we have seen that fractions can possess infinitely long decimal expansion.

For example, 13=0.3333...\dfrac{1}{3} = 0.3333... and 67=0.857142057142857142...\dfrac{6}{7} = 0.857142057142857142....

All the examples of fractions with infinitely long decimal expansions we’ve seen so far fall into a repeating pattern. This is curious.

We can even say that our finite examples eventually fall into a repeating pattern too, a repeating pattern of zeros after an initial start.

18=0.12500000...=0.1250\dfrac{1}{8} = 0.12500000... = 0.125\overline{0}

12=0.50000...=0.50\dfrac{1}{2} = 0.50000... = 0.5\overline{0}

13=0.3\dfrac{1}{3} = 0.\overline{3}

67=0.857142\dfrac{6}{7} = 0.\overline{857142}

This begs the question:

Does every fraction have a decimal representation that eventually repeats?

Question 2

Let’s go through the division process again, slowly, first with a familiar example. Let’s compute the decimal expansion of 13\dfrac{1}{3} again in a 1101 \leftarrow 10 machine. We think of 13\dfrac{1}{3} as the answer to the division problem 1÷31 \div 3, and so we need to find groups of three within a diagram of one dot.
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We unexplode the single dot to make ten dots in the tenths position. There we find three groups of three leaving a remainder of 11 in that box.
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Now we can unexplode that single dot in the tenths box and write ten dots in the hundredths box.
There we find three more groups of three, again leaving a single dot behind.
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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 13=0.333...\dfrac{1}{3} = 0.333.... The key point is that the same remainder of a single dot kept appearing.

Question 3

Here’s a more complicated example. Let’s compute the decimal expansion of 47\dfrac{4}{7} in the 1101 \leftarrow 10 machine.
That is, let’s compute 4÷74 \div 7.
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We start by unexploding the four dots to give 4040 dots in the tenths cell. There we find 55 groups of seven, leaving five dots over.
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Now unexplode those five dots to make 5050 dots in the hundredths position. There we find 77 groups of seven, leaving one dot over.
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Unexplode this single dot. This yields 11 group of seven leaving three remaining.
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Unexplode these three dots. This gives 44 groups of seven with two remaining.
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Unexplode the two dots. This gives 22 groups of seven with six remaining.
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Unexplode the six dots. This gives 88 groups of seven with four remaining.
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But this is predicament we started with: four dots in a box!

So now we are going to repeat the pattern and produce a cycle in the decimal representation. We have

47=0.571428571428571428...\dfrac {4}{7} = 0.571428 571428 571428....

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 – 0,1,2,3,4,5,0, 1, 2, 3, 4, 5, or 66 – and there is no option but to eventually repeat a remainder and so enter a cycle.

In the same way, the decimal expansion of 1837\dfrac{18}{37} must also cycle. In doing the division, there are only thirty-seven possible remainders for dots in a cell (0,1,2,3,4,5,...,360, 1, 2, 3, 4, 5, ..., 36). As we complete the division computation, we must eventually repeat a remainder and again fall into a cycle.

We have just established a very interesting fact.


(A repeating pattern of zeros is possible. In fact, as a check, conduct the division procedure for the fraction 18\dfrac{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

0.1011001110001111000011111000000...0.10 1100 111000 11110000 11111000000...

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

A number that is equivalent to the ratio of two whole numbers (that is, 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 just proved that irrational numbers exist! Not all numbers are fractions.
In fact, we can now invent all sorts of numbers that can’t be fractions!

For example




are irrational numbers.

Of course, we have all heard that numbers like 2\sqrt {2} and π\pi are irrational numbers. It is not at all obvious why they are and how you would go about proving that they are. (In fact, it took mathematicians about 2000 years to finally establish that π\pi is an irrational number. Swiss mathematician Johann Lambert finally proved it so in 1761.) But if you are willing to believe that these numbers are irrational, then you can say for sure that their decimal expansions possess no repeating patterns!

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

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