1. Converting Repeating Decimals to Fractions
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Taking the Fear
out of Math #8
Converting Repeating Decimals to Fractions
1/3 .3
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The fact that every rational number can be represented by either a decimal that terminates or else by a decimal that eventually repeats the same cycle of digits endlessly does not mean in itself that the “reverse” (called the converse) is true. In this respect, so far we have shown that every common fraction can be represented as a terminating decimal.
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3. the same cycle of digits endlessly must represent a rational number.1
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However, it still remains to be proven that any decimal that eventually repeats
the same cycle of digits endlessly must represent a rational number.1
Demonstrating this is a bit tricky in the sense that it is easier to scramble
an egg than to unscramble one.
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1 An easy way to visualize why the converse of a true statement need not be true is to consider the fact that while it is true that every bear is an animal, it is not true that every animal is a bear.
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4. For example, if we start with 1/3 and
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For example, if we start with 1/3 and
want to write it in decimal form,
we simply go through the division process
and see that the decimal is 0.3.
However, suppose that we didn’t already know that 0.3 was equal to 1/3, and we wanted to express it as an equivalent
common fraction, where would we begin?2
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2 Unlike in what happens when we have a terminating decimal, in the case of 0.3 (or in the case of any non-terminating decimal) there is no digit “furthest to the right”. In other words, the definition of endless means that we never come to the end of the decimal.
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In other words, the decimal fraction that represents 5 ÷ 6 consists of a decimal point followed by an 8 and an endless number of 3’s!
In this particular example, we saw that a problem arose when we tried to express the common fraction 5 ÷ 6 as an equivalent decimal fraction. Unfortunately, this situation turns out to be the general rule rather than the exception.
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6. in a “real world” application.
As we’ve mentioned before, from a practical point of view, there is no need to know how to do this because we can get as an exact approximation as is needed
in a “real world” application.
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However, there is a side of mathematics that belongs as much to the humanities as it does to science and technology. There are people who study mathematics for the same reason that there are people who study poetry; not to build better bridges, but because it represents a beauty that shows the heights to which the human mind can soar.
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7. and how abundantly they exist.
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In this context, it would simply be “nice” to know that the converse is true.
In fact, once we succeed in proving this we will have developed a very strong connection between rational numbers and decimals that may help us understand what irrational numbers (which will be discussed in our next presentation) are
and how abundantly they exist.
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8. 0.3 and determined that it represented 1/3.
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Having said all of this, let’s now try to see how we could have started with
0.3 and determined that it represented 1/3.
To begin with, since the repeating cycle in 0.3 consists of just a single digit (3), if we move the decimal point just 1 place to the right, the part that now follows the decimal point is still .3.
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However, if we move the decimal point 1 place to the right, we have multiplied the decimal by 10. In other words, if we let n stand for 0.3, then 10n stands for 3.3.
The strategy we will now use relies on the fact that any number subtracted from itself is 0 (even if the number is represented by an endless decimal).
Let’s subtract n = 0.3 from 10n = Recall that n means the same thing as 1n.
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10. Hence, the left hand side becomes
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Hence, the left hand side becomes
10n – 1n or 9n, and on the right hand side the .3 in the bottom line “cancels” with the .3 in top line. That is, 3.3 – 0.3 = 3.
In summary…
10n = 3.3
n = .3
9n = 3
And if 9n = 3, then n = 3 ÷ 9 = 3/9 = 1/3
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11. To convert a non-terminating repeating
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Summary
To convert a non-terminating repeating
decimal into a common fraction, move the decimal point in such a way that we get two decimals that have the same
fractional part (that is, the same repeating cycle). We then subtract the lesser decimal from the greater to get the relationship that yields the desired fraction.
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12. Illustrative Example #1
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Illustrative Example #1
Let’s write each of the following as an equivalent common fraction in lowest terms.
(a)
(b)
(c)
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13. decimal point, the denominator of that common fraction will be 1,000.
Solving part (a) is relatively easy. Namely, terminates and the digit
furthest to the right is 6. Thus, the numerator of the equivalent common
fraction will be 216, and since there are three digits (216) to the right of the
decimal point, the denominator of that common fraction will be 1,000.
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The equivalent common fraction is
216/1,000, but since the numerator and denominator of this fraction are both divisible by 8, the fraction can be written
in reduced form as 27/125.
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As a check we see that… .2 1 6
125
2 5 0
2 0 0
1 2 5
7 5 0
7 5 0
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15. the right. Moving the decimal point
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However, we cannot use the same strategy for doing part (b) because 0.216
has no “last” digit. However, it repeats the cycle of digits 216 endlessly.
Thus, the same cycle will repeat if we move the decimal point 3 places to
the right. Moving the decimal point
3 places to the right is equivalent to
multiplying the original decimal by 1,000.
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16. Hence, if we now let n denote the original decimal we have…
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Hence, if we now let n denote the original decimal we have…
1,000n =
n =
999n = 216
The fact that 999n = 216 means that
n = 216 ÷ 999, and if we reduce this to
lowest terms (27 is a common factor
of the numerator and denominator),
we see that n = 8/37.
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As a check we see that… .2 1 6 37
7 4
6 0
3 7
2 3 0
2 2 2 83
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3 We would have obtained the same result if we used the division algorithm to express 216 ÷ 999 as a common fraction. However, by using 8 ÷ 37 we were able to work with less cumbersome numbers. Notice that as far as the decimal representation is concerned it doesn’t matter whether we use 216 ÷ 999 or 8 ÷ 37.
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Part (c) is bit “trickier” because the repeating cycle consists only of the “6”. The first place we can put the decimal point so that the fractional part is .6 is after the “1”. That is, if we move the decimal point two places to the right we get the number Moving the decimal point two places to the right is the same as multiplying the original decimal by 100.
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The next place we could move the decimal point so that the fractional part is still .6 is after the first “6”. That is, if we move the decimal point three places to the right we get the number Moving the decimal point three places to the right is the same thing as multiplying the original
decimal by 1,000.
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20. The fact that 900n ÷ 195 tells us that
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Therefore, if we now let n denote the original decimal and subtract, we have…
1,000n =
100n =
900n = 195
The fact that 900n ÷ 195 tells us that
n = 195/900 and dividing numerator and denominator by 15 we see that in lowest terms n = 13/60.
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As a check we see that… .2 1 6 60
1 2 0
1 0 0
6 0
4 0 0
3 6 0
4 04
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4 Notice that the first repeated remainder (40) was not the first remainder (10).
That’s why the repetition did not begin with the first digit to the right of the
decimal point.
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22. In particular, don’t confuse 0.216 and 0.216.
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Important Note
► Notice the importance of where we place the “bar”. In this problem, all
three parts look like without the “bar”; yet all three parts are different.
In particular, don’t confuse and
► In fraction form…
0.216 = 216/1,000 while = 216/999,
and = 13/60.
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Summary
The decimal representation of any rational number either is a terminating decimal or else eventually repeats the same cycle
of digits endlessly.
The decimal equivalent of a common fraction that is represented in lowest terms will terminate if and only if the only prime factors of the denominator of the fraction are 2’s and/or 5’s.
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Conversely, any decimal that either terminates or else eventually repeats the same cycle of digits endlessly
represents a rational number.
In other words, a decimal represents a rational number (fraction) if and only if it either terminates or else eventually
repeats the same cycle of digits endlessly.
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25. The “Paradox” that Surrounds “Endlessness”
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The “Paradox” that Surrounds “Endlessness”
Suppose there are a group of people in a room and all of them decide to change their names. Does this have any bearing on how many people are in the room?
In fact to make this problem seem more mathematical, suppose there were ten people in the room, and we named them numerically from 1 through 10. Suppose we then ask each person to change his name to
twice the number he now had.
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For example, #1 would now become #2; #2 would now become #4 and so on. In this way there would still be ten people but instead of having the names 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10; they would now have the names 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20.
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The same thing would happen even if there were a billion people. If the people were numbered from 1 through 1,000,000,000 and each changed their name to twice the number they originally had, there would still be a billion people but they would now be named by the even numbers from 2 through 2,000,000,000.
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28. Now try to imagine that there are an “endless” number of people in the
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Now try to imagine that there are an “endless” number of people in the
room and that we were able to number them. Their names would now be 1,
2, 3, 4, 5, 6, etc. but now the list would never end! In fact the list of their
names would be synonymous with the
set of all the non-zero whole numbers.
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However, now suppose they all change their name to twice the number they originally had (that is, 1 becomes 2, 2 becomes 4, 3 becomes 6, 4 becomes 8, etc.). In this case all the odd numbers have
disappeared. That is…
Original name 1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20
New name
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30. A “cute” way to restate the above remark is to imagine
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A “cute” way to restate the
above remark is to imagine
a hotel that has infinitely many rooms numbered from 1 to “infinity” and that all the rooms are filled. The room clerk, to make more room, asks each guest to move to the room that is twice the number of his or her current room. In this way the even numbered rooms are still completely occupied, but every one of the odd numbered rooms is now vacant!
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The point of the above discussion is to prepare you for the fact that ideas that are quite “simple” in finite situations are mind boggling when we try to apply these ideas to infinite (endless) situations.
For example, suppose you start with the decimal 0.33 and delete one of the 3’s to get a new decimal Clearly 0.3 and 0.33 are not equal (in fact 0.33 exceeds 0.3 by 0.03).
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Suppose next that we started with and deleted one of the 3’s
to get the new decimal
Again we get two different decimals even though their difference, ,
is quite small.
Even if there were a billion (or a trillion or any finite but arbitrarily large number) of 3’s after the decimal point, deleting one of the 3’s would make the new decimal less than the original decimal.
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However, if there are an endless number of 3’s after the decimal point and you delete one of them (or any finite number of them), there are still an endless number of 3’s after the decimal point. Most people use the term “endlessly” as a form of exaggeration.
For example a person might say that a job was “endless” instead of saying that the job took a very long time.5
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5 One way to see the difference between “ a lot” and endless is to let N stand for the “greatest number you can imagine”. For example, consider the number which in place value is represented by a 1 followed by 10 billion zeroes. In exponential notation this number is 1010,000,000,000. Let N denote this number. Since a billion seconds is in excess of 30 years, at the rate of 1 digit per second, it would take over 300 years just to write this number using place value notation! However, if we start counting after N, we get N + 1, N + 2, N + 3, etc., and we are once again back to the beginning of our counting system only using N as if it were 0.
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However, in terms of what we are discussing in this presentation, what follows is a very interesting result; a result that to many may seem paradoxical or even incorrect. We will state it in the form of another illustrative example.
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35. Illustrative Example #2 Which number is greater, 0.9 or 1?
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Illustrative Example #2
Which number is greater, 0.9 or 1?
To grasp the point of this example, let’s review the procedure we used to show that 0.3 = 1/3 . That is…
10n = 3.3
n = .3
9n = 3
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36. Now let’s use the same technique to express 0.9 as an equivalent
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Now let’s use the same technique to express 0.9 as an equivalent
common fraction.
10n = 9.9
n = .9
9n = 9
... and hence n = 1
If we accept the fact that a number has one and only one value, the fact that n is equal to both 1 and 0.9 means that 0.9 = 16
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6 Another way to get this result is by starting with the fact that 1/3 × 3 = 1; and then replacing 1/3 by 0.3 to obtain 0.3 × 3 = 1. Since 0.3 × 3 = 0.9 it again follows that 0.9 = 1.
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Yet how can this be? If we write any number of 9’s after the decimal point,
whenever we stop, the resulting decimal will name a number that is less than 1.
For example…
is less than 1. In fact, 1 – 0.9 = 0.1
is less than 1. In fact, 1 – 0.99 = 0.01
is less than 1. In fact, 1 – = 0.001
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38. the decimal would still be less than 1.
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Do you see the pattern?
Every time we annex one more 9 the resulting decimal is greater than the previous decimal; yet it’s still less than 1! And even if we wrote a trillion 9’s after the decimal point, when we were finished
the decimal would still be less than 1.
The paradox here is the key phrase is “when we finish”; namely how does one finish an endless process?
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So, strange as it may seem, once we agree that the sequence of 9’s never ends, the resulting decimal, 0.9, is equal to 1.
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Using the process by which we compare the size of two decimals, what we can show is that it’s impossible to find a number that’s less than 1 but greater than 0.9.
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Specifically, we look for the first place in which the two decimals are different, in which case the greater of the two digits names the greater decimal.
However, since it is impossible for a digit to be greater than 9, there is no number less than 1 that’s greater than In other words, no number exists that is between 0.9 and 1.7
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7 The same “paradox” existed when we worked with Yet for perhaps some psychological reason we seemed to accept the fact that no number exists
that is between 0.3 and 1/3
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41. is continuous (that is, it has no “gaps”)
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And if we now make the rather “natural” assumption that the number line
is continuous (that is, it has no “gaps”)
it must be true that 0.9 = 1.
In summary, even if we have a trillion 9’s following the decimal point, there is an even larger number that is still less than 1. But when it comes to “endlessness”, even a trillion is but a drop in the bucket!
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42. Non-terminating Decimals
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At any rate, this concludes this presentation. In the next presentation we
will discuss what happens if there is a non-terminating decimal that doesn’t eventually repeat the same cycle of digits endlessly.
Non-terminating Decimals
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