General Mathematics · Class 9-10 · Chapter 1 · Real numbers
Turning a recurring decimal into a fraction.
There is one thing in this chapter you will be asked to actually do, and this is it. The last section showed that every recurring decimal is rational, which is to say it is some fraction. Now for the method of finding that fraction, and the trick behind it, which once seen makes the rule unnecessary to memorise.
1. Pure or mixed
Before the method comes a division into two kinds, because each needs a different multiplier. If there are no digits after the point besides the repeating block, it is a pure recurring fraction, as in 0.3 or 0.23. If there are one or more digits after the point before the block starts, it is a mixed recurring fraction, as in 0.23 or 0.123.
One more thing is worth saying now, because it halves the arithmetic: the whole number part never has to be touched. 2.3 means 2 + 0.3. Work on the fractional part and add the 2 back at the end.
sort themlook at the digit right after the point
Pure or mixed?
Pick up a chip, then drop it in a bin. One question only: does the dotted block start right after the point?
Right. Next one.
Not there. Do not look at the whole number part; look at the digit right after the point.
All in place. Look hard at the last two. 2.567 is pure even though it has a whole number part; and 3.07 is mixed, because a 0 sits right after the point outside the block. The whole number part plays no part in this decision at all.
2. The trick: line the tails up and cancel
Let x = 0.3 = 0.333…. The block is one digit, so multiply by 10: 10x = 3.333…. Now subtract, and because the decimal tails are identical they cancel, leaving 10x − x = 3, that is 9x = 3, so x = 3/9 = 1/3. The whole trick is in that cancelling.
With a two digit block, multiply by 100: if x = 0.23 = 0.2323…, then 100x = 23.2323…, so 99x = 23 and x = 23/99. In general: if the block is n digits, multiply by 10 to the n, and the tails will line up.
A mixed recurring decimal needs two multiplications, and this is where most mistakes happen. Take x = 0.23 = 0.2333…. First multiply by 10 to move the point past the non-repeating part: 10x = 2.333…. Then by 100 to move it one digit further: 100x = 23.333…. Subtracting gives 90x = 21, so x = 21/90 = 7/30. Put the six lines below in order.
which firsttap them in the right order
Put the working in order
The working starts by naming the thing. Which line does that?
Yes. And then?
Not yet. The smaller multiplier comes first, then the larger.
That is the order. Notice that two multiplications were needed: one to move the point past the non-repeating part (10x), another to move it past the block (100x). Subtracting cancels the tails because they match. A pure recurring decimal has no non-repeating part, so there one multiplication is enough.
The machine below does not give an answer; it shows the working, because that is where the marks are. Fill the three boxes with any recurring decimal and the full six lines come out exactly as they have to be written in a script.
the workingfill the three boxes
The 10x − x machine
What you type goes nowhere. The page is yours.
check yourself
One question
Written as a common fraction, what is 0.45?
That would be right for 0.45 = 0.4545…, where both digits repeat. Here the 4 does not.
Right. One non-repeating digit (4) and one repeating digit (5). Numerator = 45 − 4 = 41, denominator = one 9 and one 0 = 90. Check it: 41 ÷ 90 = 0.4555….
The denominator is right, but the non-repeating digit has not been taken off the numerator. It should be 45 − 4 = 41.
5/11 = 0.4545…, which is 0.45. What was asked for is 0.4555…, a different number.
3. The shortcut
You do not have to write the six lines every time; they compress into a two line rule. The numerator is the number you get by reading all the digits after the point together, minus the number you get by reading only the non-repeating ones. The denominator is as many 9s as there are repeating digits, followed by as many 0s as there are non-repeating ones.
The rule is four phrases long, and each phrase points at a particular place in the number. The picture below breaks 0.123 = 111/900 apart; tap the chips and see which phrase points where.
the partstap a chip
The parts of a recurring decimal
The non-repeating part. The digits after the point that come only once: here 12, two digits. These decide how many 0s go below the line, and how much comes off the numerator.
The repeating block. The part that keeps coming back: here 3, a single digit. This alone decides how many 9s go below the line.
The dot. A dot goes over the first and last digit of the block; with one digit there is just the one dot. Without a dot the number is terminating, so 0.123 and 0.123 are two entirely different numbers.
The 9s. One repeating digit means one 9. Two would give 99, three 999.
The 0s. Two non-repeating digits mean two 0s, and they go after the 9. So the denominator is 900, which worked out is 10² × (10¹ − 1) = 100 × 9.
All five seen. The shortcut will stay put now on its own: numerator is all the digits minus the non-repeating ones, and the denominator is as many 9s as repeating digits followed by as many 0s as non-repeating ones.
check yourself
Another question
What kind of recurring fraction is 2.567, and what denominator will it have?
A whole number part does not make it mixed. Mixed means there are digits after the point outside the block, and here there are none.
Right. The repetition starts immediately after the point, 567 over and over, so it is pure; the whole number part 2 changes nothing. Three repeating digits and no non-repeating ones give 999. The full value is 2 + 567/999 = 2565/999 = 95/37.
The denominator is right, the kind is not. There are no digits after the point outside the block, so it is pure.
A denominator of 1000 belongs to a terminating decimal. A recurring one takes 9s, and three repeating digits mean three 9s.
4. One strange result
Playing with the rule, students discover something for themselves and assume they have made a mistake. Let x = 0.9 = 0.999…. Then 10x = 9.999…, and subtracting gives 9x = 9, so x = 1.
No mistake has been made. The very rule that gives 0.3 = 1/3 gives 0.9 = 1; nothing special was done. 0.999… and 1 are two ways of writing one number, and there is a single point on the line for them. This is outside the textbook, but it cannot be avoided once the rule is applied properly, so it is worth knowing.
check yourself
Last question
What is the value of 0.9?
0.9 is a terminating decimal with a single 9. In 0.9 the 9s never stop.
0.99 has two 9s and then stops. Here the 9s never end.
It feels like the natural answer, but the arithmetic says otherwise. You cannot name a number between 1 and 0.999…, and between two different reals there is always another.
Right. Let x = 0.999…; then 10x = 9.999…; subtracting gives 9x = 9, so x = 1. The rule that gives 0.3 = 1/3 gives 1 here. They are two ways of writing one number.
The rules from this note are cut onto the cards below.
cut from the notetap to turn over
Four cards
frontHow do you tell pure from mixed?
backRepetition starting right after the point means pure, otherwise mixed. The whole number part is irrelevant.
frontWhy does the 10x − x trick work?
backThe two decimal tails become identical, so subtracting cancels them and leaves whole numbers.
frontHow do you build the denominator?
backA 9 per repeating digit, then a 0 per non-repeating one. For 0.123 that is 900.
frontWhat is 0.9?
backExactly 1. It comes out of the same rule, and there is one point on the line for both.