General Mathematics · Class 9-10 · Chapter 1 · Real numbers
Like recurring decimals, and arithmetic.
How do you add numbers that never end? There are two routes. One is to bring both to the same shape and work in columns; the other is to turn both into fractions, work as with any fractions, and turn the answer back. The first route is worth understanding, and the second is the one to use in an exam.
1. What makes two of them alike
Two or more recurring decimals are called like when both parts, the non-repeating and the repeating, have the same number of digits. If either count differs they are unlike. So 0.3 and 0.5 are alike, since neither has a non-repeating digit and each has one repeating digit.
The commonest mistake here is worth naming: the whole number part does not enter into it at all. Likeness is measured only to the right of the point. Nor does it matter that the digits themselves differ; sameness of shape is all that is asked. In the table below each number carries its shape, and tapping a cell compares the pair.
gridtap a cell
Which pairs are alike?
number
(0,1)
(0,1)
(0,2)
(1,1)
(0,2)
0.3(0,1)
✓
✓
✕
✕
✕
0.5(0,1)
✓
✓
✕
✕
✕
0.23(0,2)
✕
✕
✓
✕
✓
0.45(1,1)
✕
✕
✕
✓
✕
2.67(0,2)
✕
✕
✓
✕
✓
Alike. 0.3 and 0.5 are both (0, 1): no non-repeating digit and one repeating digit each. That the digits themselves differ does not matter.
Unlike. (0, 1) against (0, 2). The repeating counts differ. To make them alike the block has to become lcm(1, 2) = 2 digits, so 0.3 would be written as 0.33.
Unlike. (0, 1) against (1, 1). Each has one repeating digit, but the non-repeating counts differ. One part failing to match is enough.
Alike, with itself. Any number is always like itself, because the two shapes are identical.
Unlike. (0, 2) against (1, 1). Both parts differ. To make them alike the new shape would be (1, 2).
Alike, and this is the point. 0.23 and 2.67 are both (0, 2). The whole number part 2 makes no difference, because likeness is measured only to the right of the point.
Every pair seen. The ticks gather into two clumps, and inside each clump every number has the same shape. Likeness really just sorts the numbers into groups by shape.
check yourself
One question
What about the pair 2.67 and 0.23?
The whole number part does not enter into it. Likeness is measured only to the right of the point.
Right. Both have shape (0, 2). Whole number part or not, and whatever the digits, same shape means alike.
Different digits are no obstacle. 0.3 and 0.5 have different digits too and are still alike.
The answer is right, the reason is not. Both being recurring is not enough; 0.45 recurs too and is not alike to these.
2. Making unlike ones alike
Making several unlike numbers alike takes two calculations. The new count of non-repeating digits is the largest among them, and the new count of repeating digits is the lcm of their repeating counts. Then each number's block is simply written out as many times as needed.
One thing needs saying firmly, because nearly everyone suspects otherwise: this changes the value not at all, only the way it is written. 2.3 and 2.3333333 are exactly the same number, both 2.333333…. Tap through the steps below, and check at the end.
bring them to one shapetap the three steps
Make them alike
check yourself
Another question
Made alike, how many digits will each repeating block of 2.3, 4.135 and 6.234 have?
3 is the largest of the repeating counts, but here it is the lcm that is needed, not the largest. A two digit cycle does not finish in three.
Four does not fit the three digit cycle. What is wanted is a number divisible by 1, 2 and 3 alike.
Right. The repeating counts are 1, 2 and 3, whose lcm is 6. At six all three cycles finish together. The non-repeating part takes the largest count, one digit.
Nine does not fit the two digit cycle. And the lcm means the smallest common meeting point, which nine is not.
3. The four operations
Once they are alike you can add and subtract in columns, and the book shows how. But there is a catch: a carry coming off the left of the block has to be added back at its right end, and there are many places to go wrong. For multiplication and division there is no column method at all.
So all four operations have one safe route, and it has three moves: turn both numbers into fractions, work as with any fractions, then turn the answer back into a decimal. Choose two numbers and an operation below and the whole working comes out.
calculatorpick two numbers and an operation
Arithmetic on recurring decimals
One thing is worth noticing: the answer is always a fraction too, so it is rational as well, which is to say terminating or recurring. The four operations cannot take you out of the rationals, which is the closure property from an earlier note. The one prohibition is dividing by zero. And now and then the answer comes out terminating: 0.3 + 0.6 = 1/3 + 2/3 = 1.
check yourself
Last question
What is 0.3 + 0.23?
Right. 0.3 = 1/3 = 33/99 and 0.23 = 23/99. The sum is 56/99 = 0.565656…. Check it as decimals too: 0.3333… + 0.2323… = 0.5656….
Only the first pair of digits has been added. There is addition at the second decimal place too: 3 + 3 = 6.
That looks like a subtraction, and a terminating one. Adding two recurring decimals gives a recurring answer unless it lands exactly on a whole number.
The digits have been swapped. 33/99 + 23/99 = 56/99, so the block is 56, not 65.
The rules from this note are cut onto the cards below.
cut from the notetap to turn over
Four cards
frontWhen are two recurring decimals alike?
backWhen both counts match, non-repeating and repeating. The whole number part is not counted.
frontWhat is the rule for making them alike?
backThe largest non-repeating count, and the lcm of the repeating ones. The values do not change.
frontHow do you multiply recurring decimals?
backBy turning them into fractions. There is no column method for it.
frontIs the answer always rational?
backYes, except for dividing by zero. Working in fractions gives a fraction, so it stops or repeats.