2/5 written as a decimal is 0.4, and it stops right there. 1/3 is 0.333... and never stops. Both are fractions, yet one ends and one does not. And the decimal of √2 neither ends nor repeats any part of itself. Tell these three apart and the rest of the chapter gets much easier.
1. Decimal fractions come in three kinds
Every real number can be written as a decimal fraction. For example, 2 = 2.0, 2/5 = 0.4, 1/3 = 0.333... and so on.
Decimal fractions are of three types: finite, repeated and infinite. A finite decimal has only a finite number of digits after the decimal point, like 0.12, 1.023, 7.832 and 54.67.
In a repeated decimal, all of the digits after the point, or some run of them, come round again and again: 3.333..., 2.454545..., 5.12765765.... The Bangla book also calls it পৌনঃপুনিক, which means recurring.
In an infinite decimal the digits after the point never end, and no part of them ever repeats. Examples are √2 = 1.4142135624... and √7 = 2.6457513111.... A repeated decimal also runs on forever, but because a block of it keeps coming back, it is kept as a separate kind. Here “infinite” means one where no part ever returns.
Finite and repeated decimals are rational numbers. Infinite decimals are irrational numbers. Sort the seven numbers below into the three boxes.
sort themnumber, then box
Three kinds of decimal
Pick up a number, then drop it in a box. Does any part keep coming back?
Right. Finite and repeated are rational, infinite is irrational. Next one.
Not yet. Look at whether the digits end anywhere, and whether any part keeps coming back.
All in place. In 0.01050105... the block 0105 repeats, and in 5.12765765... the block 765 does. Both are repeated, so both are rational.
2. Division gives a repeated decimal, and its names
To write a fraction as a decimal, divide the numerator by the denominator. With 23/6 the division never ends and the same digit 3 keeps arriving in the quotient. So 23/6 = 3.8333..., a repeated decimal.
The part of a repeated decimal that keeps coming back is the repeating part. The rest is the non-repeating part. In 3.8333... the 3 repeats and the 8 does not.
Writing endless digits is pointless, so we mark dots. If one digit repeats, a single dot goes over it. If several repeat, dots go over only the first and the last. 2.555... is written 2.5, and 3.124124124... is written 3.124.
If nothing but the repeating part follows the decimal point, it is a pure repeated decimal, like 1.3. If one or more non-repeating digits come before the repeating part, it is a mixed repeated decimal, like 4.235112. Tap each one and check.
spot themtap one, then check
Pure or mixed?
Pure. Only one digit follows the point, the 3, and it is the one that repeats.
Pure. The first dot sits on 3, right after the point, and the last on 6. Every digit after the point repeats.
Pure. Both 2 and 7 repeat, and nothing else follows the point.
Mixed. The digits 2351 do not repeat; the dots fall on the later 1 and 2.
Mixed. The 1 does not repeat; only the 6 comes back.
Mixed. The 32 is the non-repeating part and 789 is the repeating part.
All six seen. Pure: the first three. Mixed: the last three. One test separates them: is there a non-repeating digit before the repeating part?
3. When the division stops, and when it loops
If the denominator has a prime factor other than 2 and 5, the numerator can never be divided out exactly, so a repeated decimal results: 3/11, 5/3. If the denominator has only 2s and 5s, the division stops and gives a finite decimal: 5/2 = 2.5, 5/4 = 1.25.
A remainder is always smaller than the denominator. So as you keep dividing, sooner or later the same remainder comes back, and the same remainder means the same digits again. That is why the repeating part always has fewer digits than the denominator. Pick a fraction, divide one step at a time, and watch for the remainder that returns, marked in red.
divide it yourselfwhen a remainder repeats, the digits repeat
Division that loops
Example 3 in the book does these two: 3/11 = 0.2727... = 0.27 and 95/37 = 2.567567... = 2.567.
Four more worth keeping: 1/6 = 0.16, 7/11 = 0.63, 3 2/9 = 3.2, 3 8/15 = 3.53.
4. Check yourself
Three questions, the first about the three kinds.
check yourselfpick one answer
Question 1
According to the book, how many types of decimal fraction are there?
Thinking finite vs infinite gives two, but the book counts repeated separately: finite, repeated, infinite, three in all.
Right. Book: decimal fractions are of three types: finite, repeated and infinite.
The book names no fourth type; there are three: finite, repeated, infinite.
The book has no five types; decimal fractions are of three types.
In the second, watch the dots.
check yourselfpick one answer
Question 2
Which of the following is a pure repeated (recurring) decimal?
The 1 does not repeat, only the 6, so this is mixed; the pure one is 4.3̇426̇.
32 does not repeat, so it is mixed; in 4.3̇426̇ every digit after the point repeats.
Right. Every digit from the first after the point repeats, so it is pure.
The 4 does not repeat, so it is mixed; the answer is 4.3̇426̇.
The last is about division and repeating once more.
check yourselfpick one answer
Question 3
Which fraction turns into a repeated decimal?
Only a 2 below, so 2.5 is finite; 3/11 has 11 below and repeats.
Right. The denominator 11 is not built from 2s and 5s, so 3/11 = 0.2̇7̇ repeats.
4 = 2 × 2, so 5/4 = 1.25 is finite; the answer is 3/11.
√5/2 is irrational, so endless and non-repeating; 3/11 is the repeating one.
The whole note on eight cards.
cut from the notetap to turn over
Eight cards
QuestionAccording to the book, how many types of decimal fraction are there?
AnswerThree: finite, repeated, infinite
QuestionWhich decimal fraction is called repeated?
AnswerOne where all or some digits after the point appear again and again, e.g. 2.454545...
QuestionWhich type of decimal fraction is irrational?
AnswerInfinite (non-repeating) decimals; finite and repeated decimals are rational
QuestionWhat is the repeating part of a repeated decimal?
AnswerThe part that appears over and over; the rest is non-repeating
QuestionHow is 3.124124124... written with dots?
Answer3.1̇24̇ (dots on the first and last repeating digits)
QuestionWhich of 1.3̇ and 4.23511̇2̇ is a pure repeated decimal?
Answer1.3̇, because every digit after the point repeats
QuestionWhich prime factors in the denominator make a fraction a repeated decimal?
AnswerAny prime factor other than 2 and 5
QuestionHow does the number of repeating digits compare with the denominator?
AnswerIt is always less than the denominator
Source: Mathematics, Classes 9-10 (NCTB), Chapter 1 “Real Numbers”, the “Decimal Fractions” part.