Matemáticas
What Is a Recurring Decimal?
A recurring decimal repeats a digit or block for ever, marked with dots. Whether a fraction recurs is decided by the prime factors of its denominator.
Recurring decimal
A recurring decimal is a decimal in which one digit or a block of digits repeats without end, shown by placing a dot over the repeating part.
- También llamado
- repeating decimal
- Dónde lo encuentra el alumnado
- Grade 6 to 8 when fractions are first turned into decimals, and again at GCSE Higher, where converting a recurring decimal into a fraction appears in the additional key stage 4 content.
La respuesta corta
A recurring decimal is one in which a digit or a block of digits repeats for ever, such as 0.333… or 0.127127…. A dot marks the repetition: one dot over a single repeating digit, or dots over the first and last digits of a longer block. Every recurring decimal is a rational number.
Un ejemplo
1/7 = 0.1̇42857̇ = 0.142857142857…
Six digits repeat, and the dots sit on the first and the last of them to show exactly where the block begins and ends. Writing 0.142857 without dots claims something false: that the decimal stops there. The dots are not decoration, they are the difference between an approximation and an exact value.
What the dots cover, precisely
A dot marks a digit that repeats for ever. When more than one digit repeats, two dots are used, one on the first digit of the block and one on the last, and everything between them is included. Digits outside the dots happen once.
Where the dots sit is the whole of the information, so a misplaced dot names a different number. 0.416̇ is 0.41666…, while 0.4̇16̇ is 0.416416416… — same digits, and nothing else in common.
- 0.3̇ = 0.3333… — a single dot on a single repeating digit. This is 1/3.
- 0.416̇ = 0.41666… — only the 6 recurs; the 4 and the 1 occur once each. This is 5/12.
- 0.1̇27̇ = 0.127127127… — dots on the first and last digit of a three-digit block.
- 0.1̇42857̇ = 0.142857142857… — a six-digit block, which is 1/7.
Which fractions stop and which never do
There is a clean test, and it uses the denominator alone. Put the fraction in its lowest terms and factorise the denominator. If the only prime factors are 2s and 5s, the decimal terminates. If anything else appears, it recurs, and there are no other outcomes.
The reason is that our decimals are built on ten, and ten is 2 × 5. So 1/8 has denominator 2 × 2 × 2 and stops at 0.125; 1/20 is 2 × 2 × 5 and stops at 0.05; 1/6 has a 3 in it and recurs; 1/7 has a 7 and recurs. Lowest terms matters — 3/6 looks as though it has a 3, but it is 1/2, and it stops at 0.5.
Recurring is not the same as irrational
A decimal that never ends sounds like a decimal out of control, and students frequently file recurring decimals alongside π. They belong nowhere near it. 0.333… is exactly 1/3, an ordinary fraction with small whole numbers in it. What makes π irrational is not that its decimal is endless but that it never falls into a repeating pattern.
One consequence surprises people and is worth confronting rather than dodging: 0.9̇ is exactly 1. Since 1/3 is 0.3̇, multiplying both sides by three gives 1 = 0.9̇. They are not two numbers that are very close; they are two ways of writing the same one.
Preguntas frecuentes
How do you write a recurring decimal?
With dots above the repeating digits. One repeating digit takes one dot, as in 0.3̇ for a third. A repeating block takes two dots, on its first and last digits, so a seventh is written 0.1̇42857̇. Some books use a bar over the block instead, which means exactly the same thing.
Is a recurring decimal rational?
Yes, always. Every recurring decimal can be written as a fraction of two whole numbers, and there is a standard method for finding that fraction. The converse holds too: every fraction of whole numbers gives a decimal that either stops or recurs, which is why these two categories cover all the rationals.
Which fractions give recurring decimals?
Those whose denominator, in lowest terms, has a prime factor other than 2 or 5. Sevenths, thirds, ninths and elevenths all recur. Halves, quarters, fifths, eighths, twentieths and their relatives terminate, because their denominators are built only from the factors of ten.
Is 0.9 recurring really equal to 1?
Yes. One third is 0.3̇, and three thirds is 1, so 0.9̇ must be 1. Any two different numbers have others between them, and nothing can be squeezed between 0.9̇ and 1. It is a case of one number having two decimal names rather than two numbers being nearly equal.
Fuentes
- National curriculum in England: mathematics programmes of study — Department for Education
Última actualización
Saber la palabra no es lo mismo que usarla
Un profesor puede ver al alumno usarlo en un ejercicio y detectar justo dónde se detiene la comprensión. La primera clase es gratuita.