# What do you call a fraction that cannot be written as a finite decimal?

For example, the fraction ⅓ cannot be written, because it repeats infinitely (0.33333333... etc). Is there a particular word for numbers that cannot be written directly, but must be expressed as fractions?

• They're called rational numbers with nonterminating decimal expansions. – John Lawler Jul 22 '14 at 14:21
• And what of irrational numbers, those with non-terminating but non-repeating decimal representations ? – High Performance Mark Jul 22 '14 at 14:23
• @HighPerformanceMark I think those are disqualified by "must be expressed as fractions" since irrational numbers can't be expressed that way (at least using integers). – Gob Ties Jul 22 '14 at 14:25
• en.wikipedia.org/wiki/Repeating_decimal – Pam Jul 22 '14 at 14:44
• I know what you meant, but in fact these numbers do not have to be expressed as fractions. For example, "0.333..." is just as precise as "1/3". The dots tell you that it repeats. – Rupe Jul 22 '14 at 16:06

Also Periodic decimal:

Richard Suchenwirth 2002-04-27 - Periodic decimal fractions are numbers where a sequence of digits behind the decimal point (the period) is endlessly repeated,

for example:

1/7 = 0.142857142857.. 1/3 = 0.3333..

Repeating decimal appears to be the most common definition according to : Ngram

Source:http://wiki.tcl.tk/3310

• Best answer for being the only one to include all three answers: periodic, repeating, recurring. The Ngram is helpful. – Benubird Jul 22 '14 at 16:51

Do you mean a recurring decimal?

A decimal number that has digits that repeat forever.

[Math is Fun]

"Periodic decimal" is actually a name for a representation of such numbers, just as vulgar fraction is a name for the representation `p/q`.

One could call the numbers themselves:

• Numbers which have a periodic decimal representation.
• Rationals which in their simplest form `p/q`, `q` has a factor other than 2 or 5.

This is another way of saying that the numbers themselves do not really have a name used by mathematicians. The reason for this is that the decimal representation is not usually considered particularly special.

One can for example represent ⅓ in ternary exactly as `0.1`.

Is there a word for numbers that can be expressed as fractions but not as finite decimals? Not exactly.

There is a word for numbers that cannot be expressed as fractions; they are called irrational numbers.

Any number that can be expressed as a fraction is a rational number. However, this term encompasses both numbers like 1/3 that have repeating decimal representations, and numbers like 1/2 that don't.

Any number that, like 1/3, has a decimal number that repeats endlessly, is called a repeating decimal or recurring decimal.

There are also numbers like pi and e that have decimal portions that neither terminate nor repeat. I believe, however, that these numbers are always irrational, meaning they cannot be represented as fractions.

If that is the case--and I welcome corrections from any mathematicians slumming on the ELL board--then all of the numbers you refer to would be repeating/recurring decimal numbers.

• When we say 'Any number that can be expressed as a [common] fraction is a rational number', we're conveniently assuming that, if a/b is the simplest form, the integers a and b are not so large that we'll not live long enough to write them . – Edwin Ashworth Jul 22 '14 at 16:01
• @EdwinAshworth Any number X can be expressed as a fraction: X/1. I think the ability to write the number is not the main assumption being made when we call something rational. – Benubird Jul 22 '14 at 16:50
• @Benubird 'We'? I was a maths teacher. But this is an English usage site. The modal 'can' carries different senses / nuances, and I was disambiguating. In mathsspeak, I'd use 'a rational is a number that can be expressed in the form a/b where a and b are integers' without worrying about the implications of 'can'. But the mathematical register is largely OT here. – Edwin Ashworth Jul 22 '14 at 16:59
• @EdwinAshworth I find it interesting that in the discussion on this topic I find no reference to 'vulgar fraction', which is what they were called when I was at school. Has the name been discontinued to suppress titters of laughter in classrooms? – WS2 Jul 22 '14 at 17:19
• @WS2 The usual denotation of 'vulgar' ([1] lacking sophistication or good taste; unrefined. "the vulgar trappings of wealth" synonyms: tasteless, crass, tawdry, ostentatious, flamboyant, overdone, showy, gaudy, garish, brassy, kitsch, [Google]) surely makes the less baggagey 'common' (which is equally at home in the 'usual' and 'tawdrey / vulgar' senses) the better choice. We used to have a slight laugh with 'vulgar fractions'; it's only right to point out an acceptable usage, but also that it is quite a rare one nowadays. – Edwin Ashworth Jul 22 '14 at 17:29