Fractions represent division vertically, with the dividend (numerator) above the divisor (denominator), separated by at least one horizontal line. This is their modern meaning. But fractions have come a long way, and in the past, the meaning of fractions fit their etymology.


"Originally" (in Egypt), a fraction would represent an even portion of a group, such as one third of the sheep in a flock (for some reason, the memorable references on this subject only describe sheep). If you had twelve sheep, four sheep would make up the third fraction of the flock, seven would be "a fourth and a third", et cetera. Some portions might be indescribable, and a half a sheep would be inconceivable. So people would do this sort of math problem: what fraction of a flock of 36 sheep is 7 sheep? We would say, "that's easy: it's 736!" but they would call problems like "what is 1384 + 111" similarly trivial.

In the spirit of this legacy come the names for the dividend and the divisor of a fraction. (The strange proscription against adding like fractions or dealing with parts of an individual has - thank goodness - faded into the past.) The denominator tells which kind of fraction we are working with (2 would be halves, 3 thirds, et cetera), and the numerator represents how many of that denomination we have. In this way, although fractions, quotients, and ratios are the same in the eyes of today's mathematician, behind them there are subtly different meanings:

n × 1n = 1
a ÷ b = c ≡ c × b = a
a : b :: c : d ≡ a × d = b × c

In plain text, fractions are denoted as a/b. Using superscripts and subscripts, they can be represented diagonally in HTML: a/b. There are four fraction-related named HTML entities: ¼, ½, ¾, and : the first three (e.g., ¾) represent the three fractions in the ISO 8859-1 character set (a historical oddity inherited from typewriters), and the last (⁄) is a fraction slash for you to use in fractions of your own devising. In a complex fraction with fractional operands, sometimes two horizontal lines are used.

a/b   ad
=== = --
c/d   bc

See also

More reliable information is available from, for example, the Rhind papyrus, Diophantus' Arithmetica, or a good secondary source on the history of numbers. For use of fractions in HTML, see Jukka “Yucca” Korpela's site on Math in HTML (and CSS). The usual rigorous mathematical treatment of fractions today is as elements of the field of fractions derived from the integers (warning: technical). A simpler interpretation can be taken from Webster's 1913 definition.

Fractions are a subject in mathematics that are where many people become confused. In the United States, at least, fractions are usually the first topic in mathematics that follow the basic, whole number math that students learn in elementary school. Between fourth and sixths grade, fractions, and how to add, subtract, multiply, divide and reduce them are going to probably be the biggest topic in arithmetic. (Along with the related decimal and percents.) This is also where many students get lost, and get a fear of mathematics.

I used to think that fractions were a stupid thing to teach. Beyond the colloquial fractions used to describe things like tanks of gas, cups of sugar, and Jon Bon Jovi's dependence on prayer to reach his goals, fractions aren't used much in every day life. 13/17th aren't things we use to measure sugar. And dividing fractions is probably the best example of a useless mathematical skill.

That is how I used to think. I teach ABE/GED now, and I have come to understand how important fractions are, how necessary they are to understanding algebra, and how the skills used to manipulate fractions, even the seemingly useless ones, are a prerequisite for understanding beyond arithmetic and into algebra. Because fractions are not about cups of sugar or even about algorithms. Fractions are about the relationships between numbers.

From a concrete point of view, to say that 1/2 is equal to 3/6 is untrue. If Alice has one dollar, Bob has two dollars, Carl has three dollars, and Debbie has six dollars, Alice and Carl do not have the same amount of money. One and three are very different things. But Alice and Carl both have the same amount of money in relation to Bob and Debbie. It seems like a simple point, but the leap from concrete numbers to describing the relationships between numbers is a gigantic leap, and one that is needed to go into algebra and further.

So I now believe that as seemingly useless, frustrating, and discouraging as the process of teaching people how to divide 13/17 by 50/24 is, it is also very necessary in order for people to begin to work on the more abstract forms of mathematics.

I hope that after reading this, the reader has made sense out of at least one part of fifth grade.

Frac"tion (?), n. [F. fraction, L. fractio a breaking, fr. frangere, fractum, to break. See Break.]


The act of breaking, or state of being broken, especially by violence.


Neither can the natural body of Christ be subject to any fraction or breaking up. Foxe.


A portion; a fragment.

Some niggard fractions of an hour. Tennyson.

3. Arith. or Alg.

One or more aliquot parts of a unit or whole number; an expression for a definite portion of a unit or magnitude.

Common, ∨ Vulgar, fraction, a fraction in which the number of equal parts into which the integer is supposed to be divided is indicated by figures or letters, called the denominator, written below a line, over which is the numerator, indicating the number of these parts included in the fraction; as , one half, , two fifths. -- Complex fraction, a fraction having a fraction or mixed number in the numerator or denominator, or in both. Davies & Peck. -- Compound fraction, a fraction of a fraction; two or more fractions connected by of. -- Continued fraction, Decimal fraction, Partial fraction, etc. See under Continued, Decimal, Partial, etc. -- Improper fraction, a fraction in which the numerator is greater than the denominator. -- Proper fraction, a fraction in which the numerator is less than the denominator.


© Webster 1913.

Frac"tion, v. t. Chem.

To separate by means of, or to subject to, fractional distillation or crystallization; to fractionate; -- frequently used with out; as, to fraction out a certain grade of oil from pretroleum.


© Webster 1913.

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