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.

part

"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:

fraction
n × 1n = 1
quotient
a ÷ b = c ≡ c × b = a
ratio
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., `&frac34;`) represent the three fractions in the ISO 8859-1 character set (a historical oddity inherited from typewriters), and the last (`&frasl;`) 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```