FRACTIONS

Fraction (mathematics)

A cake with one quarter removed. The remaining three quarters are shown.

A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.

The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on.^[1] A much later development were the common or "vulgar" fractions, still used today, such as 1/2, 5/8, 3/4, etc., which consist of a numerator and a denominator—the numerator representing a number of equal parts and the denominator telling how many of those parts make up a whole. An example is 3/4, in which the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole.^[2]

A still later development was the decimal fraction, now called simply a decimal, in which the denominator is a power of ten, determined by the number of digits to the right of a decimal separator, the appearance of which (e.g., a period, a raised period (•), a comma) depends on the locale (for examples, see decimal separator). Thus for 0.75 the numerator is 75 and the denominator is 10 to the second power, viz. 100, because there are two digits to the right of the decimal separator.

A third kind of fraction still in common use is the percentage, in which the denominator is always 100. Thus 75% means 75/100.

Fractional numbers can also be expressed using negative exponents, such as 10^-1 which represents 1/10, 7×5^-2 which represents 7/25, and 6.023×10^-7 which represents 0.0000006023.

Other uses for fractions are to represent ratios, and to represent division. Thus the fraction 3/4 is also used to represent the ratio 3:4 (three to four) and the division 3 ÷ 4 (three divided by four).

In mathematics, the set of all numbers which can be expressed as a fraction m/n, where m and n are integers and n is not zero, is called the set of rational numbers and is represented by the symbol Q. Some sources limit the definition of the word fraction to rational numbers.^[3][4][5][6] Other sources do not specify that fractions must be rational numbers,^[7] and the word fraction is often used for expressions written in the same form as a fraction that do include irrational numbers, for example √2/2 (see Square root of 2) and π/4 (see Proof that π is irrational). The word is also used in related expressions, such as continued fraction and algebraic fraction.