Fraction

When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters.

The term fraction and the notation ⁠a/b⁠ can also be used for mathematical expressions that do not represent a rational number (for example

In a fraction, the number of equal parts being described is the numerator (from Latin: numerātor, "counter" or "numberer"), and the type or variety of the parts is the denominator (from Latin: dēnōminātor, "thing that names or designates").

Informally, the numerator and denominator may be distinguished by placement alone, but in formal contexts they are usually separated by a fraction bar.

The denominators of English fractions are generally expressed as ordinal numbers, in the plural if the numerator is not 1.

The term over is used even in the case of solidus fractions, where the numbers are placed left and right of a slash mark.

Fractions with large denominators that are not powers of ten are often rendered in this fashion (e.g., ⁠1/117⁠ as one over one hundred seventeen), while those with denominators divisible by ten are typically read in the normal ordinal fashion (e.g., ⁠6/1000000⁠ as six-millionths, six millionths, or six one-millionths).

Decimal fractions can also be expressed using scientific notation with negative exponents, such as 6.023×10−7, a convenient alternative to the unwieldy 0.0000006023.

The related concept of permille, or parts per thousand (ppt), means a denominator of 1000, and this parts-per notation is commonly used with larger denominators, such as million and billion, e.g. 75 parts per million (ppm) means that the proportion is ⁠75/1000000⁠.

The mixed number ⁠2+3/4⁠ is spoken two and three quarters or two and three fourths, with the integer and fraction portions connected by the word and.

Any mixed number can be converted to an improper fraction by applying the rules of adding unlike quantities.

In primary school, teachers often insist that every fractional result should be expressed as a mixed number.

[19] Outside school, mixed numbers are commonly used for describing measurements, for instance ⁠2+1/2⁠ hours or 5 3/16 inches, and remain widespread in daily life and in trades, especially in regions that do not use the decimalized metric system.

[20] College students with years of mathematical training are sometimes confused when re-encountering mixed numbers because they are used to the convention that juxtaposition in algebraic expressions means multiplication.

Any positive rational number can be written as a sum of unit fractions in infinitely many ways.

Like whole numbers, fractions obey the commutative, associative, and distributive laws, and the rule against division by zero.

As another example, since the greatest common divisor of 63 and 462 is 21, the fraction ⁠63/462⁠ can be reduced to lowest terms by dividing the numerator and denominator by 21: The Euclidean algorithm gives a method for finding the greatest common divisor of any two integers.

Since a whole number can be rewritten as itself divided by 1, normal fraction multiplication rules can still apply.

Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have.

For example: If leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros: If a non-repeating set of digits precede the pattern (such as 0.1523987), one may write the number as the sum of the non-repeating and repeating parts, respectively: Then, convert both parts to fractions, and add them using the methods described above: Alternatively, algebra can be used, such as below: In addition to being of great practical importance, fractions are also studied by mathematicians, who check that the rules for fractions given above are consistent and reliable.

For example, an algebraic fraction is in lowest terms if the only factors common to the numerator and the denominator are 1 and −1.

Considering the rational fractions with real coefficients, radical expressions representing numbers, such as ⁠

In computer displays and typography, simple fractions are sometimes printed as a single character, e.g. ½ (one half).

Followers of the Greek philosopher Pythagoras (c. 530 BC) discovered that the square root of two cannot be expressed as a fraction of integers.

(This is commonly though probably erroneously ascribed to Hippasus of Metapontum, who is said to have been executed for revealing this fact.)

A modern expression of fractions known as bhinnarasi seems to have originated in India in the work of Aryabhatta (c. AD 500),[citation needed] Brahmagupta (c. 628), and Bhaskara (c. 1150).

[35] Their works form fractions by placing the numerators (Sanskrit: amsa) over the denominators (cheda), but without a bar between them.

If the fraction was marked by a small circle ⟨०⟩ or cross ⟨+⟩, it is subtracted from the integer; if no such sign appears, it is understood to be added.

[38] In discussing the origins of decimal fractions, Dirk Jan Struik states:[39] The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphlet De Thiende, published at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon Stevin (1548–1620), then settled in the Northern Netherlands.

It is true that decimal fractions were used by the Chinese many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century).