Fraction (mathematics)

denominatorfractionsfractionnumeratorvulgar fractionfractionalfraction barimproper fractionmixed numbercommon fraction
A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.wikipedia
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Rational number

rationalrational numbersrationals
A common fraction is a numeral which represents a rational number.
In mathematics, a rational number is a number that can be expressed as the quotient or fraction

Ratio

ratiosproportionratio analysis
Other uses for fractions are to represent ratios and division.
Consequently, a ratio may be considered as an ordered pair of numbers, as a fraction with the first number in the numerator and the second as denominator, or as the value denoted by this fraction.

Algebraic fraction

fractionsfractionIrrational fraction
However, the word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as √2/2 (see square root of 2) and π/4 (see proof that π is irrational).
In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions.

Integer

integersintegralZ
A common, vulgar, or simple fraction (examples: and ) consists of an integer numerator displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. Mathematicians define a fraction as an ordered pair (a,b)of integers aand b \ne 0, for which the operations addition, subtraction, multiplication, and division are defined as follows:
An integer (from the Latin integer meaning "whole") is a number that can be written without a fractional component.

Unit fraction

unit fractions
They used least common multiples with unit fractions.
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer.

Division by zero

divide by zerodividing by zerodivided by zero
The non-zero denominator in the case using a fraction to represent division is an example of the rule that division by zero is undefined.
Or, the problem with 5 cookies and 2 people can be solved by cutting one cookie in half, which introduces the idea of fractions (5⁄2 = 21⁄2).

Decimal separator

decimal pointdecimal markthousands separator
Decimal fractions are commonly expressed using decimal notation in which the implied denominator is 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).
A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form.

Slash (punctuation)

slash/solidus
These marks are respectively known as the horizontal bar; the virgule, slash (US), or stroke (UK); and the fraction bar, solidus, or fraction slash.
The fraction slash is used between two numbers to indicate a fraction or ratio.

Division (mathematics)

divisiondividingdivided
Other uses for fractions are to represent ratios and division. Mathematicians define a fraction as an ordered pair (a,b)of integers aand b \ne 0, for which the operations addition, subtraction, multiplication, and division are defined as follows:
Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a fraction bar, between them.

Percentage

percent%FG%
(For example, 2⁄5 and 3⁄5 are both read as a number of "fifths".) Exceptions include the denominator 2, which is always read "half" or "halves", the denominator 4, which may be alternatively expressed as "quarter"/"quarters" or as "fourth"/"fourths", and the denominator 100, which may be alternatively expressed as "hundredth"/"hundredths" or "percent".
In mathematics, a percentage is a number or ratio expressed as a fraction of 100.

Parts-per notation

ppmparts per millionppb
The related concept of permille or parts per thousand (ppt) has an implied denominator of 1000, while the more general parts-per notation, as in 75 parts per million (ppm), means that the proportion is 75/1,000,000.
Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement.

Egyptian fraction

Egyptian fractionssum of unit fractionsfractions
An Egyptian fraction is the sum of distinct positive unit fractions, for example.
:That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other.

Power of two

powers of twopower of 2powers of 2
A fraction that has a power of two as its denominator is called a dyadic rational.

Euclidean algorithm

Euclid's algorithmEuclideanEuclid
The Euclidean algorithm gives a method for finding the greatest common divisor of any two positive integers.
It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.

Greatest common divisor

gcdcommon factorgreatest common factor
To do this, the greatest common factor is identified, and both the numerator and the denominator are divided by this factor.
The greatest common divisor is useful for reducing fractions to be in lowest terms.

Subtraction

differencesubtrahendminuend
Mathematicians define a fraction as an ordered pair (a,b)of integers aand b \ne 0, for which the operations addition, subtraction, multiplication, and division are defined as follows:
Subtraction represents removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices.

Addition

sumaddadded
Mathematicians define a fraction as an ordered pair (a,b)of integers aand b \ne 0, for which the operations addition, subtraction, multiplication, and division are defined as follows:
* A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number.

Least common multiple

lcmcommon multiplelowest common multiple
The smallest possible denominator is given by the least common multiple of the single denominators, which results from dividing the rote multiple by all common factors of the single denominators.
The LCM is the "lowest common denominator" (LCD) that can be used before fractions can be added, subtracted or compared.

Ordinal numeral

ordinal numberordinal numbersordinal
The denominators of English fractions are generally expressed as ordinal numbers, in the plural if the numerator is not one.
When speaking the numbers in fractions, the spatial/chronological numbering system is used for denominators larger than 2 (2 as the denominator of a fraction is "half" rather than "second"), with a denominator of 4 sometimes spoken as "quarter" rather than "fourth".

Partial fraction decomposition

partial fractionpartial fractions in integrationpartial fraction expansion
The term partial fraction is used when decomposing rational expressions into sums.
In algebra, the partial fraction decomposition or partial fraction expansion of a rational function (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

Field (mathematics)

fieldfieldsfield theory
Algebraic fractions are subject to the same field properties as arithmetic fractions.
They are numbers that can be written as fractions

Dyadic rational

dyadic fractiondyadic rational numberdyadic
The Egyptians also had a different notation for dyadic fractions in the Akhmim Wooden Tablet and several Rhind Mathematical Papyrus problems.
In mathematics, a dyadic fraction or dyadic rational is a rational number whose denominator, when the ratio is in minimal (coprime) terms, is a power of two, i.e., a number of the form where a is an integer and b is a natural number; for example, 1/2 or 3/8, but not 1/3.

Rationalisation (mathematics)

rationalisationrationalizationrationalising
If the denominator contains radicals, it can be helpful to rationalize it (compare Simplified form of a radical expression), especially if further operations, such as adding or comparing that fraction to another, are to be carried out.
In elementary algebra, root rationalisation is a process by which radicals in the denominator of an algebraic fraction are eliminated.

Nth root

radicalsn''th rootroot
If the denominator contains radicals, it can be helpful to rationalize it (compare Simplified form of a radical expression), especially if further operations, such as adding or comparing that fraction to another, are to be carried out. Considering the field of fractions generated by polynomials with real coefficients, radical expressions such as are also rational fractions, as is the transcendental expression \pi/2, since all of and 2are (constant) polynomials over the reals.
In calculus, roots are treated as special cases of exponentiation, where the exponent is a fraction:

Irrational number

irrationalirrational numbersirrationality
However, the word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as √2/2 (see square root of 2) and π/4 (see proof that π is irrational).
In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.