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In arithmetic, a fraction is a quantity which represents a part or parts of a unit or a whole. A fraction is expressed by two terms, called the numerator and the denominator (Lees). The denominator indicates the number of equal parts into which the whole is divided, and the numerator indicates the number of these equal parts contained in the fraction (Scott). A fraction may be considered as the quotient arising from the division of the numerator by the denominator (Lees), and is a rational number.

In elementary algebra, a fraction (known as an algebraic fraction) is an indicated division in which the divisor, or both dividend and divisor, are algebraic expressions, and the dividend is not a multiple of the divisor. Two examples of algebraic fractions are ${\frac {3x}{x^{2}+2x-3}}$ and ${\frac {\sqrt {x+2}}{x^{2}-3}}$. If the numerator and the denominator are polynomials, as in example 1, the algebraic fraction is called a rational algebraic fraction or simply rational fraction (or rational expression). An irrational fraction is one that contains the variable under a fractional exponent, as in example 2. Since any coefficient is a polynomial of degree zero, a radical expression such as √2/2 is a rational fraction, whereas ${\frac {10}{\sqrt {a}}}$ is considered an irrational fraction.

A fraction is a rational number expressed in the form a/b (in-line notation) or ${\frac {a}{b}}$ (traditional "display" notation), where a and b are called numerator and denominator, respectively, and $b\neq 0$. For example, 2/3, 1.4/4, and ${\frac {1/2}{3/5}}$ are fractions, while π/2 and $e^{-1}$ are not (although π/2 is written in inline form, it is not a rational number).

Definitions

A common fraction represents one or more equal parts of a unit or a whole and can be written as a/b or ${\tfrac {a}{b}}$, where a and b are natural numbers and b ≠ 0. If a < b, the common fraction is called a proper fraction, otherwise it is called an improper fraction. For example, 2/3 is a proper fraction, while 4/3 is an improper fraction.

Formally, an arithmetic fraction may be represented as an ordered pair (a, b) of integers a and b ≠ 0, for which the operations addition, subtraction, multiplication, and division are defined as follows:

$(a,b)+(c,d)=(ad+bc,bd)$

$(a,b)-(c,d)=(ad-bc,bd)$

$(a,b)\cdot (c,d)=(ac,bd)$

$(a,b):(c,d)=(ad,bc)$

In addition, an equivalence relation is specified as follows: $(a,b)$ and $(c,d)$ are considered equal if $ad=bc$.

More generally, a and b may be elements of any integral domain R, in which case a fraction is an element of the field of fractions of R. For example, when a and b are polynomials in one indeterminate, the field of fractions is the field of rational fractions (also known as the field of rational functions). When a and b are integers, the field of fractions is the field of rational numbers.

Basic terminology

A common fraction (or, rarely, vulgar fraction) is a fraction where a and b are integers. For example, 2/3 is a common fraction, while 1.4/4 and ${\frac {1/2}{3/5}}$ are not. Common fractions are used to express part of a whole or a set, represent a ratio, or indicate division. Often common fractions are referred to simply as fractions.

A proper fraction is a fraction with $a,b>0$ and $a<b$. Otherwise, the fraction is called an improper fraction. For example, 2/3 is a proper fraction, while 4/3 is an improper fraction. An improper fraction is sometimes written as a mixed number (or mixed fraction) consisting of a whole number and a fraction, for example 4/3 can be written $1{\frac {1}{3}}$, which means $1+{\frac {1}{3}}$.

A common fraction is said to be in lowest terms or reduced if the integers a and b are relatively prime, i.e. their greatest common divisor is one. If $gcd(a,b)>1$, the fraction is said to be reducible, otherwise it is said to be irreducible.

A complex fraction is a fraction in which the numerator, the denominator, or both are themselves fractions, for example ${\frac {1/2}{3/5}}$ and ${\frac {2}{3/5}}$.

A compound fraction is a fraction of a fraction, as ${\frac {2}{3}}$ of ${\frac {4}{5}}$ (Lees).

Special cases

A decimal fraction is a fraction whose denominator is an integral power of ten. Decimal fractions are commonly expressed without a denominator, the decimal separator being inserted into the numerator (with leading zeros added if needed) at the position from the right corresponding to the power of ten of the denominator; for example, the decimal fractions 8/10, 83/100, 83/1000, and 8/10000 are expressed as 0.8, 0.83, 0.083, and 0.0008, respectively.

A unit fraction is a fraction whose numerator is one and whose denominator is a positive integer. A unit fraction is the reciprocal of a positive integer, 1/n. Examples are 1/1, 1/2, 1/3, 1/4, etc. A sum of unit fractions is known as an Egyptian fraction. Two fractions whose difference is a unit fraction are called adjacent fractions.

Distinction from ratio

In some ways, ratios are similar to fractions, but in some ways they are very different. Ratios are not always rational numbers, but part-whole, operator, measure and quotient fractions are always rational numbers. Consider the ratio of circumference of a circle to its radius, $C:r=2\pi$, which is not a rational number. Another example is the ratio of the side of a square to its diagonal, $1:{\sqrt {2}}$, which is also an irrational number. Ratios may have a zero as its second component, while fractions must have a non-zero denominator. For example, the ratio of men to women in a meeting attended by 10 males and no females is 10:0.

Generalizations

• Continued fraction
• Generalized continued fraction
• Algebraic fraction
• Partial fraction

From the lead

Some sources limit the definition of the word fraction to rational numbers. Every rational number p/q can be expressed as a finite continued fraction using the Euclidean algorithm.

Other sources do not specify that fractions must be rational numbers, and the word fraction is often used for expressions written in the same form as a fraction that do include irrational numbers. The term fraction may also refer to the indicated quotient of two algebraic expressions, known as an algebraic fraction.

Continued fractions

A continued fraction is an expression such as

$a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+\ddots }}}},$

where a_i are integers. Every rational number p/q has two closely related expressions as a finite continued fraction, whose coefficients a_i can be determined by applying the Euclidean algorithm to (p,q).

Algebraic fractions

In elementary algebra, an algebraic fraction is the indicated quotient of two algebraic expressions. Two examples of algebraic fractions are ${\frac {3x}{x^{2}+2x-3}}$ and ${\frac {\sqrt {x+2}}{x^{2}-3}}$. Algebraic fractions are subject to the same laws as arithmetic fractions.

If the numerator and the denominator are polynomials, as in the first example, the algebraic fraction is called a rational fraction (or rational expression). An irrational fraction is one that contains the variable under a fractional exponent, as in the second example.

Rational fractions can be characterized in a similar way as arithmetic fractions, for example as proper or improper, depending on the degrees of the polynomials in the numeratator and the denominator. An improper rational fraction can be expressed as the sum of a polynomial and a proper rational fraction. A proper rational fraction can, just like a proper arithmetic fraction, be decomposed into partial fractions. In the case of ordinary fractions, prime numbers take the role of irreducible polynomial denominators.

Rational numbers are the quotient field of integers. Rational expressions are the quotient field of the polynomials (over some integral domain). Since a coefficient is a polynomial of degree zero, a radical expression such as √2/2 is a rational fraction. An example over the field of reals is $\textstyle {\tfrac {\pi }{2}}$, the radian measure of a right angle.

The term partial fraction is used when decomposing rational expressions. The goal is to write the rational expression as the sum of other rational expressions with denominators of lesser degree. For example, the rational expression $\textstyle {2x \over x^{2}-1}$ can be rewritten as the sum of two fractions: $\textstyle {1 \over x+1}$ and $\textstyle {1 \over x-1}$. This is useful in many areas such as integral calculus and differential equations.

See also

• Ratio
• Proportionality (mathematics)

References

• George Lees (1826). Elements of arithmetic, algebra and geometry.
• William Scott (1844). Elements of arithmetic and algebra.
• Joseph Ray (1848). Elementary algebra: Ray's Algebra. Part first.
• George Hervey Hallett (1844). Elementary algebra.