Fraction

A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. 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. A simple fraction (examples: ⁠1/2⁠ and ⁠17/3⁠) consists of an integer numerator, displayed above a line (or before a slash like 1⁄2), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction ⁠3/4⁠, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates ⁠3/4⁠ of a cake.

Fractions can be used to represent ratios and division. Thus the fraction ⁠3/4⁠ can be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division 3 ÷ 4 (three divided by four).

Negative fractions represent the opposite of a positive fraction. For example, if ⁠1/2⁠ represents a half-dollar profit, then −⁠1/2⁠ represents a half-dollar loss. Because of the rules of division of signed numbers (which states in part that negative divided by positive is negative), −⁠1/2⁠, ⁠−1/2⁠ and ⁠1/−2⁠ all represent the same fraction – negative one-half. And because a negative divided by a negative produces a positive, ⁠−1/−2⁠ represents positive one-half.

In mathematics a rational number is a number that can be represented by a fraction of the form ⁠a/b⁠, where a and b are integers and b is not zero; the set of all rational numbers is commonly represented by the symbol ⁠\(\mathbb{Q}\)⁠ or Q, which stands for quotient. The term fraction and the notation ⁠a/b⁠ can also be used for mathematical expressions that do not represent a rational number (for example \(\textstyle \frac \sqrt 2 2\)), or even do not represent any number (for example the rational fraction \(\textstyle \frac 1 x\)).

A rational number, expressed as \(\frac{p}{q}\) where p and q are coprime integers and is in base \({b}\), has a terminating representation in base \({b}\) if and only if q divides a power of b, or \(\frac{p}{q} = \frac{C}{b^n}\),for some \({C}\) and some integer \({n}\) > 0.

By cross multiplying, the equality is equivalent to \({qC}={pb^n}\). Because q doesn't divide p , it must divide \(b^n\), and the expansion will not continue.

In a fraction, the number of equal parts being described is the numerator (numerātor being Latin for a "counter" or "numberer") and the type or variety of the parts is the denominator (dēnōminātor being Latin for a "namer" or "designator"). As an example, the fraction ⁠8/5⁠ amounts to eight parts, each of which is of the type named fifth. In terms of division, the numerator corresponds to the dividend and the denominator corresponds to the divisor.

Informally, the numerator and denominator may be distinguished by placement alone, but in formal contexts they are usually separated by a fraction bar. The fraction bar may be horizontal (as in ⁠1/3⁠), oblique (as in 2/5), or diagonal (as in 4⁄9). These marks are respectively known as the horizontal bar; the virgule, slash (US), or stroke (UK); and the fraction bar, solidus, or fraction slash. In typography, fractions stacked vertically are also known as en or nut fractions, and diagonal ones as em or mutton fractions, based on whether a fraction with a single-digit numerator and denominator occupies the proportion of a narrow en square, or a wider em square. In traditional typefounding, a piece of type bearing a complete fraction (e.g. ⁠1/2⁠) was known as a case fraction, while those representing only parts of fractions were called piece fractions.

The denominators of English fractions are generally expressed as ordinal numbers, pluralized when the numerator is not 1. (For example, ⁠1/5⁠ and ⁠2/5⁠ are read as one-fifth and two-fifths.) There are a few notable exceptions for some of the most common fractions. The numerator is often omitted entirely when it is 1. (For example, ⁠1/5⁠ may also be read as a fifth.) Fractions with a denominator of 1 are usually read as the numerator alone, sometimes in terms of wholes but never as firsts. (For example, ⁠5/1⁠ may be read as five or five wholes.) Fractions with the denominator 2 are read as half or halves, never seconds; the denominator 4 may be read as quarter/quarters as well as fourth/fourths; and the denominator 100 may be read as percent as well as hundredth/hundredths.

Any fraction may also be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. (For example, ⁠1/2⁠ and ⁠2/5⁠ may also be expressed as one over two and two over five.) The term over is used regardless of the form of the fraction bar. (For example, ⁠1/5⁠, 1/5, and 1⁄5 may all be read as one over five.) Fractions with unusual components or with large denominators that are not powers of ten are often rendered in this fashion (e.g., ⁠√2/2⁠, ⁠1/x⁠, and ⁠1/117⁠ as square root of two over two, one over x, and one over one hundred seventeen), while those with large denominators divisible by ten are typically read in the normal ordinal fashion (e.g., ⁠6/1000000⁠ as six-millionths).

When spelling out fractions, they are properly hyphenated, especially when used as adjectives. The fraction ⁠2/5⁠ as a single composition should be written out as two-fifths. Writing two fifths without the hyphen describes the same value but understood as two separate instances of ⁠1/5⁠.

A simple fraction (also known as a common fraction or vulgar fraction) is a rational number written as \(a \div b\) or ⁠\(\tfrac{a}{b}\)⁠, where a and b are both integers. As with other fractions, the denominator (b) cannot be zero. Examples include ⁠1/2⁠, −⁠8/5⁠, ⁠−8/5⁠, and ⁠8/−5⁠. The term was originally used to distinguish this type of fraction from the sexagesimal fraction used in astronomy.

Common fractions can be positive or negative, and they can be proper or improper (see below). Compound fractions, complex fractions, mixed numerals, and decimal expressions (see below) are not common fractions though, unless irrational, they can be evaluated to a common fraction.

• A unit fraction is a common fraction with a numerator of 1 (e.g., ⁠1/7⁠). Unit fractions can also be expressed using negative exponents, as in \(2^{-1}\), which represents \(\frac{1}{2^{1}}\), which also equals \(\frac{1}{2}\), and \(2^{-2}\) which represents \(\frac{1}{2^{2}}\) or \(\frac{1}{4}\).
• A dyadic fraction is a common fraction in which the denominator is a power of two, e.g. ⁠1/8⁠ = ⁠1/2⁠.

In Unicode, precomposed fraction characters are in the Number Forms block.

Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise. The concept of an improper fraction is a late development, with the terminology deriving from the fact that fraction means "piece", so a proper fraction must be less than 1. This was explained in the 17th century textbook The Ground of Arts.

In general, a common fraction is said to be a proper fraction if the absolute value of the fraction is strictly less than one, that is, if the fraction is greater than −1 and less than 1. It is said to be an improper fraction, or sometimes top-heavy fraction, if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are \(\frac{2}{3}\), −3/4, and 4/9, whereas examples of improper fractions are \(\frac{9}{4}\), \(\frac{-4}{3}\), and \(\frac{3}{3}\). As described below, any improper fraction can be converted to a mixed number (integer plus proper fraction), and vice versa.

The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of ⁠3/7⁠, for instance, is ⁠7/3⁠. The product of a non-zero fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 (that is, numerator and denominator are not equal) is a proper fraction.

When the numerator and denominator of a fraction are equal (for example, ⁠7/7⁠), its value is 1, and the fraction therefore is improper. Its reciprocal is identical and hence also equal to 1 and improper.

Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as ⁠17/1⁠, where 1 is sometimes referred to as the invisible denominator. Therefore, every fraction and every integer, except for zero, has a reciprocal. For example, the reciprocal of 17 is ⁠1/17⁠.

A ratio is a relationship between two or more numbers that can be sometimes expressed as a fraction. Typically, a number of items are grouped and compared in a ratio, specifying numerically the relationship between each group. Ratios are expressed as "group 1 to group 2 ... to group n". For example, if a car lot had 12 vehicles, of which

• 2 are white,
• 6 are red, and
• 4 are yellow,

then the ratio of red to white to yellow cars is 6 to 2 to 4. The ratio of yellow cars to white cars is 4 to 2 and may be expressed as 4:2 or 2:1.

A ratio is often converted to a fraction when it is expressed as a ratio to the whole. In the above example, the ratio of yellow cars to all the cars on the lot is 4:12 or 1:3. We can convert these ratios to a fraction, and say that ⁠4/12⁠ of the cars or ⁠1/3⁠ of the cars in the lot are yellow. Therefore, if a person randomly chose one car on the lot, then there is a one in three chance or probability that it would be yellow.

A decimal fraction is a fraction whose denominator is an integer power of ten, commonly expressed using decimal notation, in which the denominator is not given explicitly but is implied by the number of digits to the right of a decimal separator. The separator can be a period ⟨.⟩, interpunct ⟨·⟩, or comma ⟨,⟩, depending on locale. (For examples, see Decimal separator.) Thus, for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, namely, 100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (such as 3.75), the fractional part of the number is expressed by the digits to the right of the separator (with a value of 0.75 in this case). 3.75 can be written either as an improper fraction, ⁠375/100⁠, or as a mixed number, ⁠3+75/100⁠.

Decimal fractions can also be expressed using scientific notation with negative exponents, such as 6.023×10, a convenient alternative to the unwieldy 0.0000006023. The 10 represents a denominator of 10. Dividing by 10 moves the decimal point seven places to the left.

A decimal fraction with infinitely many digits to the right of the decimal separator represents an infinite series. For example, ⁠1/3⁠ = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ....

Another kind of fraction is the percentage (from Latin: per centum, meaning "per hundred", represented by the symbol %), in which the implied denominator is always 100. Thus, 51% means 51⁄100. Percentages greater than 100 or less than zero are treated in the same way, e.g. 311% means 311⁄100 and −27% means −27⁄100.

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 choice between fraction and decimal notation is often a matter of taste and context. Fractions are used most often when the denominator is relatively small. By mental calculation, it is easier to multiply 16 by 3⁄16 than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more precise (exact, in fact) to multiply 15 by 1⁄3, for example, than it is to multiply 15 by any decimal approximation of one third. Monetary values are commonly expressed as decimal fractions with denominator 100, i.e., with two digits after the decimal separator, for example $3.75. However, as noted above, in pre-decimal British currency, shillings and pence were often given the form (but not the meaning) of a fraction, as, for example, "3/6", commonly read three and six, means three shillings and sixpence and has no relationship to the fraction three sixths.

A mixed number (also called a mixed fraction or mixed numeral) is the sum of a non-zero integer and a proper fraction, conventionally written by juxtaposition (or concatenation) of the two parts, without the use of an intermediate plus (+) or minus (−) sign. When the fraction is written horizontally, a space is added between the integer and fraction to separate them.

As a basic example, two entire cakes and three quarters of another cake might be written as \(2\tfrac{3}{4}\) cakes or \(2\ \,3/4\) cakes, with the numeral \(2\) representing the whole cakes and the fraction \(\tfrac34\) representing the additional partial cake juxtaposed; this is more concise than the more explicit notation \(2+\tfrac{3}{4}\) cakes. 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. Subtraction or negation is applied to the entire mixed numeral, so \(-2\tfrac{3}{4}\) means \(-\bigl(2+\tfrac{3}{4}\bigr).\)

Any mixed number can be converted to an improper fraction by applying the rules of adding unlike quantities: change the measure of one of the unlike quantities so it has the same measure as the other. For example, \(2 + \tfrac34 = \tfrac84 + \tfrac34 = \tfrac{11}4.\) Alternatively, one can multiply the whole number part by the denominator, then add the numerator, to get the numerator of the improper fraction, which keeps the same denominator as the improper fraction. Conversely, an improper fraction can be converted to a mixed number using division with remainder, with the mixed number consisting the whole number part as the quotient and the fraction part (if any) being the remainder divided by denominator of the original improper fraction. For example, since 4 goes into 11 twice, with 3 left over, \(\tfrac{11}4 = 2 + \tfrac{3}{4}.\)

In primary school, teachers often insist that every fractional result should be expressed as a mixed number. 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. However, scientific measurements typically use the metric system, which is based on decimal fractions, and starting from the secondary school level, mathematics pedagogy treats every fraction uniformly as a rational number, the quotient ⁠p/q⁠ of integers, leaving behind the concepts of improper fraction and mixed number. 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.

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

Mixed-number arithmetic can be performed either by converting each mixed number to an improper fraction, or by treating each as a sum of integer and fractional parts.

Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. This is true because for any non-zero number \(n\), the fraction \(\tfrac{n}{n}\) equals 1. Therefore, multiplying by \(\tfrac{n}{n}\) is the same as multiplying by one, and any number multiplied by one has the same value as the original number. By way of an example, start with the fraction ⁠\(\tfrac{1}{2}\)⁠. When the numerator and denominator are both multiplied by 2, the result is ⁠2/4⁠, which has the same value (0.5) as ⁠1/2⁠. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together (⁠2/4⁠) make up half the cake (⁠1/2⁠).

Comparing fractions with the same positive denominator yields the same result as comparing the numerators:

\(\tfrac{3}{4}>\tfrac{2}{4}\) because 3 > 2, and the equal denominators \(4\) are positive.

If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions:

\(\tfrac{3}{-4}<\tfrac{2}{-4} \text{ because } \tfrac{a}{-b}= \tfrac{-a}{b} \text{ and } -3 < -2.\)

If two positive fractions have the same numerator, then the fraction with the smaller denominator is the larger number. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger.

One way to compare fractions with different numerators and denominators is to find a common denominator. To compare \(\tfrac{a}{b}\) and \(\tfrac{c}{d}\), these are converted to \(\tfrac{a\cdot d}{b\cdot d}\) and \(\tfrac{b\cdot c}{b\cdot d}\) (where the dot signifies multiplication and is an alternative symbol to ×). Then bd is a common denominator and the numerators ad and bc can be compared. It is not necessary to determine the value of the common denominator to compare fractions. One can just compare ad and bc, without evaluating bd, e.g., comparing \(\tfrac{2}{3}\) ? \(\tfrac{1}{2}\) gives \(\tfrac{4}{6}>\tfrac{3}{6}\).

For the more laborious question \(\tfrac{5}{18}\) ? \(\tfrac{4}{17},\) multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding \(\tfrac{5 \times 17}{18 \times 17}\) ? \(\tfrac{18 \times 4}{18 \times 17}\). It is not necessary to calculate \(18 \times 17\), only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is ⁠\(\tfrac{5}{18}>\tfrac{4}{17}\)⁠.

Because every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, it follows that any negative fraction is less than any positive fraction. This allows, together with the above rules, to compare all possible fractions.

The first rule of addition is that only like quantities can be added; for example, various quantities of quarters. Unlike quantities, such as adding thirds to quarters, must first be converted to like quantities as described below: Imagine a pocket containing two quarters, and another pocket containing three quarters; in total, there are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows:

\(\tfrac24+\tfrac34=\tfrac54=1\tfrac14\).

The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance,

\(\tfrac23-\tfrac12=\tfrac46-\tfrac36=\tfrac16.\)

To subtract a mixed number, an extra one can be borrowed from the minuend, for instance

\(4 - 2\tfrac34 = (4-2-1) + \bigl(1 - \tfrac34\bigr) = 1\tfrac14.\)

To divide a fraction by a whole number, you may either divide the numerator by the number, if it goes evenly into the numerator, or multiply the denominator by the number. For example, \(\tfrac{10}{3} \div 5\) equals \(\tfrac{2}{3}\) and also equals \(\tfrac{10}{3 \cdot 5} = \tfrac{10}{15}\), which reduces to \(\tfrac{2}{3}\). To divide a number by a fraction, multiply that number by the reciprocal of that fraction. Thus, \(\tfrac{1}{2} \div \tfrac{3}{4} = \tfrac{1}{2} \times \tfrac{4}{3} = \tfrac{1 \cdot 4}{2 \cdot 3} = \tfrac{2}{3}\).