Fraction

A fraction 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, A common, vulgar, or simple fraction consists of a numerator displayed above a line (or before a slash), and a non-zero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction 3⁄4, 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. The picture to the right illustrates.or 3⁄4 of a cake.

A common fraction is a numeral which represents a rational number.That same number can also be represented as a decimal, a percent, or with a negative exponent. For example, 0.01, 1%, and 10−2 are all equal to the fraction 1/100. An integer can be thought of as having an implicit denominator of one (for example, 7 equals 7/1). Other uses for fractions are to represent ratios and division.Thus the fraction 3/4 can also 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). The non-zero denominator rule, which applies when representing a division as a fraction, is an example of the rule that division by zero is undefined. We can also write negative fractions, which 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 the set of all numbers that can be expressed in the form a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. A number is a rational number precisely when it can be written in that form (i.e., as a common fraction). However, the word fraction can also be used to describe mathematical expressions that are not rational numbers. Examples of these usages include algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as √2/2 . In a fraction, the number of equal parts being described is the numerator (  numerātor, "counter" or "numberer"), and the type or variety of the parts is the denominator, "thing that names or designates"). 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 part of fraction were called "piece fractions." The denominators of English fractions are generally expressed as ordinal numbers, in the plural if the numerator is not one. (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". When the denominator is 1, it may be expressed in terms of "wholes" but is more commonly ignored, with the numerator read out as a whole number. For example, 3/1 may be described as "three wholes", or simply as "three". When the numerator is one, it may be omitted (as in "a tenth" or "each quarter"). The entire fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. (For example, "two-fifths" is the fraction 2/5 and "two fifths" is the same fraction understood as 2 instances of 1⁄5.) Fractions should always be hyphenated when used as adjectives. Alternatively, a fraction may be described by reading it out as the numerator "over" the denominator, with the denominator expressed as a cardinal number. (For example, 3/1 may also be expressed as "three over one".) The term "over" is used even in the case of solidus fractions, where the numbers are placed left and right of a slash mark. (For example, 1/2 may be read "one-half", "one half", or "one over two".) 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").

The word ‘fraction’ has been derived from the Latin ‘fractus’ which means “broken”. Fraction has been into existence from the Egyptian era which is known to be one of the oldest civilizations in the world. However, fractions were not regarded as numbers, in fact, they were used to compare the whole numbers with one another.

A fraction has two parts. The number on the top of the line is called the numerator. It tells how many equal parts of the whole or collection are taken.  The number below the line is called the denominator.  It shows the total divisible number of equal parts the whole into or the total number of equal parts which are there in a collection. 

Fractions on a number line: Fractions can be represented on a number line, as shown below. 

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. Compound fractions, complex fractions, mixed numerals, and decimals (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. Unit fractions can also be expressed using negative exponents, as in 2−1, which represents 1/2, and 2−2, which represents 1/(22) or 1/4. A dyadic fraction is a common fraction in which the denominator is a power of two. 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 "a piece", so a proper fraction must be less than 1.This was explained in the 17th century. 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 2/3, –3/4, and 4/9, whereas examples of improper fractions are 9/4, –4/3, and 3/3. The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The product of a 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, its value is 1, and the fraction therefore is improper. Its reciprocal also has the value 1, and is improper, too. Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as where 1 is sometimes referred to as the invisible denominator. Therefore, every fraction or integer, except for zero, has a reciprocal. For example. the reciprocal of 17 is. A decimal fraction is a fraction whose denominator is not given explicitly, but is understood to be an integer power of ten. 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. Thus for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, 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 decimal (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

Decimal fractions can also be expressed using scientific notation  with negative exponents, such as 6.023×10−7, which represents 0.0000006023. The 10−7 represents a denominator of 107. Dividing by 107 moves the decimal point 7 places to the left. Decimal fractions with infinitely many digits to the right of the decimal separator represent 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 (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% equals 311/100, and −27% equals −27/100. The related concept of permille  or parts per thousand 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. Whether common fractions or decimal fractions are used is often a matter of taste and context. Common 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 accurate 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 decimals, for example $3.75. However, as noted above, in per-decimal British currency, shillings and pence were often given the form (but not the meaning) of a fraction, as, for example 3/6 (read "three and six") meaning 3 shillings and 6 pence, and having no relationship to the fraction 3/6. A mixed numeral (also called a mixed fraction or mixed number) is a traditional denotation of the sum of a non-zero integer and a proper fraction (having the same sign). It is used primarily in measurements.

An improper fraction can be converted to a mixed number as follows:Using Euclidean division (division with remainder), divide the numerator by the denominator. In the example,divide 11 by 4. 11 ÷ 4 = 2 remainder 3. The quotient (without the remainder) becomes the whole number part of the mixed number. The remainder becomes the numerator of the fractional part. In the example, 2 is the whole number part and 3 is the numerator of the fractional part.The new denominator is the same as the denominator of the improper fraction. In the example, it is 4

. An Egyptian fraction is the sum of distinct positive unit fractions, for example 

. This definition derives from the fact that the ancient Egyptians  expressed all fractions

 and

 in this manner. Every positive rational number can be expanded as an Egyptian fraction. For example,

 can be written as

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

 are a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number,corresponding to division of fractions and are complex fractions. To reduce a complex fraction to a simple fraction, treat the longest fraction line as representing division. If, in a complex fraction, there is no unique way to tell which fraction lines takes precedence, then this expression is improperly formed, because of ambiguity. So 5/10/20/40 is not a valid mathematical expression, because of multiple possible interpretations,A compound fraction is a fraction of a fraction, or any number of fractions connected with the word of, corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication. The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other. ( the compound fraction is equivalent to the complex fraction.) Nevertheless, "complex fraction" and "compound fraction" may both be considered outdated and now used in no well-defined manner, partly even taken synonymous for each other or for mixed numerals.They have lost their meaning as technical terms and the attributes "complex" and "compound" tend to be used in their every day meaning of "consisting of 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, the fraction equals A compound fraction is a fraction of a fraction, or any number of fractions connected with the word of, corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication. For example, of  is a compound fraction, corresponding to . The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other. (For example, the compound fraction  is equivalent to the complex fraction.) Nevertheless, "complex fraction" and "compound fraction" may both be considered outdated and now used in no well-defined manner, partly even taken synonymous for each other or for mixed numerals.They have lost their meaning as technical terms and the attributes "complex" and "compound" tend to be used in their every day meaning of "consisting of 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, the fraction  equals . Therefore, multiplying by  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 . When the numerator and denominator are both multiplied by 2, the result is, which has the same value (0.5) as . To picture this visually, imagine cutting a cake into four pieces; two of the pieces together make up half the cake. Dividing the numerator and denominator of a fraction by the same non-zero number will also yield an equivalent fraction. If the numerator and the denominator of a fraction are both divisible by a number (called a factor) greater than 1, then the fraction can be reduced to an equivalent fraction with a smaller numerator and a smaller denominator. To do this, the greatest common factor is identified, and both the numerator and the denominator are divided by this factor. For example, if both the numerator and the denominator of the fraction are divisible by   then they can be written as and  so the fraction becomes, which can be reduced by dividing both the numerator and denominator by  to give the reduced fraction  If the numerator and the denominator do not share any factor greater than 1, then the fraction is said to be irreducible, in lowest terms, or in simplest terms. For example, is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, is in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1. Using these rules, we can show that, for example. As another example, since the greatest common divisor of 63 and 462 is 21, the fraction  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 positive integers.Comparing fractions with the same positive denominator yields the same result as comparing the numerators: because 3 > 2, and the equal denominators  are positive. If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions. 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  and, these are converted to (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 multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is. 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 is useful for the computation of antiderivatives of rational functions. 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. The Egyptians used Egyptian fractions. 1000 BC. About 4000 years ago, Egyptians divided with fractions using slightly different methods. They used least common multiples with unti fractions. Their methods gave the same answer as modern methods. The Egyptians also had a different notation for dyadic fractions in the Akhmin Wooden Tablet and several Rhind Mathematical Papyrus problems. The Greeks used unit fractions. 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.) In 150 BC Jain mathematicians in India wrote the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, and operations with fractions. A modern expression of fractions known as Bhinna Rasi seems to have originated in India in the work of Aryabhatta (c. AD 500), Brahmagupta (c. 628), and Bhaskara (c. 1150). Their works form fractions by placing the numerators (Sanskrit: amsa) over the denominators (cheda), but without a bar between them. In Sanskrit literature, fractions were always expressed as an addition to or subtraction from an integer. The integer was written on one line and the fraction in its two parts on the next line. 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.

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