Fraction
A fraction takes its name from the Latin fractus, meaning broken. The word carries the idea of something split into equal parts. When you say one-half, eight-fifths, or three-quarters out loud, you are describing how many parts of a certain size there are. Above a line sits an integer numerator. Below it sits a non-zero integer denominator. In three-quarters, the 3 counts the equal parts you have, and the 4 says how many parts make a whole. A simple picture of a sliced cake makes the idea plain.
But this small notation hides a long and tangled story. Why do mathematicians read one denominator as a half and never as a second? Why was the idea of an improper fraction a late arrival, treated almost as a contradiction? Who first drew the horizontal bar that everyone now takes for granted? And how did a Flemish pamphlet from 1585 change the way the world calculates? The answers reach from ancient Egypt to a mathematician in Fez, and from primary-school cake-cutting to the abstract field of rational numbers.
The numerator gets its name from numerator, Latin for a counter or numberer, while the denominator comes from denominator, a namer or designator. So the numerator counts the parts and the denominator names their type. The fraction eight-fifths amounts to eight parts, each of the type called a fifth. In the language of division, the numerator is the dividend and the denominator is the divisor.
English reads these denominators as ordinal numbers, pluralized when the numerator is not 1, so you hear one-fifth and two-fifths. A handful of common fractions break the pattern. A denominator of 2 is read as half or halves, never as seconds. A denominator of 4 may be a quarter as well as a fourth. A denominator of 100 may be a percent as well as a hundredth. When the numerator is 1, it often vanishes entirely, so one-fifth can simply be a fifth.
The bar between the two numbers has its own vocabulary. It may be horizontal, oblique like 2/5, or diagonal. Those marks are called, in turn, the horizontal bar; the virgule, slash, or stroke; and the fraction bar, solidus, or fraction slash. Typographers go further still. Fractions stacked vertically are en or nut fractions, diagonal ones are em or mutton fractions, depending on whether the shape fills a narrow en square or a wider em square. In traditional typefounding, a single piece of type bearing a whole fraction was a case fraction, while pieces representing only parts were piece fractions. Spelling matters too: two-fifths with a hyphen names one value, while two fifths without it reads as two separate instances of a fifth.
When numerator and denominator are both positive, a fraction is proper if the numerator is smaller than the denominator, and improper otherwise. The improper fraction was a late development, and the very name betrays unease. Since fraction means piece, a proper fraction was held to be less than 1. A 17th-century textbook called The Ground of Arts explained the point. More generally, a fraction is proper when its absolute value is strictly less than one, and improper, or top-heavy, when that value is at least 1.
Negative fractions represent the opposite of positive ones. If a half stands for a half-dollar profit, then negative one-half stands for a half-dollar loss. The rules for signed division mean a negative divided by a positive is negative, so several written forms all collapse to the same negative one-half. A negative divided by a negative turns positive, giving back one-half. And because every negative number sits below zero while every positive number sits above it, any negative fraction is smaller than any positive fraction.
Flipping a fraction gives its reciprocal, with numerator and denominator swapped. The product of a non-zero fraction and its reciprocal is 1, which makes the reciprocal a multiplicative inverse. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 is proper. Any integer can wear a denominator of 1, sometimes called the invisible denominator, so 17 becomes seventeen over one. That means every fraction and every integer except zero has a reciprocal of its own.
A decimal fraction has a denominator that is an integer power of ten, hidden behind the digits to the right of the separator. In 0.75, the numerator is 75 and the implied denominator is 100, because two digits follow the separator. That separator may be a period, an interpunct, or a comma, depending on locale. A number like 3.75 can appear as an improper fraction or as a mixed number. Scientific notation with negative exponents handles the unwieldy ones, turning 0.0000006023 into a compact form. A decimal with infinitely many digits stands for an infinite series, as 0.333... unfolds into three-tenths plus three-hundredths and onward.
Percentage comes from per centum, per hundred, marked by the symbol %, with an implied denominator always 100. So 51% means fifty-one hundredths, and values above 100 or below zero follow the same rule, like 311% and negative 27%. The related permille, or parts per thousand, uses a denominator of 1000, and this parts-per notation scales up to parts per million and beyond.
Choosing between fraction and decimal is often a matter of taste. Fractions shine when the denominator is small. Multiplying 16 by three-sixteenths in your head beats wrestling with 0.1875, and multiplying 15 by one-third is exact in a way no decimal approximation can match. Money usually takes two decimal places, as in $3.75. But pre-decimal British currency borrowed only the look of a fraction. The form 3/6, read three and six, meant three shillings and sixpence, and had nothing to do with the value three sixths.
A mixed number is the sum of a non-zero integer and a proper fraction, set side by side with no plus or minus sign between them. Two and three-quarters describes two whole cakes and three quarters of a third cake, the integer and fraction joined by the word and. Any mixed number converts to an improper fraction: multiply the whole part by the denominator, add the numerator, and keep the same denominator. The reverse uses division with remainder, since 4 goes into 11 twice with 3 left over.
In primary school, teachers often demand that every fractional result be written as a mixed number. Outside school, mixed numbers describe measurements, like hours or inches, and stay common in the trades, especially where the decimalized metric system is not used. From secondary school onward, mathematics treats every fraction uniformly as a rational number, the quotient of integers, and leaves improper fractions and mixed numbers behind. College students trained for years sometimes stumble when they meet mixed numbers again, because juxtaposition in algebra usually means multiplication.
A ratio is a relationship between two or more numbers, sometimes written as a fraction. Picture a car lot with 12 vehicles: 2 white, 6 red, and 4 yellow. The ratio of red to white to yellow is 6 to 2 to 4, and yellow to white reduces to 4:2, or 2:1. Compared to the whole, yellow cars to all cars is 4:12, which reduces to 1:3. So one car in three is yellow, and a random choice carries a one in three chance of landing on yellow.
Multiplying a numerator and denominator by the same non-zero number leaves a fraction's value unchanged, because that ratio equals 1 and multiplying by 1 changes nothing. Double the top and bottom of one-half and you get two-quarters, still worth 0.5. Dividing both by a shared factor reverses the move and reduces the fraction. Using the greatest common divisor strips it to its lowest terms. Since the greatest common divisor of 63 and 462 is 21, dividing both by 21 reduces that fraction at once. The Euclidean algorithm supplies a method for finding that divisor.
Addition demands like quantities. Two quarters plus three quarters make five quarters, and since four quarters equal a dollar, the sum follows directly. Unlike quantities must first be converted: to add quarters and thirds, recast both as twelfths by multiplying the denominators together. The smallest workable denominator is the least common multiple of the originals, called the least common denominator. Subtraction runs the same way, finding a common denominator and subtracting the numerators, with an extra one borrowed from the minuend when needed.
Multiplying fractions means multiplying numerators and multiplying denominators. One third of one quarter is one twelfth, because twelve small slices make a whole. A shortcut called cancellation reduces the answer to lowest terms during the multiplication itself, dividing out common factors top and bottom before finishing. Division flips the script: to divide by a fraction, multiply by its reciprocal. To divide a fraction by a whole number, either divide the numerator if it goes evenly or multiply the denominator instead.
The earliest fractions were reciprocals of integers, symbols for one part of two, one part of three, one part of four. The Egyptians used Egyptian fractions around 1000 BC, and about 4000 years ago they divided using least common multiples with unit fractions, reaching the same answers modern methods give. They kept a separate notation for dyadic fractions in certain systems of weights and measures. An Egyptian fraction is a sum of distinct positive unit fractions, and every positive rational number can be expanded that way, in infinitely many forms.
The Greeks worked with unit fractions and later with simple continued fractions. Followers of Pythagoras, around 530 BC, discovered that the square root of two cannot be written as a fraction of integers. The finding is often, though probably wrongly, ascribed to Hippasus of Metapontum, said to have been executed for revealing it. In 150 BC, Jain mathematicians in India wrote the Sthananga Sutra, covering number theory, arithmetic, and operations with fractions.
A modern form of fractions called bhinna seems to have begun in India, described in the Bakhshali manuscript around AD 400, then by Brahmagupta around 628 and Bhaskara around 1150. They placed numerators over denominators with no bar between, the integer on one line and the two parts of the fraction on the next. The horizontal bar is first attested in the work of Al-Hassar, a mathematician from Fez, Morocco, around 1200, who specialized in Islamic inheritance jurisprudence. He instructed readers to write three-fifths and a third of a fifth in a stacked form. The same notation, with the fraction given before the integer, appears soon after in Leonardo Fibonacci in the 13th century.
Dirk Jan Struik traced the common use of decimal fractions to a Flemish pamphlet, De Thiende, published at Leyden in 1585 alongside a French translation, La Disme, by Simon Stevin, who lived from 1548 to 1620 and had settled in the Northern Netherlands. Yet decimals predated him. The Chinese used them many centuries earlier, and the Persian astronomer Al-Kashi handled both decimal and sexagesimal fractions in his Key to arithmetic at Samarkand in the early 15th century. Al-Kashi claimed to have discovered decimal fractions himself, but J. Lennart Berggren notes he was mistaken, since the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi used them five centuries earlier, as early as the 10th century.
Mathematicians study fractions to check that the familiar rules stay consistent and reliable. They define a fraction as an ordered pair of integers, with addition, subtraction, multiplication, and division specified on those pairs. An equivalence relation groups fractions into classes, and each class behaves as a single abstract fraction, independent of which representative you pick. When numerator and denominator are coprime, that pair is the unique representative, and these fractions of integers form the field of the rational numbers, written Q for quotient. Push the idea further and a and b may come from any integral domain, producing its field of fractions. Polynomials over an integral domain yield the field of rational fractions, also called rational functions.
An algebraic fraction is the quotient of two algebraic expressions, again with a non-zero denominator. When both parts are polynomials, it is a rational fraction; one with a variable under a root is irrational. Partial fraction decomposition breaks a rational fraction into a sum of simpler ones, a tool useful for finding antiderivatives. Even the youngest learners meet these ideas first through Cuisenaire rods, fraction bars, pie-shaped pieces, and grid paper, long before the Common Core State Standards Initiative defines a fraction in its own careful terms.
Common questions
What does the word fraction mean and where does it come from?
The word fraction comes from the Latin fractus, meaning broken. A fraction represents a part of a whole or any number of equal parts, such as one-half, eight-fifths, or three-quarters. A simple fraction has an integer numerator above the line and a non-zero integer denominator below it.
What is the difference between a proper and improper fraction?
When the numerator and denominator are both positive, a fraction is proper if the numerator is less than the denominator and improper otherwise. More generally, a fraction is proper if its absolute value is strictly less than one and improper, or top-heavy, if that value is at least 1. The improper fraction was a late development, since fraction means piece and a proper fraction was held to be less than 1.
Who invented the horizontal fraction bar?
The horizontal fraction bar is first attested in the work of Al-Hassar, a Muslim mathematician from Fez, Morocco, who was active around 1200 and specialized in Islamic inheritance jurisprudence. The same notation, with the fraction given before the integer, appears soon after in the work of Leonardo Fibonacci in the 13th century.
When were decimal fractions first introduced?
Decimal fractions became a common computational practice through the Flemish pamphlet De Thiende, published at Leyden in 1585 with a French translation, La Disme, by Simon Stevin, who lived from 1548 to 1620. Decimals predated him, however, used by the Chinese many centuries earlier and by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.
What is an Egyptian fraction and who used it?
An Egyptian fraction is the sum of distinct positive unit fractions, and every positive rational number can be expanded this way in infinitely many forms. The Egyptians used Egyptian fractions around 1000 BC, dividing with least common multiples and unit fractions about 4000 years ago.
How are mixed numbers converted to improper fractions?
A mixed number is the sum of a non-zero integer and a proper fraction written side by side. To convert it to an improper fraction, multiply the whole number by the denominator, add the numerator to get the new numerator, and keep the same denominator. Conversely, an improper fraction converts to a mixed number using division with remainder, as when 4 goes into 11 twice with 3 left over.