As the ‘Fire’ in the King James Version (bible) can be applicable as in the verse of reference it is really Prometheus that values this time to date. As the Greek, Prometheus gave fire to man, and, the Olympics must be considered as for arithmetic the gathering of two to three, for effect is important.

# Matthew 18:20

“For where two or three are gathered together in my name, there am I in the midst of them.”

The importance for arithmetic in Cantore Physics will be the total fashion to dress as the Adam and Eve story involved more than the fruit. In regards to such the importance of adding Prometheus and the fire is the fact that now the books chapter and verse may introduce the Book of Numbers as its denominator. The denominator: See below and entertain “The fraction”!

# Olympic flame

The **Olympic flame** is a symbol used in the Olympic movement. It is also a symbol of continuity between ancient and modern games.^{[1]} Several months before the Olympic Games, the Olympic flame is lit at Olympia, Greece. This ceremony starts the Olympic torch relay, which formally ends with the lighting of the Olympic cauldron during the opening ceremony of the Olympic Games. The flame then continues to burn in the cauldron for the duration of the Games, until it is extinguished during the Olympic closing ceremony.

## Origins[edit]

This section needs additional citations for verification. (March 2021) |

The Olympic flame as a symbol of the modern Olympic movement was introduced by architect Jan Wils who designed the stadium for the 1928 Summer Olympics in Amsterdam.

The idea for the Olympic flame was derived from ancient Greek ceremonies where a sacred fire was kept burning throughout the celebration of the ancient Olympics on the altar of the sanctuary of Hestia.^{[2]}^{[3]} In Ancient Greek mythology, fire had divine connotations and it was thought to have been stolen from the gods by Prometheus. Sacred fires were present at many ancient Greek sanctuaries, including those at Olympia. Every four years, when Zeus was honoured at the Olympic Games, additional fires were lit at his temple and that of his wife, Hera. The modern Olympic flame is ignited every two years in front of the ruins of the temple of Hera.

When the tradition of an Olympic fire was reintroduced during the 1928 Summer Olympics, an employee of the Electric Utility of Amsterdam lit the first modern Olympic flame in the Marathon Tower of the Olympic Stadium in Amsterdam.^{[4]} The Olympic flame has been part of the Summer Olympics ever since. The Olympic torch relay was first introduced to the Summer Olympics at the 1936 Summer Olympics in Berlin.

## Main ceremonies[edit]

### Olympic flame lighting[edit]

The Olympic fire is ignited several months before the opening ceremony of the Olympic Games at the site of the ancient Olympics in Olympia, Greece.

Eleven women representing the Vestal Virgins,^{[notes 1]} perform a celebration at the Temple of Hera in which the first torch of the Olympic Torch Relay is kindled by the light of the Sun. Its rays are concentrated by a parabolic mirror. A female performer acting as a priestess presents the torch and an olive branch to the first relay bearer (who also is a Greek athlete) followed by a dove release to symbolize peace.

At the beginning of the ceremony, the Olympic anthem is sung first followed by the national anthem of the country hosting the Olympics and the national anthem of Greece along with the hoisting of the flags.

After the ceremony at Olympia the Olympic flame first travels around Greece, and is then transferred during a ceremony in the Panathenaic Stadium in Athens from the previous Olympic city to the current year's host city.^{[5]}^{[6]}

By contrast to the Olympic flame, the Olympic torch relay, which transports the flame from Olympia, Greece to the various designated sites of the Games, had no ancient precedent and was introduced by Carl Diem at the 1936 Summer Olympics in Berlin, Germany.^{[7]}

At the first Olympic torch relay, the flame was transported from Olympia to Berlin over 3,187 kilometers (1,980 miles) by 3,331 runners in twelve days and eleven nights. There were minor protests in Yugoslavia and Czechoslovakia on the way, which were suppressed by the local security forces.^{[8]}

In the 1956 Melbourne Games in Australia, local veterinary student Barry Larkin protested against the relay when he tricked onlookers by carrying a fake flame, consisting of a pair of underpants set on fire in a plum pudding can, attached to a chair leg. He successfully managed to hand over the fake flame to the Mayor of Sydney, Pat Hills and escape without being noticed.^{[9]}^{[10]}^{[11]}

The Olympic torch travels routes that symbolise human achievement. Although most of the time the torch with the Olympic flame is still carried by runners, it has been transported in many different ways. The fire travelled by boat in 1948 and 2012 to cross the English Channel and was carried by rowers in Canberra as well as by dragon boatin Hong Kong in 2008.^{[12]}

It was first transported by airplane in 1952 when the flame travelled to Helsinki. In 1956, all carriers in the torch relay to Stockholm, where the equestrian events were held instead of in Melbourne, travelled on horseback.

Remarkable means of transportation were used in 1976, when the flame was transformed to a radio signal and transmitted from Europe to the New World: Heat sensors in Athens detected the flame, the signal was sent to Ottawa via satellite where it was received and used to trigger a laser beam to re-light the flame.^{[13]}^{[14]} The torch, but not the flame, was taken into space by astronauts in 1996, 2000 and 2013.^{[15]} Other unique means of transportation include a Native American canoe, a camel, and Concorde.^{[16]} The torch has been carried across water; the 1968 Grenoble Winter Games was carried across the port of Marseilles by a diver holding it aloft above the water.^{[17]} In 2000, an underwater flare was used by a diver across the Great Barrier Reef en route to the Sydney Games.^{[18]} In 2012 it was carried by boat across Bristol Harbour in the UK and on the front of a London Underground train to Wimbledon.

In 2004, the first global torch relay was undertaken, a journey that lasted 78 days. The Olympic flame covered a distance of more than 78,000 km in the hands of some 11,300 torchbearers, travelling to Africa and South America for the first time, visiting all previous Olympic cities, and finally returning to Athens for the 2004 Summer Olympics.

The 2008 Summer Olympics torch relay spanned all six inhabited continents before proceeding through China. However there was protests against China's human rights record in London where a "ring of steel" was formed around the flame to protect it, but one protester managed to grab hold of the torch while it was being held by television presenter Konnie Huq.^{[19]} In Paris the torch was extinguished at least twice by Chinese officials (five times according to French police^{[20]}) so that it could be transported in a bus amid protests while it was being paraded through Paris.^{[21]}^{[22]} This eventually led to the cancellation of the relay's last leg in the city.^{[23]}Demonstrations were also held in San Francisco and the route the torch would take was cut in half.^{[24]}

As a result in 2009, the International Olympic Committee announced that future torch relays could be held only within the country hosting the Olympics after the initial Greek leg was finished.^{[25]} Although this rule took effect with the 2014 Winter Olympics, the organizers of the 2010 Winter Olympics in Vancouver and the 2012 Summer Olympics in London chose to hold their torch relays only in their respective hosting countries of Canada and the United Kingdom (except for brief stops in the United States and Ireland, respectively).^{[26]}^{[27]} In 2016, ten days before the beginning of the 2016 Summer Olympics in Rio de Janeiro, citizens of Angra dos Reis, a city near Rio de Janeiro, managed to extinguish the Olympic flame during a protest against the city spending money on hosting the Olympics despite an economic crisis in Brazil.^{[28]}

The Olympic torch relay in the host country ends with the lighting of the Olympic cauldron during the opening ceremony in the central host stadium of the Games. The final carrier is often kept unannounced until the last moment. Over the years, it has become a tradition to let a famous athlete of the host nation, former athletes or athletes with significant achievements and milestones be the last runner in the Olympic torch relay.

#### Re-igniting the flame[edit]

It is not uncommon for the Olympic flame to be accidentally or deliberately extinguished during the course of the torch relay (and on at least one occasion the cauldron itself has gone out during the Games). To guard against this eventuality, multiple copies of the flame are transported with the relay or maintained in backup locations. When a torch goes out, it is re-lit (or another torch is lit) from one of the backup sources. Thus, the fires contained in the torches and Olympic cauldrons all trace a common lineage back to the same Olympia lighting ceremony.

- One of the more memorable extinguishings occurred at the 1976 Summer Olympics held in Montreal, Quebec, Canada. After a rainstorm doused the Olympic flame a few days after the games had opened, an official re-lit the flame using a cigarette lighter. Organizers quickly doused it again and re-lit it using a backup of the original flame.
^{[14]} - At the 2004 Summer Olympics, when the Olympic flame came to the Panathinaiko Stadium to start the global torch relay, the night was very windy and the torch, lit by Gianna Angelopoulos-Daskalaki of the Athens 2004 Organizing Committee, blew out due to the wind, but was re-lit from the backup flame taken from the original flame lit at Olympia.
- In October 2013 in Russia, the Olympic flame was blown out at the Kremlin and was reignited from a security officer's lighter instead of the back up flame.
^{[29]}

The current design of the torch has a safeguard built into it: There are two flames inside the torch. There is a highly visible (yellow flame) portion that burns cooler and is more prone to extinguish in wind and rain, but there is also a smaller hotter (blue in the candle's wick) flame akin to a pilot light hidden inside the torch which is protected from wind and rain and is capable of relighting the cooler, more visible portion if it is extinguished. The fuel contained inside the torch is able to keep it lit for approximately 15 minutes before it would be extinguished.^{[30]}

#### Selected relays in detail[edit]

The flame is transported from Greece to the host country where the flame is transported by torch around the host nation to the main stadium.

### Olympic cauldron lighting[edit]

During the opening ceremony the final bearer of the torch runs towards the cauldron, often placed at the top of a grand staircase, and then uses the torch to start the flame in the stadium. The climactic transfer of the Olympic flame from the final torch to the cauldron at the central host stadium marks the symbolic commencement of the Games.

As with being the final runner of the Olympic torch relay, it is considered to be a great honor to light the Olympic cauldron, and in the same way it has become a tradition to select notable athletes to conduct this part of the ceremony. On other occasions, the people who lit the cauldron in the stadium are not famous but nevertheless symbolize Olympic ideals. Japanese runner Yoshinori Sakai was born on the day of the atomic bombing of Hiroshima. He was chosen for the role to symbolize Japan's postwar reconstruction and peace, opening the 1964 Tokyo Games. At the 1976 Games in Montreal, two teenagers — one from the French-speaking part of the country, one from the English-speaking part — symbolized the unity of Canada.

At the 2012 Games in London, the torch was carried by Sir Steve Redgrave to a group of seven young British athletes (Callum Airlie, Jordan Duckitt, Desiree Henry, Katie Kirk, Cameron MacRitchie, Aidan Reynolds and Adelle Tracey) — each nominated by a British Olympic champion — who then each lit a single tiny flame on the ground, igniting 204 copper petals before they converged to form the cauldron for the Games.

The first well-known athlete to light the cauldron in the stadium was the ninefold Olympic Champion Paavo Nurmi, who excited the home crowd in Helsinki in 1952. In 1968, Enriqueta Basilio became the first woman to light the Olympic Cauldron at the Olympic Games in Mexico City.

Perhaps one of the most spectacular of Olympic cauldron lighting ceremonies took place at the 1992 Summer Olympics, when Paralympic archer Antonio Rebollo lit the cauldron by shooting a burning arrow over it, which ignited gas rising from the cauldron.^{[31]}^{[32]} Unofficial videos seem to indicate that the flame was lit from below.^{[33]}^{[34]} Twenty years after the Barcelona Games one of those involved said that the flame was "switched on" ("Se encendió con un botón", in Spanish).^{[35]} Two years later, the Olympic fire was brought into the stadium of Lillehammer by a ski jumper. In Beijing 2008, Li Ning "ran" on air around the interior edge of the Beijing National Stadium's roof, and lit a cauldron attached to it.

### Olympic cauldron designs[edit]

The cauldron and the pedestal are always the subjects of unique and often dramatic design. These also tie in with *how* the cauldron is lit during the Opening Ceremony.

- In Los Angeles in 1984, Rafer Johnson lit a wick at the top of the archway after having climbed a big flight of steps. The flame flared up a pipe, through the Olympic Rings, and on up the side of the tower to ignite the cauldron.
- In Atlanta in 1996, the cauldron was an artistic scroll decorated in red and gold. It was lit by Muhammad Ali, using a mechanical, self-propelling fuse ball that transported the flame up a wire from the stadium to its cauldron.
^{[36]}At the 1996 Summer Paralympics, the scroll was lit by paraplegic climber Mark Wellman, hoisting himself up a rope to the cauldron. - For the 2000 Summer Olympics in Sydney, Cathy Freeman walked across a circular pool of water and ignited the cauldron through the water, surrounding herself within a ring of fire. The planned spectacular climax to the ceremony was delayed by the technical glitch of a computer switch that malfunctioned, causing the sequence to shut down by giving a false reading. This meant that the Olympic flame was suspended in mid-air for about four minutes, rather than immediately rising up a water-covered ramp to the top of the stadium. When it was discovered what the problem was, the program was overridden and the cauldron continued up the ramp, where it finally rested on a tall silver pedestal.
- For the 2002 Winter Olympics in Salt Lake City, Utah, United States, the cauldron was lit by the members of the winning 1980 US hockey team. After being skated around the centre ice rink there in the stadium, the flame was carried up a staircase to the team members, who then lit a wick of sorts at the bottom of the cauldron tower which set off a line of flames that travelled up inside the tower until it reached the cauldron at the top which ignited. This cauldron was the first to use glass and incorporated running water to prevent the glass from heating and to keep it clean.
- For the 2004 Summer Olympics in Athens, the cauldron was in the shape of a giant olive leaf which bowed down to accept the flame from windsurfer Nikolaos Kaklamanakis.
^{[37]} - In the 2006 Winter Olympics in Turin, Stefania Belmondo placed the flame on an arched lighting apparatus, which initiated a series of fireworks before lighting the top of the 57-metre-high (187 ft) Olympic cauldron, the highest in the history of the Winter Olympic Games.
^{[38]} - In the 2008 Summer Olympics in Beijing, the cauldron resembled the end of a scroll that lifted out from the stadium rim and spiralled upwards. It was lit by Li Ning, who was raised to the rim of the stadium by wires. He ran around the rim of the stadium while suspended and as he ran, an unrolling scroll was projected showing film clips of the flame's journey around the world from Greece to Beijing. As he approached the cauldron, he lit an enormous wick, which then transferred the flame to the cauldron. The flame then spiralled up the structure of the cauldron before lighting it at the top.
^{[39]} - In the 2010 Winter Olympics at Vancouver, a team of athletes (Catriona Le May Doan, Steve Nash, Nancy Greene and Wayne Gretzky) were to simultaneously light the base of poles, which would then carry the flames upwards to the cauldron. However, only three out of four poles came out of the ground due to mechanical problems, resulting in inadvertently excluding Le May Doan from lighting it with the other three athletes. Because the site of the ceremonies - BC Place - was a domed stadium, Gretzky was sent via the back of a pick-up truck to a secondary site — the Vancouver Convention Centrewhich served as the International Broadcast Centre for these Olympics — to light a larger cauldron of a similar design located outdoors, as Olympic rules state that the flame must be in public view for the entirety of the Olympics. In the closing ceremonies, Le May Doan took part in a joke about the mechanical glitch, and she was able to light the fully raised fourth pole and have the indoor cauldron relit.
- At the 2012 Summer Olympics in London, the flame was passed to a group of seven young British athletes (Callum Airlie, Jordan Duckitt, Desiree Henry, Katie Kirk, Cameron MacRitchie, Aidan Reynolds, and Adelle Tracey) who then each lit a single tiny flame on the ground, igniting 204 petals (one for each delegation in the Games) that rose up to form a single cauldron. The cauldron that traditionally flames continuously from the opening until the closing ceremony was temporarily extinguished (the flame itself was transferred to a lantern) prior to the athletics events while the cauldron was moved to the southern side of the stadium. It was relit by Austin Playfoot, a torchbearer from the 1948 Olympics.
^{[40]}In contrast to the cauldrons in Vancouver, the cauldron was not visible to the public outside the stadium. Instead, monitors had been placed throughout the Olympic Park showing the public live footage of the flame. - For the 2014 Winter Olympics in Sochi, Russia, the cauldron was situated directly outside Fisht Olympic Stadium, the ceremonial venue for the Games. After the torch's lap around the stadium, triple gold medalists Irina Rodnina and Vladislav Tretiak carried the torch outside the stadium to light a larger version of the "celebration cauldron" used in the main torch relay at the center of the Olympic Park. A line of gas jets carried the flame from the celebration cauldron up the main cauldron tower, eventually lighting it at the top.
- For the 2016 Summer Olympics in Rio de Janeiro, Brazil, the cauldron was lit inside the Maracanã Stadium, the ceremonial venue for the Games, by Vanderlei de Lima. As part of these Games' appeal towards environmental protection, organizers deliberately chose to use a basic design with a smaller flame than past cauldrons. To compensate for the smaller cauldron, it is accompanied by a larger kinetic sculpture designed by Anthony Howe. A very similar public cauldron was lit in a plaza outside the Candelária Church following the opening ceremony.
^{[41]}^{[42]}^{[43]} - For the 2018 Winter Olympics in Pyeongchang, South Korea, the flame was eventually handed to Yuna Kim, who was at the top of a set of steps. She then lit a wick of sorts, which lit a large metal flaming pillar. The pillar rose to the top of the cauldron, lighting it. The cauldron was a large white sculpture with a large sphere on the top, acting for the cauldron. The cauldron's design was inspired by Joseon white porcelain.
- For the 2020 Summer Olympics in Tokyo, Japan, the flame was handed to Naomi Osaka, who stood in front of a large mountain-like structure resembling Mount Fuji. At the top of the mountain was a large ball, resembling the sun. This ball unfolded into the petals of a flower representing hope and vitality, forming the cauldron. Osaka then walked up a set of steps revealed as the ball unfolded and lit the cauldron.
- The 2022 Winter Olympics in Beijing eschewed a cauldron entirely, with the final two torchbearers—skiers Zhao Jiawen and Dinigeer Yilamujiang—inserting the torch into a mount at the centre of a large snowflake, constructed from crystalline placards with the names of each competing delegation.
^{[44]}^{[45]}

After being lit the flame in the Olympic cauldron continues to burn during the Games, until the closing ceremony, when it is finally put out symbolizing the official end of the Games.

## Coinage[edit]

The Olympic flame has been used as a symbol and a main motif numerous times in different commemorative coins. A recent example was the 50th anniversary of the Helsinki Olympic Games commemorative coin, minted in 2002. In the obverse, the Olympic flame above the Earth can be seen. Finland is the only country highlighted; it was the host of the 1952 games.

# 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 *common*, *vulgar*, or *simple* fraction (examples: and ) consists of a **numerator**, displayed above a line (or before a slash like 1⁄2), 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 34, 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 34of 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.^{[1]} Thus the fraction 34 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 12 represents a half-dollar profit, then −12 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), −12, −12 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 (see square root of 2) and π4 (see proof that π is irrational).

## Vocabulary

In a fraction, the number of equal parts being described is the **numerator** (from Latin: *numerātor*, "counter" or "numberer"), and the type or variety of the parts is the **denominator** (from Latin: *dēnōminātor*, "thing that names or designates").^{[2]}^{[3]} As an example, the fraction 85 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 13), oblique (as in 2/5), or diagonal (as in 4⁄9).^{[4]} These marks are respectively known as the horizontal bar; the virgule, slash (US), or stroke (UK); and the fraction bar, solidus,^{[5]} or fraction slash.^{[n 1]} 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.^{[4]} In traditional typefounding, a piece of type bearing a complete fraction (e.g. 12) 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 1. (For example, 25 and 35 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, 31 may be described as "three wholes", or simply as "three". When the numerator is 1, 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 25 and "two fifths" is the same fraction understood as 2 instances of 15.) 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, 31 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., 1117 as "one over one hundred seventeen"), while those with denominators divisible by ten are typically read in the normal ordinal fashion (e.g., 61000000 as "six-millionths", "six millionths", or "six one-millionths").

## Forms of fractions

### Simple, common, or vulgar fractions

A **simple fraction** (also known as a **common fraction** or **vulgar fraction**, where vulgar is Latin for "common") is a rational number written as *a*/*b* or , where *a* and *b* are both integers.^{[9]} As with other fractions, the denominator (*b*) cannot be zero. Examples include , , , and . The term was originally used to distinguish this type of fraction from the sexagesimal fraction used in astronomy.^{[10]}

*Common fractions* can be positive or negative, and they can be proper or improper (see below). 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 (e.g., ). Unit fractions can also be expressed using negative exponents, as in 2
^{−1}, which represents 1/2, and 2^{−2}, which represents 1/(2^{2}) or 1/4. - A dyadic fraction is a common fraction in which the denominator is a power of two, e.g. .

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

### Proper and improper fractions

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.^{[11]}^{[12]} 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.^{[10]}This was explained in the 17th century textbook *The Ground of Arts*.^{[13]}^{[14]}

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.^{[15]}^{[16]} It is said to be an **improper fraction**, or sometimes **top-heavy fraction**,^{[17]} 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.

### Reciprocals and the "invisible denominator"

The **reciprocal** of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of , for instance, is . 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 (for example, ), 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 , 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 .

### Ratios

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 412 of the cars or 13 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.

### Decimal fractions and percentages

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, an interpunct (·), a comma) depends on the 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 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 10^{7}. Dividing by 10^{7}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, 13 = 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% equals 311/100, and −27% equals −27/100.

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.

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 pre-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.

### Mixed numbers

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 measurement: inches, for example. Scientific measurements almost invariably use decimal notation rather than mixed numbers. The sum can be implied without the use of a visible operator such as the appropriate "+". For example, in referring to two entire cakes and three quarters of another cake, the numerals denoting the integer part and the fractional part of the cakes can be written next to each other as instead of the unambiguous notation Negative mixed numerals, as in , are treated like Any such sum of a *whole* plus a *part* can be converted to an improper fraction by applying the rules of adding unlike quantities.

This tradition is, formally, in conflict with the notation in algebra where adjacent symbols, without an explicit infix operator, denote a product. In the expression , the "understood" operation is multiplication. If x is replaced by, for example, the fraction , the "understood" multiplication needs to be replaced by explicit multiplication, to avoid the appearance of a mixed number.

When multiplication is intended, may be written as

- or or

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. Thus, .

### Historical notions

#### Egyptian fraction

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 except , 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 and .

#### Complex and compound fractions

In a **complex fraction**, either the numerator, or the denominator, or both, is a fraction or a mixed number,^{[18]}^{[19]} corresponding to division of fractions. For example, and are complex fractions. To reduce a complex fraction to a simple fraction, treat the longest fraction line as representing division. For example:

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, e.g. as

- or as

A **compound fraction** is a fraction of a fraction, or any number of fractions connected with the word *of*,^{[18]}^{[19]} corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see the section on 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^{[20]} and now used in no well-defined manner, partly even taken synonymously for each other^{[21]} or for mixed numerals.^{[22]} 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".

## Arithmetic with fractions

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

### Equivalent fractions

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 ().

#### Simplifying (reducing) fractions

Dividing the numerator and denominator of a fraction by the same non-zero number yields 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. For example, if both the numerator and the denominator of the fraction are divisible by then they can be written as and and the fraction becomes , which can be reduced by dividing both the numerator and denominator by to give the reduced fraction

If one takes for c the greatest common divisor of the numerator and the denominator, one gets the equivalent fraction whose numerator and denominator have the lowest absolute values. One says that the fraction has been reduced to its *lowest terms*.

If the numerator and the denominator do not share any factor greater than 1, the fraction is already reduced to its lowest terms, and it is said to be *irreducible*, *reduced*, 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 integers.

### Comparing fractions

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 and (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 ? gives .

For the more laborious question ? multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding ? . It is not necessary to calculate – 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.

### Addition

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:

- .

#### Adding unlike quantities

To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the chosen type of fraction to convert to; simply multiply together the two denominators (bottom number) of each fraction. In case of an integer number apply the invisible denominator

For adding quarters to thirds, both types of fraction are converted to twelfths, thus:

Consider adding the following two quantities:

First, convert into fifteenths by multiplying both the numerator and denominator by three: . Since equals 1, multiplication by does not change the value of the fraction.

Second, convert into fifteenths by multiplying both the numerator and denominator by five: .

Now it can be seen that:

is equivalent to:

This method can be expressed algebraically:

This algebraic method always works, thereby guaranteeing that the sum of simple fractions is always again a simple fraction. However, if the single denominators contain a common factor, a smaller denominator than the product of these can be used. For example, when adding and the single denominators have a common factor and therefore, instead of the denominator 24 (4 × 6), the halved denominator 12 may be used, not only reducing the denominator in the result, but also the factors in the numerator.

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. This is called the least common denominator.

### Subtraction

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,

### Multiplication

#### Multiplying a fraction by another fraction

To multiply fractions, multiply the numerators and multiply the denominators. Thus:

To explain the process, consider one third of one quarter. Using the example of a cake, if three small slices of equal size make up a quarter, and four quarters make up a whole, twelve of these small, equal slices make up a whole. Therefore, a third of a quarter is a twelfth. Now consider the numerators. The first fraction, two thirds, is twice as large as one third. Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The second fraction, three quarters, is three times as large as one quarter, so two thirds of three quarters is three times as large as two thirds of one quarter. Thus two thirds times three quarters is six twelfths.

A short cut for multiplying fractions is called "cancellation". Effectively the answer is reduced to lowest terms during multiplication. For example:

A two is a common factor in both the numerator of the left fraction and the denominator of the right and is divided out of both. Three is a common factor of the left denominator and right numerator and is divided out of both.

#### Multiplying a fraction by a whole number

Since a whole number can be rewritten as itself divided by 1, normal fraction multiplication rules can still apply.

This method works because the fraction 6/1 means six equal parts, each one of which is a whole.

#### Multiplying mixed numbers

When multiplying mixed numbers, it is considered preferable to convert the mixed number into an improper fraction.^{[23]} For example:

In other words, is the same as , making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total) and 33 quarters is , since 8 cakes, each made of quarters, is 32 quarters in total.

### Division

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, equals and also equals , which reduces to . To divide a number by a fraction, multiply that number by the reciprocal of that fraction. Thus, .

### Converting between decimals and fractions

To change a common fraction to a decimal, do a long division of the decimal representations of the numerator by the denominator (this is idiomatically also phrased as "divide the denominator into the numerator"), and round the answer to the desired accuracy. For example, to change 14 to a decimal, divide 1.00 by 4 ("4 into 1.00"), to obtain 0.25. To change 13 to a decimal, divide 1.000... by 3 ("3 into 1.000..."), and stop when the desired accuracy is obtained, e.g., at 4 decimals with 0.3333. The fraction 14 can be written exactly with two decimal digits, while the fraction 13 cannot be written exactly as a decimal with a finite number of digits. To change a decimal to a fraction, write in the denominator a 1 followed by as many zeroes as there are digits to the right of the decimal point, and write in the numerator all the digits of the original decimal, just omitting the decimal point. Thus

#### Converting repeating decimals to fractions

Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infinite repeating decimal is required to reach the same precision. Thus, it is often useful to convert repeating decimals into fractions.

A conventional way to indicate a repeating decimal is to place a bar (known as a vinculum) over the digits that repeat, for example 0.789 = 0.789789789... For repeating patterns that begin immediately after the decimal point, the result of the conversion is the fraction with the pattern as a numerator, and the same number of nines as a denominator. For example:

- 0.5 = 5/9
- 0.62 = 62/99
- 0.264 = 264/999
- 0.6291 = 6291/9999

If leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros:

- 0.05 = 5/90
- 0.000392 = 392/999000
- 0.0012 = 12/9900

If a non-repeating set of decimals precede the pattern (such as 0.1523987), one may write the number as the sum of the non-repeating and repeating parts, respectively:

- 0.1523 + 0.0000987

Then, convert both parts to fractions, and add them using the methods described above:

- 1523 / 10000 + 987 / 9990000 = 1522464 / 9990000

Alternatively, algebra can be used, such as below:

- Let
*x*= the repeating decimal:*x*= 0.1523987

- Multiply both sides by the power of 10 just great enough (in this case 10
^{4}) to move the decimal point just before the repeating part of the decimal number:- 10,000
*x*= 1,523.987

- 10,000
- Multiply both sides by the power of 10 (in this case 10
^{3}) that is the same as the number of places that repeat:- 10,000,000
*x*= 1,523,987.987

- 10,000,000
- Subtract the two equations from each other (if
*a*=*b*and*c*=*d*, then*a*−*c*=*b*−*d*):- 10,000,000
*x*− 10,000*x*= 1,523,987.987 − 1,523.987

- 10,000,000
- Continue the subtraction operation to clear the repeating decimal:
- 9,990,000
*x*= 1,523,987 − 1,523 - = 1,522,464

- 9,990,000
- Divide both sides by 9,990,000 to represent
*x*as a fraction*x*= 15224649990000

## Fractions in abstract mathematics

In addition to being of great practical importance, fractions are also studied by mathematicians, who check that the rules for fractions given above are consistent and reliable. Mathematicians define a fraction as an ordered pair of integers and for which the operations addition, subtraction, multiplication, and division are defined as follows:^{[24]}

These definitions agree in every case with the definitions given above; only the notation is different. Alternatively, instead of defining subtraction and division as operations, the "inverse" fractions with respect to addition and multiplication might be defined as:

Furthermore, the relation, specified as

is an equivalence relation of fractions. Each fraction from one equivalence class may be considered as a representative for the whole class, and each whole class may be considered as one abstract fraction. This equivalence is preserved by the above defined operations, i.e., the results of operating on fractions are independent of the selection of representatives from their equivalence class. Formally, for addition of fractions

- and imply

and similarly for the other operations.

In the case of fractions of integers, the fractions ab with a and b coprime and *b* > 0 are often taken as uniquely determined representatives for their *equivalent* fractions, which are considered to be the *same*rational number. This way the fractions of integers make up the field of the rational numbers.

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, polynomials in one indeterminate, with coefficients from some integral domain *D*, are themselves an integral domain, call it *P*. So for *a* and *b* elements of *P*, the generated *field of fractions* is the field of rational fractions (also known as the field of rational functions).

## Algebraic fractions

An algebraic fraction is the indicated quotient of two algebraic expressions. As with fractions of integers, the denominator of an algebraic fraction cannot be zero. Two examples of algebraic fractions are and . Algebraic fractions are subject to the same field properties as arithmetic fractions.

If the numerator and the denominator are polynomials, as in , the algebraic fraction is called a *rational fraction* (or *rational expression*). An *irrational fraction* is one that is not rational, as, for example, one that contains the variable under a fractional exponent or root, as in .

The terminology used to describe algebraic fractions is similar to that used for ordinary fractions. For example, an algebraic fraction is in lowest terms if the only factors common to the numerator and the denominator are 1 and −1. An algebraic fraction whose numerator or denominator, or both, contain a fraction, such as , is called a **complex fraction**.

The field of rational numbers is the field of fractions of the integers, while the integers themselves are not a field but rather an integral domain. Similarly, the rational fractions with coefficients in a field form the field of fractions of polynomials with coefficient in that field. Considering the rational fractions with real coefficients, radical expressions representing numbers, such as are also rational fractions, as are a transcendental numbers such as since all of and are real numbers, and thus considered as coefficients. These same numbers, however, are not rational fractions with *integer* coefficients.

The term partial fraction is used when decomposing rational fractions into sums of simpler fractions. For example, the rational fraction can be decomposed as the sum of two fractions: This is useful for the computation of antiderivatives of rational functions (see partial fraction decomposition for more).

## Radical expressions

A fraction may also contain radicals in the numerator or the denominator. 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. It is also more convenient if division is to be done manually. When the denominator is a monomial square root, it can be rationalized by multiplying both the top and the bottom of the fraction by the denominator:

The process of rationalization of binomial denominators involves multiplying the top and the bottom of a fraction by the conjugate of the denominator so that the denominator becomes a rational number. For example:

Even if this process results in the numerator being irrational, like in the examples above, the process may still facilitate subsequent manipulations by reducing the number of irrationals one has to work with in the denominator.

## Typographical variations

In computer displays and typography, simple fractions are sometimes printed as a single character, e.g. ½ (one half). See the article on Number Forms for information on doing this in Unicode.

Scientific publishing distinguishes four ways to set fractions, together with guidelines on use:^{[25]}

**Special fractions**: fractions that are presented as a single character with a slanted bar, with roughly the same height and width as other characters in the text. Generally used for simple fractions, such as: ½, ⅓, ⅔, ¼, and ¾. Since the numerals are smaller, legibility can be an issue, especially for small-sized fonts. These are not used in modern mathematical notation, but in other contexts.**Case fractions**: similar to special fractions, these are rendered as a single typographical character, but with a horizontal bar, thus making them*upright*. An example would be , but rendered with the same height as other characters. Some sources include all rendering of fractions as*case fractions*if they take only one typographical space, regardless of the direction of the bar.^{[26]}**Shilling**or**solidus fractions**: 1/2, so called because this notation was used for pre-decimal British currency (£sd), as in "2/6" for a half crown, meaning two shillings and six pence. While the notation "two shillings and six pence" did not represent a fraction, the forward slash is now used in fractions, especially for fractions inline with prose (rather than displayed), to avoid uneven lines. It is also used for fractions within fractions (complex fractions) or within exponents to increase legibility. Fractions written this way, also known as**piece fractions**,^{[27]}are written all on one typographical line, but take 3 or more typographical spaces.**Built-up fractions**: . This notation uses two or more lines of ordinary text and results in a variation in spacing between lines when included within other text. While large and legible, these can be disruptive, particularly for simple fractions or within complex fractions.

## History

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.^{[28]} The Egyptians used Egyptian fractions c. 1000 BC. About 4000 years ago, Egyptians divided with fractions using slightly different methods. They used least common multiples with unit fractions. Their methods gave the same answer as modern methods.^{[29]} The Egyptians also had a different notation for dyadic fractions in the Akhmim Wooden Tablet and several Rhind Mathematical Papyrus problems.

The Greeks used unit fractions and (later) continued 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 **bhinnarasi** seems to have originated in India in the work of Aryabhatta (c. AD 500),^{[citation needed]} Brahmagupta (c. 628), and Bhaskara (c. 1150).^{[30]} Their works form fractions by placing the numerators (Sanskrit: amsa) over the denominators (*cheda*), but without a bar between them.^{[30]} In Sanskrit literature, fractions were always expressed as an addition to or subtraction from an integer.^{[citation needed]} 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. For example, Bhaskara I writes:^{[31]}

- ६ १ २
- १ १ १
_{०} - ४ ५ ९

which is the equivalent of

- 6 1 2
- 1 1 −1
- 4 5 9

and would be written in modern notation as 614, 115, and 2 − 19 (i.e., 189).

The horizontal fraction bar is first attested in the work of Al-Hassār (fl. 1200),^{[30]} a Muslim mathematician from Fez, Morocco, who specialized in Islamic inheritance jurisprudence. In his discussion he writes: "for example, if you are told to write three-fifths and a third of a fifth, write thus, ".^{[32]} The same fractional notation—with the fraction given before the integer^{[30]}—appears soon after in the work of Leonardo Fibonacci in the 13th century.^{[33]}

In discussing the origins of decimal fractions, Dirk Jan Struik states:^{[34]}

While the Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggren notes that he was mistaken, as decimal fractions were first used five centuries before him by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.^{[36]}^{[n 2]}

## In formal education

### Pedagogical tools

In primary schools, fractions have been demonstrated through Cuisenaire rods, Fraction Bars, fraction strips, fraction circles, paper (for folding or cutting), pattern blocks, pie-shaped pieces, plastic rectangles, grid paper, dot paper, geoboards, counters and computer software.

### Documents for teachers

Several states in the United States have adopted learning trajectories from the Common Core State Standards Initiative's guidelines for mathematics education. Aside from sequencing the learning of fractions and operations with fractions, the document provides the following definition of a fraction: "A number expressible in the form where is a whole number and is a positive whole number. (The word *fraction*in these standards always refers to a non-negative number.)"^{[38]} The document itself also refers to negative fractions.