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Islamic calendar

The Hijri calendar (ٱلتَّقْوِيم ٱلْهِجْرِيّ), or Arabic calendar also known in English as the Muslim calendar and Islamic calendar, is a lunar calendar consisting of 12 lunar months in a year of 354 or 355 days. It is used to determine the proper days of Islamic holidays and rituals, such as the annual fasting and the annual season for the great pilgrimage. In almost all countries where the predominant religion is Islam, the civil calendar is the Gregorian calendar, with Syriac month-names used in the Levant and Mesopotamia (Iraq, Syria, Jordan, Lebanon and Palestine) but the religious calendar is the Hijri one.

This calendar enumerates the Hijri era, whose epoch was established as the Islamic New Year in 622 CE. During that year, Muhammad and his followers migrated from Mecca to Medina and established the first Muslim community (ummah), an event commemorated as the Hijrah. In the West, dates in this era are usually denoted AH (Anno Hegirae). In Muslim countries, it is also sometimes denoted as H from its Arabic form (سَنَة هِجْرِيَّة, abbreviated ھ). In English, years prior to the Hijra are denoted as BH ("Before the Hijra").

Since 7 July 2024 CE,. In the Gregorian calendar reckoning, 1446 AH runs from 7 July 2024 to approximately 26 June 2025.

Pre-Islamic calendar
For central Arabia, especially Mecca, there is a lack of epigraphical evidence but details are found in the writings of Muslim authors of the Abbasid era. Inscriptions of the ancient South Arabian calendars reveal the use of a number of local calendars. At least some of these South Arabian calendars followed the lunisolar system. Both al-Biruni and al-Mas'udi suggest that the ancient Arabs used the same month names as the Muslims, though they also record other month names used by the pre-Islamic Arabs.

The Islamic tradition is unanimous in stating that Arabs of Tihamah, Hejaz, and Najd distinguished between two types of months, permitted (ḥalāl) and forbidden (ḥarām) months. The forbidden months were four months during which fighting is forbidden, listed as Rajab and the three months around the pilgrimage season, Dhu al-Qa‘dah, Dhu al-Hijjah, and Muharram. A similar if not identical concept to the forbidden months is also attested by Procopius, where he describes an armistice that the Eastern Arabs of the Lakhmid al-Mundhir respected for two months in the summer solstice of 541 CE. However, Muslim historians do not link these months to a particular season. The Qur'an links the four forbidden months with Nasī', a word that literally means "postponement". According to Muslim tradition, the decision of postponement was administered by the tribe of Kinanah, by a man known as the al-Qalammas of Kinanah and his descendants (pl. qalāmisa).

Different interpretations of the concept of Nasī' have been proposed. Some scholars, both Muslim and Western,  maintain that the pre-Islamic calendar used in central Arabia was a purely lunar calendar similar to the modern Islamic calendar. According to this view, Nasī' is related to the pre-Islamic practices of the Meccan Arabs, where they would alter the distribution of the forbidden months within a given year without implying a calendar manipulation. This interpretation is supported by Arab historians and lexicographers, like Ibn Hisham, Ibn Manzur, and the corpus of Qur'anic exegesis.

This is corroborated by an early Sabaic inscription, where a religious ritual was "postponed" (ns'w) due to war. According to the context of this inscription, the verb ns'’ has nothing to do with intercalation, but only with moving religious events within the calendar itself. The similarity between the religious concept of this ancient inscription and the Qur'an suggests that non-calendaring postponement is also the Qur'anic meaning of Nasī'. The Encyclopaedia of Islam concludes "The Arabic system of [Nasī'] can only have been intended to move the Hajj and the fairs associated with it in the vicinity of Mecca to a suitable season of the year. It was not intended to establish a fixed calendar to be generally observed." The term "fixed calendar" is generally understood to refer to the non-intercalated calendar.

Others concur that it was originally a lunar calendar, but suggest that about 200 years before the Hijra it was transformed into a lunisolar calendar containing an intercalary month added from time to time to keep the pilgrimage within the season of the year when merchandise was most abundant. This interpretation was first proposed by the medieval Muslim astrologer and astronomer Abu Ma'shar al-Balkhi, and later by al-Biruni, al-Mas'udi, and some western scholars. This interpretation considers Nasī' to be a synonym to the Arabic word for "intercalation" (kabīsa). The Arabs, according to one explanation mentioned by Abu Ma'shar, learned of this type of intercalation from the Jews. The Jewish Nasi was the official who decided when to intercalate the Jewish calendar. Some sources say that the Arabs followed the Jewish practice and intercalated seven months over nineteen years, or else that they intercalated nine months over 24 years; there is, however, no consensus among scholars on this issue.

Prohibiting Nasī'

Nasi' is interpreted to signify either the postponement of the pre-Islamic month of Hajj, or the (also pre-Islamic) practice of intercalation – periodic insertion of an additional month to reset the calendar into accordance with the seasons.

In the tenth year of the Hijra, as documented in the Qur'an (Surah At-Tawbah (9):36–37), Muslims believe God revealed the "prohibition of the Nasī.

"Indeed, the number of months ordained by Allah is twelve—in Allah's Record since the day He created the heavens and the earth—of which four are sacred. That is the Right Way. So do not wrong one another during these months. And together fight the polytheists as they fight against you together. And know that Allah is with those mindful ˹of Him˺. Reallocating the sanctity of ˹these˺ months is an increase in disbelief, by which the disbelievers are led ˹far˺ astray. They adjust the sanctity one year and uphold it in another, only to maintain the number of months sanctified by Allah, violating the very months Allah has made sacred. Their evil deeds have been made appealing to them. And Allah does not guide the disbelieving people."

The prohibition of Nasī' would presumably have been announced when the intercalated month had returned to its position just before the month of Nasi' began. If Nasī' meant intercalation, then the number and the position of the intercalary months between AH 1 and AH 10 are uncertain; western calendar dates commonly cited for key events in early Islam such as the Hijra, the Battle of Badr, the Battle of Uhud and the Battle of the Trench should be viewed with caution as they might be in error by one, two, three or even four lunar months. This prohibition was mentioned by Muhammad during the farewell sermon which was delivered on 9 Dhu al-Hijjah AH 10 (Julian date Friday 6 March 632 CE) on Mount Arafat during the farewell pilgrimage to Mecca.

"Certainly the Nasi' is an impious addition, which has led the infidels into error. One year they authorise the Nasi', another year they forbid it. They observe the divine precept with respect to the number of the sacred months, but in fact they profane that which God has declared to be inviolable, and sanctify that which God has declared to be profane. Assuredly time, in its revolution, has returned to such as it was at the creation of the heavens and the earth. In the eyes of God the number of the months is twelve. Among these twelve months four are sacred, namely, Rajab, which stands alone, and three others which are consecutive."

The three successive sacred (forbidden) months mentioned by Muhammad (months in which battles are forbidden) are Dhu al-Qa'dah, Dhu al-Hijjah, and Muharram, months 11, 12, and 1 respectively. The single forbidden month is Rajab, month 7. These months were considered forbidden both within the new Islamic calendar and within the old pagan Meccan calendar.

Days of the week
Traditionally, the Islamic day begins at sunset and ends at the next sunset. Each Islamic day thus begins at nightfall and ends at the end of daylight.

The days in the seven-day week are, with the exception of the last two days, named after their ordinal place in the week.

On the sixth day of the week, the "gathering day", Muslims assemble for the Friday-prayer at a local mosque at noon. The "gathering day" is often regarded as the weekly day off. This is frequently made official, with many Muslim countries adopting Friday and Saturday (e.g., Egypt, Saudi Arabia) or Thursday and Friday as official weekends, during which offices are closed; other countries (e.g., Iran) choose to make Friday alone a day of rest. A few others (e.g., Turkey, Pakistan, Morocco, Nigeria, Malaysia) have adopted the Saturday-Sunday weekend while making Friday a working day with a long midday break to allow time off for worship.

Months
Each month of the Islamic calendar commences on the birth of the new lunar cycle. Traditionally, this is based on actual observation of the moon's crescent (hilal) marking the end of the previous lunar cycle and hence the previous month, thereby beginning the new month. Consequently, each month can have 29 or 30 days depending on the visibility of the Moon, astronomical positioning of the Earth and weather conditions.

Four of the twelve Hijri months are considered sacred: (7), and the three consecutive months of  (11),  (12) and  (1).

Afghan lunar calendar
The "Afghan lunar calendar" refers to two distinct naming systems for the months of the Hijri calendar, one of which was used by the Pashtuns and the other by the Hazaras. They were in use until the time of Amanullah Khan's reign, when the usage of the Solar Hijri Calendar was formalized across Afghanistan.

Turki lunar calendar
In Xinjiang, the Uyghur Muslims traditionally had different names for the months of the Hijri calendar, which were in use until the adoption of the Gregorian calendar in the 20th century. These names were collectively referred to as the "Turki lunar year" or "Turki lunar calendar".

Alternative order
Twelver Shia Muslims believe the Islamic new year and first month of the Hijri calendar is Rabi' al-Awwal rather than Muharram, due to it being the month in which the Hijrah took place. This has led to difference regarding description of the years in which some events took place, such as the Muharram-occurring battle of Karbala, which Shias say took place in 60 AH, while Sunnis say it took place in 61 AH.

Length of year
The mean duration of a tropical year is 365.24219 days, while the long-term average duration of a synodic month is 29.530587981 days. Thus the average lunar year (twelve new moons) is 10.87513 days shorter than the average solar year (365.24219 − (12 × 29.530587981)), causing months of the Hijri calendar to advance about eleven days earlier each year, relative to the equinoxes. "As a result," says the Astronomical Almanac, "the cycle of twelve lunar months regresses through the seasons over a period of about 33 [solar] years".

Year numbering
In pre-Islamic Arabia, it was customary to identify a year after a major event which took place in it. Thus, according to Islamic tradition, Abraha, governor of Yemen, then a province of the Christian Kingdom of Aksum of Northeast Africa and South Arabia, attempted to destroy the Kaaba with an army which included several elephants. The raid was unsuccessful, but that year became known as the Year of the Elephant, during which Muhammad was born (surah al-Fil). Most equate this to the year 570 CE, but a minority use 571 CE.

The first ten years of the Hijra were not numbered, but were named after events in the life of Muhammad according to al-Biruni:
 * 1) The year of permission.
 * 2) The year of the order of fighting.
 * 3) The year of the trial.
 * 4) The year of congratulation on marriage.
 * 5) The year of the earthquake.
 * 6) The year of enquiring.
 * 7) The year of gaining victory.
 * 8) The year of equality.
 * 9) The year of exemption.
 * 10) The year of farewell.

In c. 638 (17 AH), Abu Musa al-Ash'ari, one of the officials of the Rashid Caliph Umar ((r. 634 – 644)) in Basra, complained about the absence of any years on the correspondence he received from Umar, making it difficult for him to determine which instructions were most recent. This report convinced Umar of the need to introduce an era for Muslims. After debating the issue with his counsellors, he decided that the first year should be the year of Muhammad's arrival at Medina (known as Yathrib, before Muhammad's arrival). Uthman then suggested that the months begin with Muharram, in line with the established custom of the Arabs at that time. The years of the Islamic calendar thus began with the month of Muharram in the year of Muhammad's arrival at the city of Medina, even though the actual emigration took place in Safar and Rabi' I of the intercalated calendar, two months before the commencement of Muharram in the new fixed calendar. Because of the Hijra, the calendar was named the Hijri calendar.

F A Shamsi (1984) postulated that the Arabic calendar was never intercalated. According to him, the first day of the first month of the new fixed Islamic calendar (1 Muharram AH 1) was no different from what was observed at the time. The day the Prophet moved from Quba' to Medina was originally 26 Rabi' I on the pre-Islamic calendar. 1 Muharram of the new fixed calendar corresponded to Friday, 16 July 622 CE, the equivalent civil tabular date (same daylight period) in the Julian calendar. The Islamic day began at the preceding sunset on the evening of 15 July. This Julian date (16 July) was determined by medieval Muslim astronomers by projecting back in time their own tabular Islamic calendar, which had alternating 30- and 29-day months in each lunar year plus eleven leap days every 30 years. For example, al-Biruni mentioned this Julian date in the year 1000 CE. Although not used by either medieval Muslim astronomers or modern scholars to determine the Islamic epoch, the thin crescent moon would have also first become visible (assuming clouds did not obscure it) shortly after the preceding sunset on the evening of 15 July, 1.5 days after the associated dark moon (astronomical new moon) on the morning of 14 July.

Though Michael Cook and Patricia Crone in their book Hagarism cite a coin from AH 17, the first surviving attested use of a Hijri calendar date alongside a date in another calendar (Coptic) is on a papyrus from Egypt in AH 22, PERF 558.

Astronomical considerations
Due to the Islamic calendar's reliance on certain variable methods of observation to determine its month-start-dates, these dates sometimes vary slightly from the month-start-dates of the astronomical lunar calendar, which are based directly on astronomical calculations. Still, the Islamic calendar roughly approximates the astronomical-lunar-calendar system, seldom varying by more than three days from it. Both the Islamic calendar and the astronomical-lunar-calendar take no account of the solar year in their calculations, and thus both of these strictly lunar based calendar systems have no ability to reckon the timing of the four seasons of the year.

In the astronomical-lunar-calendar system, a year of 12 lunar months is 354.37 days long. In this calendar system, lunar months begin precisely at the time of the monthly "conjunction", when the Moon is located most directly between the Earth and the Sun. The month is defined as the average duration of a revolution of the Moon around the Earth (29.53 days). By convention, months of 30 days and 29 days succeed each other, adding up over two successive months to 59 full days. This leaves only a small monthly variation of 44 minutes to account for, which adds up to a total of 24 hours (i.e., the equivalent of one full day) in 2.73 years. To settle accounts, it is sufficient to add one day every three years to the lunar calendar, in the same way that one adds one day to the Gregorian calendar every four years. The technical details of the adjustment are described in Tabular Islamic calendar.

The Islamic calendar, however, is based on a different set of conventions being used for the determination of the month-start-dates. Each month still has either 29 or 30 days, but due to the variable method of observations employed, there is usually no discernible order in the sequencing of either 29 or 30-day month lengths. Traditionally, the first day of each month is the day (beginning at sunset) of the first sighting of the hilal (crescent moon) shortly after sunset. If the hilal is not observed immediately after the 29th day of a month (either because clouds block its view or because the western sky is still too bright when the moon sets), then the day that begins at that sunset is the 30th. Such a sighting has to be made by one or more trustworthy men testifying before a committee of Muslim leaders. Determining the most likely day that the hilal could be observed was a motivation for Muslim interest in astronomy, which put Islam in the forefront of that science for many centuries. Still, due to the fact that both lunar reckoning systems are ultimately based on the lunar cycle itself, both systems still do roughly correspond to one another, never being more than three days out of synchronisation with one another.

This traditional practice for the determination of the start-date of the month is still followed in the overwhelming majority of Muslim countries. For instance, Saudi Arabia uses the sighting method to determine the beginning of each month of the Hijri calendar. Since AH 1419 (1998/99), several official hilal sighting committees have been set up by the government to determine the first visual sighting of the lunar crescent at the beginning of each lunar month. Nevertheless, the religious authorities also allow the testimony of less experienced observers and thus often announce the sighting of the lunar crescent on a date when none of the official committees could see it.

Each Islamic state proceeds with its own monthly observation of the new moon (or, failing that, awaits the completion of 30 days) before declaring the beginning of a new month on its territory. However, the lunar crescent becomes visible only some 17 hours after the conjunction, and only subject to the existence of a number of favourable conditions relative to weather, time, geographic location, as well as various astronomical parameters. Given the fact that the moon sets progressively later than the sun as one goes west, with a corresponding increase in its "age" since conjunction, Western Muslim countries may, under favorable conditions, observe the new moon one day earlier than eastern Muslim countries. Due to the interplay of all these factors, the beginning of each month differs from one Muslim country to another, during the 48-hour period following the conjunction. The information provided by the calendar in any country does not extend beyond the current month.

A number of Muslim countries try to overcome some of these difficulties by applying different astronomy-related rules to determine the beginning of months. Thus, Malaysia, Indonesia, and a few others begin each month at sunset on the first day that the moon sets after the sun (moonset after sunset). In Egypt, the month begins at sunset on the first day that the moon sets at least five minutes after the sun. A detailed analysis of the available data shows, however, that there are major discrepancies between what countries say they do on this subject, and what they actually do. In some instances, what a country says it does is impossible.

Due to the somewhat variable nature of the Islamic calendar, in most Muslim countries, the Islamic calendar is used primarily for religious purposes, while the Solar-based Gregorian calendar is still used primarily for matters of commerce and agriculture.

Theological considerations
If the Islamic calendar were prepared using astronomical calculations, Muslims throughout the Muslim world could use it to meet all their needs, the way they use the Gregorian calendar today. But, there are divergent views on whether it is licit to do so.

A majority of theologians oppose the use of calculations (beyond the constraint that each month must be not less than 29 nor more than 30 days) on the grounds that the latter would not conform with Muhammad's recommendation to observe the new moon of Ramadan and Shawal in order to determine the beginning of these months.

However, some Islamic jurists see no contradiction between Muhammad's teachings and the use of calculations to determine the beginnings of lunar months. They consider that Muhammad's recommendation was adapted to the culture of the times, and should not be confused with the acts of worship.

Thus the jurists Ahmad Muhammad Shakir and Yusuf al-Qaradawi both endorsed the use of calculations to determine the beginning of all months of the Islamic calendar, in 1939 and 2004 respectively. So did the Fiqh Council of North America (FCNA) in 2006 and the European Council for Fatwa and Research (ECFR) in 2007.

The major Muslim associations of France also announced in 2012 that they would henceforth use a calendar based on astronomical calculations, taking into account the criteria of the possibility of crescent sighting in any place on Earth. But, shortly after the official adoption of this rule by the French Council of the Muslim Faith (CFCM) in 2013, the new leadership of the association decided, on the eve of Ramadan 2013, to follow the Saudi announcement rather than to apply the rule just adopted. This resulted in a division of the Muslim community of France, with some members following the new rule, and others following the Saudi announcement.

Isma'ili-Taiyebi Bohras having the institution of da'i al-mutlaq follow the tabular Islamic calendar (see section below) prepared on the basis of astronomical calculations from the days of Fatimid imams.

Islamic calendar of Turkey
Turkish Muslims use an Islamic calendar which is calculated several years in advance by the Turkish Presidency of Religious Affairs (Diyanet İşleri Başkanlığı). From 1 Muharrem 1400 AH (21 November 1979) until 29 Zilhicce 1435 (24 October 2014) the computed Turkish lunar calendar was based on the following rule: "The lunar month is assumed to begin on the evening when, within some region of the terrestrial globe, the computed centre of the lunar crescent at local sunset is more than 5° above the local horizon and (geocentrically) more than 8° from the Sun." In the current rule the (computed) lunar crescent has to be above the local horizon of Ankara at sunset.

Saudi Arabia's Umm al-Qura calendar
Saudi Arabia has traditionally used the Umm al-Qura calendar, which is based on astronomical calculations, for administrative purposes. The parameters used in the establishment of this calendar underwent significant changes during the decade to AH 1423.

Before AH 1420 (before 18 April 1999), if the moon's age at sunset in Riyadh was at least 12 hours, then the day ending at that sunset was the first day of the month. This often caused the Saudis to celebrate holy days one or even two days before other predominantly Muslim countries, including the dates for the Hajj, which can only be dated using Saudi dates because it is performed in Mecca.

From AH 1420–22, if moonset occurred after sunset at Mecca, then the day beginning at that sunset was the first day of a Saudi month, essentially the same rule used by Malaysia, Indonesia, and others (except for the location from which the hilal was observed).

Since the beginning of AH 1423 (16 March 2002), the rule has been clarified a little by requiring the geocentric conjunction of the sun and moon to occur before sunset, in addition to requiring moonset to occur after sunset at Mecca. This ensures that the moon has moved past the sun by sunset, even though the sky may still be too bright immediately before moonset to actually see the crescent.

In 2007, the Islamic Society of North America, the Fiqh Council of North America and the European Council for Fatwa and Research announced that they would henceforth use a calendar based on calculations using the same parameters as the Umm al-Qura calendar to determine (well in advance) the beginning of all lunar months (and therefore the days associated with all religious observances). This was intended as a first step on the way to unify, at some future time, Muslims' calendars throughout the world.

On 14 February 2016, Saudi Arabia adopted the Gregorian calendar for payment of the monthly salaries of government employees (as a cost cutting measure), while retaining the Islamic calendar for religious purposes.

Other calendars using the Islamic era
The Solar Hijri calendar is a solar calendar used in Iran which counts its years from the Hijra or migration of Muhammad from Mecca to Medina in 622 CE.

Tabular Islamic calendar
The Tabular Islamic calendar is a rule-based variation of the Islamic calendar, in which months are worked out by arithmetic rules rather than by observation or astronomical calculation. It has a 30-year cycle with 11 leap years of 355 days and 19 years of 354 days. In the long term, it is accurate to one day in about 2,500 solar years or 2,570 lunar years. It also deviates up to about one or two days in the short term.

Kuwaiti algorithm
Microsoft uses the "Kuwaiti algorithm", a variant of the tabular Islamic calendar, to convert Gregorian dates to the Islamic ones. Microsoft claimed that the variant is based on a statistical analysis of historical data from Kuwait, however it matches a known tabular calendar.

Notable dates
Important dates in the Islamic (Hijri) year are:
 * 1 Muharram: the Islamic New Year.
 * 10 Muharram: Day of Ashura. For both Shias and Sunnis, the martyrdom of Husayn ibn Ali, the grandson of Muhammad, and his followers. For Sunnis, the crossing of the Red Sea by Moses occurred on this day, along with many other significant events in the lives of prophets and that have to do with Creation.
 * 12 Rabi al-Awwal: Mawlid or Birth of the Prophet for Sunnis.
 * 17 Rabi al-Awwal: Mawlid for Shias.
 * 27 Rajab: Isra and Mi'raj for the majority of Muslims.
 * 15 Sha'ban: Mid-Sha'ban, or Night of Forgiveness. For Shiites, also the birthday of Muhammad al-Mahdi, the Twelfth Imam.
 * 1 Ramadan: The first day of fasting in Islam
 * 27 Ramadan: Start of the Revelation of the Qur’an. The most probable day Muhammad received the first verses of the Quran (17 Ramadan in Indonesia and Malaysia).
 * Last third of Ramadan which includes Laylat al-Qadr.
 * Last Friday of Ramadan: Jumu'atul-Wida
 * 1 Shawwal: Eid ul-Fitr.
 * 8–13 Dhu al-Hijjah: The Hajj pilgrimage to Mecca.
 * 9 Dhu al-Hijjah: Day of Arafa.
 * 10 Dhu al-Hijjah: Eid al-Adha.

Days considered important predominantly for Shia Muslims:
 * 9 Rabi' al-Awwal: Omar Koshan (Mukhtar al-Thaqafi avenges the events of Ashura).
 * 13 Rajab: Birthday of Ali ibn Abi Talib
 * 3 Sha'ban: Birthday of Husayn ibn Ali.
 * 21 Ramadan: Martyrdom of Ali ibn Abi Talib.
 * 18 Dhu al-Hijjah: the Eid al-Ghadir

Uses
The Islamic calendar is now used primarily for religious purposes, and for official dating of public events and documents in Muslim countries. Because of its nature as a purely lunar calendar, it cannot be used for agricultural purposes and historically Islamic communities have used other calendars for this purpose: the Egyptian calendar was formerly widespread in Islamic countries, and the Iranian calendar and the 1789 Ottoman calendar (a modified Julian calendar) were also used for agriculture in their countries. In the Levant and Iraq the Aramaic names of the Babylonian calendar are still used for all secular matters. In the Maghreb, Berber farmers in the countryside still use the Julian calendar for agrarian purposes. These local solar calendars have receded in importance with the near-universal adoption of the Gregorian calendar for civil purposes. Saudi Arabia uses the lunar Islamic calendar. In Indonesia, the Javanese calendar combines elements of the Islamic and pre-Islamic Saka calendars.

British author Nicholas Hagger writes that after seizing control of Libya, Muammar Gaddafi "declared" on 1 December 1978 "that the Muslim calendar should start with the death of the prophet Mohammed in 632 rather than the hijra (Mohammed's 'emigration' from Mecca to Medina) in 622". This put the country ten solar years behind the standard Muslim calendar. However, according to the 2006 Encyclopedia of the Developing World, "More confusing still is Qaddafi's unique Libyan calendar, which counts the years from the Prophet's birth, or sometimes from his death. The months July and August, named after Julius and Augustus Caesar, are now Nasser and Hannibal respectively." Reflecting on a 2001 visit to the country, American reporter Neil MacFarquhar observed, "Life in Libya was so unpredictable that people weren't even sure what year it was. The year of my visit was officially 1369. But just two years earlier Libyans had been living through 1429. No one could quite name for me the day the count changed, especially since both remained in play. ... Event organizers threw up their hands and put the Western year in parentheses somewhere in their announcements."

Computer support

 * Hijri support was available in later versions of traditional Visual Basic, and is also available in the .NET Framework.
 * Since the release of Java 8, the Islamic calendar is supported in the new Date and Time API.

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Interquartile range

In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data. The IQR may also be called the midspread, middle 50%, fourth spread, or H‑spread. It is defined as the difference between the 75th and 25th percentiles of the data. To calculate the IQR, the data set is divided into quartiles, or four rank-ordered even parts via linear interpolation. These quartiles are denoted by Q1 (also called the lower quartile), Q2 (the median), and Q3 (also called the upper quartile). The lower quartile corresponds with the 25th percentile and the upper quartile corresponds with the 75th percentile, so IQR = Q3 −  Q1.

The IQR is an example of a trimmed estimator, defined as the 25% trimmed range, which enhances the accuracy of dataset statistics by dropping lower contribution, outlying points. It is also used as a robust measure of scale It can be clearly visualized by the box on a box plot.

Use
Unlike total range, the interquartile range has a breakdown point of 25% and is thus often preferred to the total range.

The IQR is used to build box plots, simple graphical representations of a probability distribution.

The IQR is used in businesses as a marker for their income rates.

For a symmetric distribution (where the median equals the midhinge, the average of the first and third quartiles), half the IQR equals the median absolute deviation (MAD).

The median is the corresponding measure of central tendency.

The IQR can be used to identify outliers (see below). The IQR also may indicate the skewness of the dataset.

The quartile deviation or semi-interquartile range is defined as half the IQR.

Algorithm
The IQR of a set of values is calculated as the difference between the upper and lower quartiles, Q3 and Q1. Each quartile is a median calculated as follows.

Given an even 2n or odd 2n+1 number of values
 * first quartile Q1 = median of the n smallest values
 * third quartile Q3 = median of the n largest values

The second quartile Q2 is the same as the ordinary median.

Data set in a table
The following table has 13 rows, and follows the rules for the odd number of entries.

For the data in this table the interquartile range is IQR = Q3 − Q1 = 119 - 31 = 88.

Data set in a plain-text box plot
 +−−−−−+−+                   * |−−−−−−−−−−−|     | |−−−−−−−−−−−|                             +−−−−−+−+     +−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+   number line 0  1   2   3   4   5   6   7   8   9   10  11  12

For the data set in this box plot: This means the 1.5*IQR whiskers can be uneven in lengths. The median, minimum, maximum, and the first and third quartile constitute the Five-number summary.
 * Lower (first) quartile Q1 = 7
 * Median (second quartile) Q2 = 8.5
 * Upper (third) quartile Q3 = 9
 * Interquartile range, IQR = Q3 - Q1 = 2
 * Lower 1.5*IQR whisker = Q1 - 1.5 * IQR = 7 - 3 = 4. (If there is no data point at 4, then the lowest point greater than 4.)
 * Upper 1.5*IQR whisker = Q3 + 1.5 * IQR = 9 + 3 = 12. (If there is no data point at 12, then the highest point less than 12.)
 * Pattern of latter two bullet points: If there are no data points at the true quartiles, use data points slightly "inland" (closer to the median) from the actual quartiles.

Distributions
The interquartile range of a continuous distribution can be calculated by integrating the probability density function (which yields the cumulative distribution function—any other means of calculating the CDF will also work). The lower quartile, Q1, is a number such that integral of the PDF from -∞ to Q1 equals 0.25, while the upper quartile, Q3, is such a number that the integral from -∞ to Q3 equals 0.75; in terms of the CDF, the quartiles can be defined as follows:

 * $$Q_1 = \text{CDF}^{-1}(0.25) ,$$

 * $$Q_3 = \text{CDF}^{-1}(0.75) ,$$

where CDF−1 is the quantile function.

The interquartile range and median of some common distributions are shown below

Interquartile range test for normality of distribution
The IQR, mean, and standard deviation of a population P can be used in a simple test of whether or not P is normally distributed, or Gaussian. If P is normally distributed, then the standard score of the first quartile, z1, is −0.67, and the standard score of the third quartile, z3, is +0.67. Given mean = $$\bar{P}$$ and standard deviation = σ for P, if P is normally distributed, the first quartile

 * $$Q_1 = (\sigma \, z_1) + \bar{P}$$

and the third quartile

 * $$Q_3 = (\sigma \, z_3) + \bar{P}$$

If the actual values of the first or third quartiles differ substantially from the calculated values, P is not normally distributed. However, a normal distribution can be trivially perturbed to maintain its Q1 and Q2 std. scores at 0.67 and −0.67 and not be normally distributed (so the above test would produce a false positive). A better test of normality, such as Q–Q plot would be indicated here.

Outliers
The interquartile range is often used to find outliers in data. Outliers here are defined as observations that fall below Q1 − 1.5 IQR or above Q3 + 1.5 IQR. In a boxplot, the highest and lowest occurring value within this limit are indicated by whiskers of the box (frequently with an additional bar at the end of the whisker) and any outliers as individual points.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Interquartile range


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Indiana Jones (character)

Dr. Henry Walton "Indiana" Jones, Jr. is the title character and protagonist of the Indiana Jones franchise. George Lucas created the character in homage to the action heroes of 1930s film serials. The character first appeared in the 1981 film Raiders of the Lost Ark, to be followed by Indiana Jones and the Temple of Doom in 1984, Indiana Jones and the Last Crusade in 1989, The Young Indiana Jones Chronicles from 1992 to 1996, Indiana Jones and the Kingdom of the Crystal Skull in 2008, and Indiana Jones and the Dial of Destiny in 2023. The character is also featured in novels, comics, video games, and other media. Jones is also the inspiration for several Disney theme park attractions, including Indiana Jones and the Temple of Peril, the Indiana Jones Adventure, and Epic Stunt Spectacular! attractions.

Jones is most famously portrayed by Harrison Ford and has also been portrayed by River Phoenix (as the young Jones in The Last Crusade), and by Corey Carrier, Sean Patrick Flanery, and George Hall in the television series The Young Indiana Jones Chronicles. Doug Lee has supplied the voice of Jones for two LucasArts video games, Indiana Jones and the Fate of Atlantis and Indiana Jones and the Infernal Machine, David Esch supplied his voice for Indiana Jones and the Emperor's Tomb, and John Armstrong for Indiana Jones and the Staff of Kings. Troy Baker provides the voice and motion capture for the character in Indiana Jones and the Great Circle (2024).

Jones is characterized by his iconic accoutrements (bullwhip, fedora, satchel, and leather jacket), wry, witty and sarcastic sense of humor, deep knowledge of ancient civilizations and languages, and fear of snakes.

Since his first appearance in Raiders of the Lost Ark, Indiana Jones has become one of cinema's most famous characters. In 2003, the American Film Institute ranked him the second-greatest film hero of all time. He was also named the greatest movie character by Empire magazine. Entertainment Weekly ranked Jones 2nd on their list of The All-Time Coolest Heroes in Pop Culture. Premiere magazine also placed Jones at number 7 on their list of The 100 Greatest Movie Characters of All Time.

Films and television installments in the chronological order

 * The Young Indiana Jones Chronicles (1992–1996)
 * Indiana Jones and the Temple of Doom (1984)
 * Raiders of the Lost Ark (1981)
 * Indiana Jones and the Last Crusade (1989)
 * Indiana Jones and the Kingdom of the Crystal Skull (2008)
 * Indiana Jones and the Dial of Destiny (2023)

A native of Princeton, New Jersey, Indiana Jones was introduced as a tenured professor of archaeology in the 1981 film Raiders of the Lost Ark, set in 1936. The Joneses are a family of paternal Scottish descent. The character is an adventurer reminiscent of the 1930s film serial treasure hunters and pulp action heroes. His research is funded by Marshall College (a fictional school named after producer Frank Marshall), where he is a professor of archaeology. He studied under the Egyptologist and archaeologist Abner Ravenwood at the Oriental Institute at the University of Chicago.

Raiders of the Lost Ark (1981)
In the first adventure, Raiders of the Lost Ark, set in 1936, Indiana Jones is pitted against Nazis commissioned by Hitler to recover artifacts of great power from the Old Testament (see Nazi archaeology). In consequence, Jones travels the world to prevent them from recovering the Ark of the Covenant (see also Biblical archaeology). He is aided by Marion Ravenwood and Sallah. The Nazis are led by Jones' archrival, a Nazi-sympathizing French archaeologist named René Belloq, and Arnold Toht, a sinister Gestapo agent.

The Temple of Doom (1984)
In the 1984 prequel, Indiana Jones and the Temple of Doom, set in 1935, Jones travels to India and attempts to free enslaved children and the three Sankara stones from the bloodthirsty Thuggee cult. He is aided by Wan "Short Round" Li, a boy played by Ke Huy Quan, and is accompanied by singer Willie Scott (Kate Capshaw). The prequel is not as centered on archaeology as Raiders of the Lost Ark and is considerably darker.

The Last Crusade (1989)
The third film, 1989's Indiana Jones and the Last Crusade, set in 1938, returned to the formula of the original, reintroducing characters such as Sallah and Marcus Brody, a scene from Professor Jones' classroom (he now teaches at Barnett College), the globe-trotting element of multiple locations, and the return of the infamous Nazi mystics, this time trying to find the Holy Grail. The film's introduction, set in 1912, provided some backstory to the character, specifically the origin of his fear of snakes, his use of a bullwhip, the scar on his chin, and his hat; the film's epilogue also reveals that "Indiana" is not Jones' first name, but a nickname he took from the family dog. The film was a buddy movie of sorts, teaming Indiana with his father, Henry Jones, Sr., often to comical effect. Although Lucas intended to make five Indiana Jones films, Indiana Jones and the Last Crusade was the last for over 18 years, as he could not think of a good plot element to drive the next installment.

The Young Indiana Jones Chronicles (1992–1996)
From 1992 to 1996, Lucas wrote and executive-produced The Young Indiana Jones Chronicles, a television series aimed mainly at teenagers and children, which showed many of the important events and historical figures of the early 20th century through the prism of Jones' life.

The show initially featured the formula of an elderly (93 to 94 years of age) Indiana Jones played by George Hall introducing a story from his youth by way of an anecdote: the main part of the episode then featured an adventure with either a young adult Indy (16 to 21 years of age) played by Sean Patrick Flanery or a child Indy (8 to 10 years) played by Corey Carrier. One episode, "Young Indiana Jones and the Mystery of the Blues", is bookended by Harrison Ford as Indiana Jones, rather than Hall. Later episodes and telemovies did not have this bookend format.

The bulk of the series centers around the young adult Indiana Jones and his activities during World War I as a 16- to 17-year-old soldier in the Belgian Army and then as an intelligence officer and spy seconded to French intelligence. The child Indiana episodes follow the boy's travels around the globe as he accompanies his parents on his father's worldwide lecture tour from 1908 to 1910.

The show provided some backstory for the films, as well as new information regarding the character. Indiana Jones was born July 1, 1899, and his middle name is Walton (Lucas's middle name). It is also mentioned that he had a sister called Suzie who died as an infant of fever, and that he eventually has a daughter and grandchildren who appear in some episode introductions and epilogues. His relationship with his father, first introduced in Indiana Jones and the Last Crusade, was further fleshed out with stories about his travels with his father as a young boy. Jones damages or loses his right eye sometime between the events in 1969 and the early 1990s, when the "Old Indy" segments take place, as the elderly Indiana Jones wears an eyepatch.

In 1999, Lucas removed the episode introductions and epilogues by George Hall for the VHS and DVD releases and re-edited the episodes into chronologically ordered feature-length stories. The series title was also changed to The Adventures of Young Indiana Jones.

The Kingdom of the Crystal Skull (2008)
The 2008 film, Indiana Jones and the Kingdom of the Crystal Skull, is the fourth film in the series. Set in 1957, nineteen years after the third film, it pits an older, wiser Indiana Jones against Soviet KGB agents bent on harnessing the power of an extraterrestrial device discovered in South America. Jones is aided in his adventure by his former lover, Marion Ravenwood (Karen Allen), and her son—a young greaser named Henry "Mutt" Williams (Shia LaBeouf), later revealed to be Jones' unknown child. There were rumors that Harrison Ford would not return for any future installments and LaBeouf would take over the franchise. This film also reveals that Jones was recruited by the Office of Strategic Services during World War II, attaining the rank of colonel in the United States Army, and that in 1947 he was forced to investigate the Roswell UFO incident, and the investigation saw that he was involved in affairs related to Hangar 51. He is tasked with conducting covert operations with MI6 agent George McHale against the Soviet Union.

The Dial of Destiny (2023)
The 2023 film, Indiana Jones and the Dial of Destiny, is the fifth and final film in the series. Set in 1969—twelve years after the fourth film and during the height of the Space Race—Jones has moved to New York City, teaching at Hunter College with plans to retire, after his marriage with Marion collapsed following Mutt's death in the Vietnam War. Once his estranged goddaughter Helena Shaw (Phoebe Waller-Bridge) arrives asking for Archimedes' Dial, a relic Jones and her father Basil retrieved from the Nazis in 1944 during the Allied liberation of Europe in World War II, a Nazi-turned-NASA scientist Jürgen Voller (Mads Mikkelsen) starts pursuing Jones, wanting to exploit the Dial's unusual properties to change the outcome of World War II.

As the film concludes, Indiana and Marion are reconciled.

Attractions
Indiana Jones is featured at several Walt Disney theme park attractions. The Indiana Jones Adventure attractions at Disneyland and Tokyo DisneySea ("Temple of the Forbidden Eye" and "Temple of the Crystal Skull," respectively) place Indy at the forefront of two similar archaeological discoveries. These two temples each contain a wrathful deity who threatens the guests who ride through World War II troop transports. The attractions, some of the most expensive of their kind at the time, opened in 1995 and 2001, respectively, with sole design credit attributed to Walt Disney Imagineering. Ford was approached to reprise his role as Indiana Jones, but ultimately negotiations to secure Ford's participation broke down in December 1994, for unknown reasons. Instead, Dave Temple provided the voice of Jones. Ford's physical likeness, however, has nonetheless been used in subsequent Audio-animatronic figures for the attractions.

Disneyland Paris also features an Indiana Jones-titled ride where people speed off through ancient ruins in a runaway mine wagon similar to that found in Indiana Jones and the Temple of Doom. Indiana Jones and the Temple of Peril is a looping roller coaster engineered by Intamin, designed by Walt Disney Imagineering, and opened in 1993.

The Indiana Jones Epic Stunt Spectacular! is a live show that has been presented in the Disney's Hollywood Studios theme park of the Walt Disney World Resort with few changes since the park's 1989 opening, as Disney-MGM Studios. The 25-minute show presents various stunts framed in the context of a feature film production and recruits members of the audience to participate in the show. Stunt artists in the show re-create and ultimately reveal some of the secrets of the stunts of the Raiders of the Lost Ark films, including the well-known "running-from-the-boulder" scene. Stunt performer Anislav Varbanov was fatally injured in August 2009, while rehearsing the show. Also formerly at Disney's Hollywood Studios, an audio-animatronic Indiana Jones appeared in another attraction; during The Great Movie Ride's Raiders of the Lost Ark segment.

Comic books
Indiana Jones has appeared in numerous comic books, from two different publishers. Marvel Comics initially held the comic book licensing rights, leading to adaptations of the films Raiders of the Lost Ark, Indiana Jones and the Temple of Doom, and Indiana Jones and the Last Crusade. Following the Raiders of the Lost Ark adaptation, Marvel published The Further Adventures of Indiana Jones from 1983 to 1986. This ongoing monthly series ran for thirty-four issues and featured the character's first original adventures in comic book form.

After Marvel's licensing of the character ended, Dark Horse Comics acquired publishing rights and adapted the Indiana Jones and the Fate of Atlantis video game. From 1992 to 1996, following the Fate of Atlantis adaptation, Dark Horse published seven limited series, as well comics based on The Young Indiana Jones Chronicles television series. In 2004, Indiana Jones appeared in the non-canon story, "Into the Great Unknown", first published in Star Wars Tales #19. The story sees Indiana Jones and Short Round discover a crashed Millennium Falcon in the Pacific Northwest, along with Han Solo's skeleton and the realization that a rumored nearby Sasquatch is Chewbacca. With the franchise's revival in 2008, Dark Horse published an adaptation of Indiana Jones and the Kingdom of the Crystal Skull. Dark Horse followed this with Indiana Jones Adventures, a short-lived series of digest-sized comics aimed at children. An additional limited series, titled Indiana Jones and the Tomb of the Gods, was also published from 2008 to 2009.

Movie tie-in novelizations
The first four Indiana Jones film scripts were novelized and published in the time frame of the films' initial releases. Raiders of the Lost Ark was novelized by Campbell Black based on the script by Lawrence Kasdan that was based on the story by George Lucas and Philip Kaufman and published in April 1981 by Ballantine Books; Indiana Jones and the Temple of Doom was novelized by James Kahn and based on the script by Willard Huyck & Gloria Katz that was based on the story by George Lucas and published May 1984 by Ballantine Books; Indiana Jones and the Last Crusade was novelized by Rob MacGregor based on the script by Jeffrey Boam that was based on a story by George Lucas and Menno Meyjes and published June 1989 by Ballantine Books.

Nearly 20 years later Indiana Jones and the Kingdom of the Crystal Skull was novelized by James Rollins based on the script by David Koepp based on the story by George Lucas and Jeff Nathanson and published in May 2008 by Ballantine Books. In addition, in 2008 to accompany the release of Kingdom of the Crystal Skull, Scholastic Books published juvenile novelizations of the four scripts written, successively in the order above, by Ryder Windham, Suzanne Weyn, Ryder Windham, and James Luceno. All these books have been reprinted, with Raiders of the Lost Ark being retitled Indiana Jones and the Raiders of the Lost Ark. While these are the principal titles and authors, there are numerous other volumes derived from the four film properties.

Original novels
From February 1991 through February 1999, 12 original Indiana Jones-themed adult novels were licensed by Lucasfilm, Ltd. and written by three genre authors of the period. Ten years afterward, a 13th original novel was added, also written by a popular genre author. The first 12 were published by Bantam Books; the last by Ballantine Books in 2009. (See Indiana Jones (franchise) for broad descriptions of these original adult novels.) The novels are:

Written by Rob MacGregor

 * Indiana Jones and the Peril at Delphi, February 1991.
 * Indiana Jones and the Dance of the Giants, June 1991.
 * Indiana Jones and the Seven Veils, December 1991.
 * Indiana Jones and the Genesis Deluge, February 1992.
 * Indiana Jones and the Unicorn's Legacy, September 1992.
 * Indiana Jones and the Interior World, December 1992.

Written by Martin Caidin

 * Indiana Jones and the Sky Pirates, December 1993.
 * Indiana Jones and the White Witch, April 1994.

Written by Max McCoy

 * Indiana Jones and the Philosopher's Stone, May 1995.
 * Indiana Jones and the Dinosaur Eggs, March 1996.
 * Indiana Jones and the Hollow Earth, March 1997.
 * Indiana Jones and the Secret of the Sphinx, February 1999.

Written by Steve Perry

 * Indiana Jones and the Army of the Dead, September 2009.

Video games
The character has appeared in several officially licensed games, beginning with adaptations of Raiders of the Lost Ark, Indiana Jones and the Temple of Doom, two adaptations of Indiana Jones and the Last Crusade (one with purely action mechanics, one with an adventure- and puzzle-based structure) and Indiana Jones' Greatest Adventures, which included the storylines from all three of the original films.

Following this, the games branched off into original storylines with Indiana Jones in the Lost Kingdom, Indiana Jones and the Fate of Atlantis, Indiana Jones and the Infernal Machine, Indiana Jones and the Emperor's Tomb and Indiana Jones and the Staff of Kings. Emperor's Tomb sets up Jones' companion Wu Han and the search for Nurhaci's ashes seen at the beginning of Temple of Doom. The first two games were developed by Hal Barwood and starred Doug Lee as the voice of Indiana Jones; Emperor's Tomb had David Esch fill the role and Staff of Kings starred John Armstrong.

Indiana Jones and the Infernal Machine was the first Indy-based game presented in three dimensions, as opposed to 8-bit graphics and side-scrolling games before.

There is also a small game from Lucas Arts Indiana Jones and His Desktop Adventures. A video game was made for young Indy called Young Indiana Jones and the Instruments of Chaos, as well as a video game version of The Young Indiana Jones Chronicles.

Two Lego Indiana Jones games have also been released. Lego Indiana Jones: The Original Adventures was released in 2008 and follows the plots of the first three films. It was followed by Lego Indiana Jones 2: The Adventure Continues in late 2009. The sequel includes an abbreviated reprise of the first three films, but focuses on the plot of Indiana Jones and the Kingdom of the Crystal Skull. However, before he got his own Lego games, he appeared as a secret character in Lego Star Wars: The Complete Saga as a playable character. He also makes a brief appearance in a minigame in Lego Star Wars III: The Clone Wars during the level "Hostage Crisis", and also made a cameo alongside Henry Jones Sr. in the level "Legacy of Terror". Social gaming company Zynga introduced Indiana Jones to their Adventure World game in late 2011.

Indiana Jones appears in Fortnite Battle Royale as part of the Chapter 3 Season 3 Battle pass.

The world building game Disney Magic Kingdoms includes Indiana Jones as a playable character to unlock for a limited time.

An original Indiana Jones game, Indiana Jones and the Great Circle, is currently in development by MachineGames and set to be published by Bethesda Softworks for Windows and Xbox Series X/S, with Troy Baker voicing the character. It will feature a standalone narrative expected to take place in Italy during 1937, placing it between the events of Raiders of the Lost Ark and Indiana Jones and the Last Crusade.

Character description and formation
"Indiana" Jones's full name is Dr. Henry Walton Jones, Jr., and his nickname is often shortened to "Indy".

In his role as a college professor of archaeology Jones is scholarly, wears a tweed suit, and lectures on ancient civilizations. At the opportunity to recover important artifacts, Dr. Jones transforms into "Indiana," a "non-superhero superhero" image he has concocted for himself. Producer Frank Marshall said, "Indy [is] a fallible character. He makes mistakes and gets hurt. ... That's the other thing people like: He's a real character, not a character with superpowers." Spielberg said there "was the willingness to allow our leading man to get hurt and to express his pain and to get his mad out and to take pratfalls and sometimes be the butt of his own jokes. I mean, Indiana Jones is not a perfect hero, and his imperfections, I think, make the audience feel that, with a little more exercise and a little more courage, they could be just like him." According to Spielberg biographer Douglas Brode, Indiana created his heroic figure so as to escape the dullness of teaching at a school. Both of Indiana's personas reject one another in philosophy, creating a duality. Harrison Ford said the fun of playing the character was that Indiana is both a romantic and a cynic, while scholars have analyzed Indiana as having traits of a lone wolf; a man on a quest; a noble treasure hunter; a hardboiled detective; a human superhero; and an American patriot.

Like many characters in his films, Jones has some autobiographical elements of Spielberg. Indiana lacks a proper father figure because of his strained relationship with his father, Henry Jones, Sr. His own contained anger is misdirected towards Professor Abner Ravenwood, his mentor at the University of Chicago, leading to a strained relationship with Marion Ravenwood. The teenage Indiana bases his own look on a figure from the prologue of Indiana Jones and the Last Crusade, after being given his hat. Marcus Brody acts as Indiana's positive role model at the college. Indiana's own insecurities are made worse by the absence of his mother. In Indiana Jones and the Temple of Doom, he becomes the father figure to Short Round, to survive; he is rescued from Kali's evil by Short Round's dedication. In Raiders of the Lost Ark, he is wise enough to close his eyes in the presence of God in the Ark of the Covenant. By contrast, his rival Rene Belloq is killed for having the audacity to try to communicate directly with God.

In the prologue of Indiana Jones and the Last Crusade, Jones is seen as a teenager, establishing his look when given a fedora hat. Indiana's intentions are revealed as prosocial, as he believes artifacts "belong in a museum." In the film's climax, Indiana undergoes "literal" tests of faith to retrieve the Grail and save his father's life. He also remembers Jesus as a historical figure—a humble carpenter—rather than an exalted figure when he recognizes the simple nature and tarnished appearance of the real Grail amongst a large assortment of much more ornately decorated ones. Henry Senior rescues his son from falling to his death when reaching for the fallen Grail, telling him to "let it go," overcoming his mercenary nature. The Young Indiana Jones Chronicles explains how Indiana becomes solitary and less idealistic following his service in World War I. In Indiana Jones and the Kingdom of the Crystal Skull, Jones is older and wiser, whereas his sidekicks Mutt and Mac are youthfully arrogant, and greedy, respectively.

Origins and inspirations
Indiana Jones is modeled after the strong-jawed heroes of the matinée serials and pulp magazines that George Lucas and Steven Spielberg enjoyed in their childhoods (such as the Republic Pictures serials, and the Doc Savage series). Sir H. Rider Haggard's safari guide and big game hunter Allan Quatermain, who appeared in King Solomon's Mines (1885) and its seventeen sequels and prequels, is a notable template for Jones. The two friends first discussed the project in Hawaii around the time of the release of the first Star Wars film. Spielberg told Lucas how he wanted his next project to be something fun, like a James Bond film (this would later be referenced when they cast Sean Connery as Henry Jones, Sr.). According to sources, Lucas responded to the effect that he had something "even better", or that he'd "got that beat."

One of the possible bases for Indiana Jones is Professor Challenger, created by Sir Arthur Conan Doyle in 1912 for his novel, The Lost World. Challenger was based on Doyle's physiology professor, William Rutherford, an adventuring academic, albeit a zoologist/anthropologist.

Another important influence on the development of the character Indiana Jones is the Disney character Scrooge McDuck. Carl Barks created Scrooge in 1947 as a one-off relation for Donald Duck in the latter's self-titled comic book. Barks realized that the character had more potential, so a separate Uncle Scrooge comic book series full of exciting and strange adventures in the company of his duck nephews was developed. This Uncle Scrooge comic series strongly influenced George Lucas. This appreciation of Scrooge as an adventurer influenced the development of Jones, with the prologue of Raiders of the Lost Ark containing homage to Barks' Scrooge adventure "The Seven Cities of Cibola", published in Uncle Scrooge #7 from September 1954. This homage in the film takes the form of playfully mimicking the removal-of-the-statuette-from-its-pedestal and the falling-stone sequences of the comic book.

The character was originally named Indiana Smith, after an Alaskan Malamute called Indiana that Lucas owned in the 1970s and on which he based the Star Wars character Chewbacca. Spielberg disliked the name Smith, and Lucas casually suggested Jones as an alternative. The Last Crusade script references the name's origin, with Jones' father revealing his son's birth name to be Henry and explaining that "we named the dog Indiana", to his son's chagrin. Some have also posited that C.L. Moore's science fiction character Northwest Smith may have also influenced Lucas and Spielberg in their naming choice.

Lucas has said on various occasions that Sean Connery's portrayal of British secret agent James Bond was one of the primary inspirations for Jones, a reason Connery was chosen for the role of Indiana's father in Indiana Jones and the Last Crusade. Spielberg earned the rank of Eagle Scout and Ford the Life Scout badge in their youth, which gave them the inspiration to portray Indiana Jones as a Life Scout at age 13 in The Last Crusade.

Historical models
Many people are said to be the real-life inspiration of the Indiana Jones character—although none of the following have been confirmed as inspirations by Lucas or Spielberg. There are some suggestions listed here in alphabetical order by last name:
 * Beloit College professor and paleontologist Roy Chapman Andrews.
 * Edgar James Banks (May 23, 1866 – May 5, 1945) – American diplomat, antiquarian and novelist. Banks is credited with the sale of an ancient cuneiform tablet famously known as Plimpton 322 proving the Babylonians beat the Greeks to the invention of trigonometry—the study of triangles—by more than 1,000 years.
 * Italian archaeologist and circus strongman Giovanni Battista Belzoni (1778–1823).
 * Yale University professor, historian, US senator, and explorer Hiram Bingham III, (1875–1956) who rediscovered and excavated the lost city of Machu Picchu, and chronicled his find in the bestselling book The Lost City of the Incas in 1948.
 * University of Chicago archaeologist Robert Braidwood.
 * University of Chicago archaeologist James Henry Breasted.
 * Frederick Russell Burnham, the celebrated American scout and British Army spy who heavily influenced Haggard's fictional Allan Quatermain character and also became the inspiration for the Scouting movement.
 * British archaeologist Percy Fawcett, who spent much of his life exploring the jungles of northern Brazil, and who was last seen in 1925 returning to the Amazon Basin to look for the Lost City of Z. A fictionalized version of Fawcett appears to Jones in the book Indiana Jones and the Seven Veils.
 * American archaeologist Walter Fairservis.
 * Harvard University paleontologist Farish Jenkins.
 * Lintel - Nabratein Synagogue.jpg Duke University biblical scholar, archeologist, and Bernice and Morton Lerner Emeritus Professor in Judaic Studies Eric M. Meyers, who with his wife and Mary Grace Wilson Professor Emerita of Religious Studies Carol Lyons Meyers uncovered the oldest known remnant of an ark found to date. It was unearthed at [[Nabratein synagogue]] in Israel around the time Raiders was released, prompting media interest and a cosplay photo of the Meyers as Indy and Marion in People magazine.
 * Northwestern University political scientist, anthropologist, professor and adventurer William Montgomery McGovern.
 * American archaeologist and adventurer Wendell Phillips led well-publicized expeditions in Africa and the Arabian Peninsula in the 1940s and 1950s.
 * American chess expert and adventurer Albert Pincus, a Manhattan Chess Club member who innovated the 2 Knights Defense, and went on expeditions into South America.
 * German archaeologist Otto Rahn.
 * Harvard University archaeologist and art historian Langdon Warner.
 * Vendyl Jones (1930–2010) led digs in Israel searching for the holy ark. He discovered items identified as the Temple incense and a clay vessel for holy anointing oil. In his 2005 book A Door of Hope: My Search for the Treasures of the Copper Scroll, he discusses the similarities.

Costume
Upon requests by Spielberg and Lucas, costume designer Deborah Nadoolman gave the character a distinctive silhouette through the styling of the hat; after examining many hats, Nadoolman chose a tall-crowned, wide-brimmed fedora. As a documentary of Raiders pointed out, the hat served a practical purpose. Following the lead of the old "B"-movies that inspired the Indiana Jones series, the fedora hid the actor's face sufficiently to allow doubles to perform the more dangerous stunts seamlessly. Examples in Raiders include the wider-angle shot of Indy and Marion crashing a statue through a wall, and Indy sliding under a fast-moving vehicle from front to back. Thus it was necessary for the hat to stay in place much of the time.

The hat became so iconic that the filmmakers could only come up with very good reasons or jokes to remove it. If it ever fell off during a take, filming would have to stop to put it back on. In jest, Ford put a stapler against his head to stop his hat from falling off when a documentary crew visited during shooting of Indiana Jones and the Last Crusade. This created the urban legend that Ford stapled the hat to his head. Anytime Indy's hat accidentally came off as part of the storyline (blown off by the wind, knocked off, etc.) and seemed almost irretrievable, filmmakers would make sure Indy and his hat were always reunited, regardless of the implausibility of its return. Although other hats were also used throughout the films, the general style and profile remained the same. Elements of the outfit include:
 * The fedora was supplied by Herbert Johnson Hatters in England for the first three films. An Australian model was used by costume designer Deborah Landis to show hat maker Richard Swales the details when making the iconic hat from "the Poets" parts. The fedora for Crystal Skull was made by Steve Delk and Marc Kitter of the Adventurebilt Hat Company of Columbus, Mississippi.
 * The leather jacket, a hybrid of the "Type 440" and the A-2 jacket, was made by Leather Concessionaires (now known as Wested Leather Co.) for Raiders of the Lost Ark and Indiana Jones and the Last Crusade. For Indiana Jones and the Temple of Doom, jackets were made in-house at Bermans & Nathans in London based on a stunt jacket they provided for Raiders of the Lost Ark. Tony Nowak made the jacket for Indiana Jones and the Kingdom of the Crystal Skull.
 * The Indiana Jones shirt is based on a typical safari-style shirt. Its distinctive feature is two vertical strips running from the shoulders to the bottom of the shirt tails and continued over both breast pockets. A common debate regards the original shirt color. Surviving samples of the original shirts seem to be darker in reality than they appear on screen. Most fans look for an off-white "stone" color for their replicas. The original shirts, however, may have been more of a "tan" or "natural" color. The shirt varied little from film to film, the only notable difference being the darker buttons in Temple of Doom and Last Crusade. Originally designed by Andreas Dometakis for the films, this shirt was once one of the hardest pieces of gear to find.
 * The trousers worn by Indiana Jones in all five films were based on original World War II Army and Army Air Corps officer trousers. Although not original Pinks they are based on the same basic design and do carry a slight pinkish hue. The trousers made for Raiders are said to be more of a greyish-brown whereas the trousers made for Temple of Doom and Last Crusade were supposedly a purer reddish brown. The trousers were made of a khaki wool-twill, pleated with seven belt loops, two scalloped button flap rear pockets, a button fly and a four-inch military style hem. They were all most likely subcontracted by the costume department and made by famed London based cinema costumers, Angels and Bermans, to be tailored perfectly for Harrison Ford for the production.
 * The satchel was a modified Mark VII gas mask bag that was used by British troops and civilians during World War II.
 * The whip was an 8- to 10-foot (2.4 to 3.0 m) bullwhip crafted by David Morgan for the first three films. The whips for Crystal Skull were crafted by a variety of people, including Terry Jacka, Joe Strain and Morgan (different lengths and styles were likely used in specific stunts).
 * The pistol was usually a World War I-era revolver, including the Webley Government (WG) Revolver (Last Crusade and Crystal Skull), or a Smith & Wesson Second Model Hand Ejector revolver (Raiders). He has also used a Colt Official Police revolver (Temple of Doom), a Nagant M1883 (Young Indiana Jones), and a 9 mm Browning Hi-Power (Raiders). The weapon is carried in a military pattern flap holster.
 * The shoes were made by Alden. A stock style (model 405) that had been a favorite of Ford's before the films, they are still sold today (though in a redder (brick) shade of brown than seen in the films) and are popularly known as "Indy Boots."

The fedora and leather jacket from Indiana Jones and the Last Crusade are on display at the Smithsonian Institution's National Museum of American History in Washington, D.C. The collecting of props and clothing from the films has become a thriving hobby for some aficionados of the franchise. Jones' whip was the third most popular film weapon, as shown by a 2008 poll held by 20th Century Fox, which surveyed approximately two thousand film fans.

Casting
Originally, Spielberg suggested Harrison Ford; Lucas resisted the idea, since he had already cast the actor in American Graffiti, Star Wars and The Empire Strikes Back, and did not want Ford to become known as his "Bobby De Niro" (in reference to the fact that fellow director Martin Scorsese regularly casts Robert De Niro in his films). During an intensive casting process, Lucas and Spielberg auditioned many actors, and finally cast actor Tom Selleck as Indiana Jones. Shortly afterward pre-production began in earnest on Raiders of the Lost Ark. CBS refused to release Selleck from his contractual commitment to Magnum, P.I., forcing him to turn down the role. Shooting for the film could have overlapped with the pilot for Magnum, P.I. but it later turned out that filming of the pilot episode was delayed and Selleck could have done both.

Subsequently, Peter Coyote and Tim Matheson both auditioned for the role. After Spielberg suggested Ford again, Lucas relented, and Ford was cast in the role less than three weeks before filming began.

Archaeological influence
The industry magazine Archaeology named eight past and present archaeologists who they felt "embodied [Jones's] spirit" as recipients of the Indy Spirit Awards in 2008. That same year Ford himself was elected to the board of directors for the Archaeological Institute of America. Commenting that "understanding the past can only help us in dealing with the present and the future," Ford was praised by the association's president for his character's "significant role in stimulating the public's interest in archaeological exploration."

He is perhaps the most influential character in films that explore archaeology. Since the release of Raiders of the Lost Ark in 1981, the very idea of archaeology and archaeologists has fundamentally shifted. Prior to the film's release, the stereotypical image of an archaeologist was that of an older, lackluster professor type. In the early years of films involving archaeologists, they were portrayed as victims who would need to be rescued by a more masculine or heroic figure. Following 1981, the stereotypical archaeologist was thought of as an adventurer consistently engaged in fieldwork.

Archeologist Anne Pyburn described the influence of Indiana Jones as elitist and sexist, and argued that the film series had caused new discoveries in the field of archaeology to become oversimplified and overhyped in an attempt to gain public interest, which negatively influences archaeology as a whole. Eric Powell, an editor with the magazine Archaeology, said "O.K., fine, the movie romanticizes what we do", and that "Indy may be a horrible archeologist, but he's a great diplomat for archeology. I think we'll see a spike in kids who want to become archeologists". Kevin McGeough, associate professor of archaeology, describes the original archaeological criticism of the film as missing the point of the film: "dramatic interest is what is at issue, and it is unlikely that film will change in order to promote and foster better archaeological techniques".

Other characters inspired by Jones
While himself an homage to various prior adventurers, aspects of Indiana Jones also directly influenced some subsequent characterizations:
 * Lara Croft, the female archaeologist of the Tomb Raider series, was originally designed as a man but was changed to a woman, partly because the developers felt the original design was too similar to Indiana Jones. Paramount Pictures, which distributed the first four installments of the Indiana Jones film series, would later make two films based on the Tomb Raider games.
 * Rick O'Connell from The Mummy films has often been compared to the likes of Indiana Jones.
 * The producer of the Prince of Persia (2008) video game, Ben Mattes, explained that its "inspiration was anything Harrison Ford has ever done: Indiana Jones, Han Solo."
 * Nathan Drake, the protagonist from the video game series Uncharted, shares many similarities with Jones himself, both visually and personality-wise.
 * Dr. Smolder Bravestone, the main protagonist's video game avatar in the Jumanji film series, shares similarities to Jones.

References in other media
Indiana Jones has been widely referenced and parodied throughout different media. Some notable franchises which have referenced Indiana Jones include:
 * Star Wars: Indiana Jones has been referenced multiple times, including in Star Wars: The Force Awakens when Han Solo is chased by a spherical rathtar, which uses sound effects from Raiders of the Lost Ark.
 * Marvel Cinematic Universe: Has made various references including in the opening scene of Guardians of the Galaxy referencing Raiders.
 * The Simpsons: Has made reference to Indiana Jones many times, including Milhouse's uncle, Norbert Van Houten, dressing as the character.
 * The Big Bang Theory: Makes numerous references to Indiana Jones, having an episode, "The Raiders Minimization", named in reference to Raiders.
 * World of Warcraft: Parodies the character with the non-player character named "Harrison Jones" paraphrasing Jones's lecture from Indiana Jones and the Last Crusade.

Psychology
In psychology, the behavior of a person's denial of an obviously proven finding against overwhelming evidence and the person's own observations is also referred to as the Indiana Jones effect. This title refers to Indiana Jones' repeated quote "I don't believe in magic.", even though the character has seen magical rituals, the Ark of the Covenant and the Holy Grail with his own eyes.

title: Indiana Jones (character)


text:
Talk:Integer

Huh?
In the section "Constuction" the follwing sentence needs fixing: "There exist at least a tenth of such constructions of signed integers" (?). Paul August ☎ 01:24, 1 May 2018 (UTC)

 * Replaced "at least a tenth of" with "at least ten".

--Vasywriter (talk) 11:25, 1 May 2018 (UTC)

 * Thanks, Paul August ☎ 14:55, 1 May 2018 (UTC)

 * I'm not so sure that this was the intended meaning. According to Purgy's edit summary, he made a guess as to the number (and this was mangled into the "tenth" statement) and he replaced it by "umpteen". I think, and I don't want to speak for Purgy, that he might have meant something like "zillions" which would also be unacceptable. Perhaps a simple "many" is what is called for. --Bill Cherowitzo (talk) 18:12, 1 May 2018 (UTC)
 * Well I think that Vasywriter was the one added the that section. Paul August ☎ 19:27, 1 May 2018 (UTC)

 * Just to reduce guessing about my guess: I guessed the quantity, addressed by "at least a tenth of", to be "at least tens of", and so to be roughly in the interval $$[20, 199]$$ for which in German the term "Zig" (the word ending of the first integer multiples of ten: zwanzig, vierzig, fünfzig, ...) is used. Furthermore, I recall to have seen and heard the use of the word "umpteen" for this purpose, synonymous perhaps to the rare "tens of" and analogous to "thousands of". Please, note that this formulation, in contrast to "at least" or "more than", supplies not only a lower, but also an upper bound in a vague manner (not hundreds). Cheers, Purgy (talk) 06:37, 2 May 2018 (UTC)

 * Oops. Yes he did, so I guess that "at least ten" is the intended meaning. However, that statement could be considered WP:SYNTH, so maybe "many" should still be considered. --Bill Cherowitzo (talk) 19:55, 1 May 2018 (UTC)
 * I would suggest being vaguer than "at least ten". "Many" is OK but maybe "a number of" is even better. --Trovatore (talk) 20:48, 1 May 2018 (UTC)

Thanks to all of you for your wise comments. It is encouraging for Wikipedia that clever people take time to carefully examine the meaning of words. Here are some elements: So, "more than ten", "many", "an infinity" would all be correct, although they are useful/small algebras and useless/large ones. --Vasywriter (talk) 22:56, 2 May 2018 (UTC)
 * 1. "A tenth of" was indeed incorrect; it should have been: "a ten of", but this was too vague. Thanks for pointing out the issue.
 * 2. "At least ten" is correct, because the cited paper lists ten different term algebras that can be used to build signed integers.
 * 3. The suggestions "umpteem" or "a number of" are also correct and, even more: they are inspiring. Indeed, given any $$n >= 3$$, I think it exists a term algebra having $$n$$ free constructors, such that there is a bijection between signed integers and the ground terms of this algebra. So, there is an infinity of algebras, which is more than ten.
 * 4. However, only those term algebras with a small number of constructors (such as the ten listed in the cited paper) are of practical interest, as the higher the number of constructors, the more complex the proofs (as the number of case disjunctions increases).

Subset Dilemma
Can someone explain why the naturals are a subset of the integers given the distinct set theoretic definitions? If an integer is defined by an entire equivalence class of ordered pairs of natural numbers, then the naturals are not themselves a subset of the integers. The article does go on to say that the naturals are embedded in the integers by a mapping n to [(n,0)], but that is just a convoluted way of saying they arent actually the same thing. 50.35.103.217 (talk) 07:28, 1 September 2018 (UTC)

 * The amount of convolutedness is no hindrance to state something, as long as it is well formed, and I believe that it is not necessary to permanently belabor the difference between "being a subset" and "being embedded", as long as the second is strictly shown once. I consider it a fair use to call the identified objects "the naturals within the integers", abbreviated to "the naturals". Purgy (talk) 07:52, 1 September 2018 (UTC)

Meaning of integers
I think you should tell us what is the meaning of integers 41.116.100.253 (talk) 18:45, 18 January 2022 (UTC)

Meaning of integer
Integer-is colloquially defined as a number that can be written without fractional (component for example ,21,04,0and-2048 are integers 41.116.100.253 (talk) 18:50, 18 January 2022 (UTC)

"Entier relatif" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Entier relatif and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 February 15 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. User:1234qwer1234qwer4 (talk) 20:25, 15 February 2022 (UTC)

Lead section
@D.Lazard reverted my edits as "controversial", which I guess means he disagrees with them. Lazard, what exactly is your problem? The lead has never been discussed before besides the recent comments by 41.116.100.253 that the article does not explain "the meaning of integers", which my edits presumably fix.

Also the comments from Cabillon are not unsourced, they are from the "Earliest Uses of Symbols of Number Theory" page. I guess I also could cite page 114 of https://www.amazon.com/Apprenticeship-Mathematician-Andre-Weil/dp/3764326506 for the part that's the André Weil quote. But Cabillon is listed as a source both on Wikipedia and in published scholarly books like. He was a moderator of the Historia Mathematica mailing list so presumably has at least some authority in this area. . Mathnerd314159 (talk) 21:06, 21 August 2022 (UTC)

Mathematics
What are integers in mathematics 190.80.50.12 (talk) 13:15, 5 October 2022 (UTC)

 * Read the article. Dhrm77 (talk) 14:52, 5 October 2022 (UTC)

Semi-protected edit request on 10 February 2023
Please change the word "number" to "numbers" (explanation of the german word "Zahlen" which means "numbers" in plural and not "number" in singular) Manloeste (talk) 23:41, 10 February 2023 (UTC)
 * ✅ small jars 11:44, 11 February 2023 (UTC)

Limit?
Are there infinitely many integers or is the negative limit -2147483648 and the positive limit 2147483647? 84.151.244.223 (talk) 17:52, 10 August 2023 (UTC)

 * There are infinitely many integers, as stated in the first paragraph of this article. –jacobolus (t) 17:59, 10 August 2023 (UTC)

Lead sentence
I had a look at the Collins source, I would not consider it too reliable - assuming that the first definition listed must be correct is essentially the tertiary source fallacy. Actually the second sentence is "a member of the set {..., -2, -1, 0, 1, 2, ...}", very close to the current lead (the lead I wrote). The "sum or difference of two natural numbers" and "closure of the natural numbers under subtraction" properties are implied by the current lead's second sentence, about the negative numbers being the additive inverses of the positive numbers. The "rational numbers with denominator 1" property is sort of implied by "a real number that can be written without a fractional component", but maybe it is worth adding to the second paragraph. Regarding "signed number" and "directed number", it seems like they are more associated with the real numbers. Even the Collins source states under the respective definitions that these terms may refer to any number, not just integers. Mathnerd314159 (talk) 05:27, 23 April 2024 (UTC)

 * The other definition includes the word "integer" when defining "integer". There's no way that this "Science and Technology Encyclopedia" is somehow better than the "Collins Dictionary of Mathematics" on the topic of integers. Thiagovscoelho (talk) 06:02, 29 May 2024 (UTC)
 * The circularity of defining integer as including negative integers is only apparent. A "negative integer" is defined as "the additive inverse of a natural number", not as an integer that is negative. We could remove the circularity completely by using a different term like "negation of natural number", or a more specific description like "string of minus sign in front of digits", but in practice people call them negative integers.
 * The Science and Technology Encyclopedia is intended for a broad audience, as opposed to Collins which is aimed at undergraduates but includes material for even advanced scholars. You have not responded to Lazard's point that your definition is too WP:TECHNICAL. Mathnerd314159 (talk) 21:00, 29 May 2024 (UTC)
 * The term negative integer would intuitively be read as "the subset of integers which are less than 0". This makes the definition of integer unclear without special knowledge. To remedy this, negative integer needs to link to the definition or an article. But there is another problem; This definition is not supported -- either by the cited reference Science and Technology Encyclopedia or Collins Dictionary of Mathematics.
 * Unless it is somehow incorrect, it would be best to simply use the definition cited and remove the "apparent" circularity.
 * I can't edit, but I would suggest:
 * An integer is the number zero (0), a positive natural number  whole number (1, 2, 3, etc.) or a  negative integer  negative whole number (−1, −2, −3, etc.). 24.20.59.206 (talk) 18:25, 27 June 2024 (UTC)
 * "Whole number" is colloquial. "Natural number" is a better choice. I have edited the lead to hopefully make it simpler, more accurate, and less circular.—Anita5192 (talk) 19:06, 27 June 2024 (UTC)
 * To me "the negation of a positive natural number" seems a lot more awkward and less concise than "negative integer", but I guess following this train of thought we should define the negative integers in the next sentence, which I have done.
 * I still think though this whole argument is stupid, the source I cited starts out by stating the integers consist of the positive integers, 0, and the negative integers. This definition is not controversial, and is supported by the sources, however much 24.20.59.206 says otherwise. Mathnerd314159 (talk) 04:05, 28 June 2024 (UTC)
 * One of my objections to the way the lead was previously worded was that defining the integers as consisting, in part, of the negative integers, when we haven't finished defining the integers, is circular.—Anita5192 (talk) 04:19, 28 June 2024 (UTC)
 * As I said, the circularity is only apparent. I think repeating "the negation of a positive natural number" is more confusing than using "negative integer" and then defining it. Arguably it was defined as the sequence -1, -2, -3, ... simply by writing "negative integers (-1, -2, -3, ...)". And even so, there is no prohibition in Wikipedia on circular definitions - in fact circular definition mentions that many dictionary-style definitions are circular. Mathnerd314159 (talk) 05:24, 28 June 2024 (UTC)

title: Talk:Integer


text:
Irreducible fraction

An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). In other words, a fraction $a⁄b$ is irreducible if and only if a and b are coprime, that is, if a and b have a greatest common divisor of 1. In higher mathematics, "irreducible fraction" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials. Every rational number can be represented as an irreducible fraction with positive denominator in exactly one way.

An equivalent definition is sometimes useful: if a and b are integers, then the fraction $a⁄b$ is irreducible if and only if there is no other equal fraction $c⁄d$ such that $|c|$ < $|a|$ or $|d|$ < $|b|$, where $|a|$ means the absolute value of a. (Two fractions $a⁄b$ and $c⁄d$ are equal or equivalent if and only if ad = bc.)

For example, $1⁄4$, $5⁄6$, and $−101⁄100$ are all irreducible fractions. On the other hand, $2⁄4$ is reducible since it is equal in value to $1⁄2$, and the numerator of $1⁄2$ is less than the numerator of $2⁄4$.

A fraction that is reducible can be reduced by dividing both the numerator and denominator by a common factor. It can be fully reduced to lowest terms if both are divided by their greatest common divisor. In order to find the greatest common divisor, the Euclidean algorithm or prime factorization can be used. The Euclidean algorithm is commonly preferred because it allows one to reduce fractions with numerators and denominators too large to be easily factored.

Examples

 * $$ \frac{120}{90}=\frac{12}{9}=\frac{4}{3}$$

In the first step both numbers were divided by 10, which is a factor common to both 120 and 90. In the second step, they were divided by 3. The final result, $4⁄3$, is an irreducible fraction because 4 and 3 have no common factors other than 1.

The original fraction could have also been reduced in a single step by using the greatest common divisor of 90 and 120, which is 30. As 120 ÷ 30 = 4, and 90 ÷ 30 = 3, one gets
 * $$ \frac{120}{90}=\frac{4}{3}$$

Which method is faster "by hand" depends on the fraction and the ease with which common factors are spotted. In case a denominator and numerator remain that are too large to ensure they are coprime by inspection, a greatest common divisor computation is needed anyway to ensure the fraction is actually irreducible.

Uniqueness
Every rational number has a unique representation as an irreducible fraction with a positive denominator (however $2⁄3$ = $−2⁄−3$ although both are irreducible). Uniqueness is a consequence of the unique prime factorization of integers, since $a⁄b$ = $c⁄d$ implies ad = bc, and so both sides of the latter must share the same prime factorization, yet a and b share no prime factors so the set of prime factors of a (with multiplicity) is a subset of those of c and vice versa, meaning a = c and by the same argument b = d.

Applications
The fact that any rational number has a unique representation as an irreducible fraction is utilized in various proofs of the irrationality of the square root of 2 and of other irrational numbers. For example, one proof notes that if √2 could be represented as a ratio of integers, then it would have in particular the fully reduced representation $a⁄b$ where a and b are the smallest possible; but given that $a⁄b$ equals √2, so does $2b − a⁄a − b$ (since cross-multiplying this with $a⁄b$ shows that they are equal). Since a > b (because √2 is greater than 1), the latter is a ratio of two smaller integers. This is a contradiction, so the premise that the square root of two has a representation as the ratio of two integers is false.

Generalization
The notion of irreducible fraction generalizes to the field of fractions of any unique factorization domain: any element of such a field can be written as a fraction in which denominator and numerator are coprime, by dividing both by their greatest common divisor. This applies notably to rational expressions over a field. The irreducible fraction for a given element is unique up to multiplication of denominator and numerator by the same invertible element. In the case of the rational numbers this means that any number has two irreducible fractions, related by a change of sign of both numerator and denominator; this ambiguity can be removed by requiring the denominator to be positive. In the case of rational functions the denominator could similarly be required to be a monic polynomial.

title: Irreducible fraction