10

Ten is the fifth composite number. It is also the smallest noncototient, a number that cannot be expressed as the difference between any integer and the total number of coprimes below it.[1] It is the second discrete semiprime (${\displaystyle 2\times 5}$), as well as the second member of the ${\displaystyle 2\times q}$ discrete semiprime family. Ten is the only number whose sum and difference of its prime divisors yield prime numbers (${\displaystyle 2+5=7}$ and ${\displaystyle 5-2=3}$). In general, powers of 10 contain ${\displaystyle n^{2}}$ divisors, where ${\displaystyle n}$ is the number of digits: 10 has 2² = 4 divisors, 100 has 3² = 9 divisors, 1,000 has 4² = 16 divisors, 10,000 has 5² = 25 divisors, and so forth. Ten is the smallest number whose status as a possible friendly number is unknown.[2]

As important sums,

• ${\displaystyle 10=1+2+3+4}$ the sum of the first four positive integers.
• ${\displaystyle 10=2+3+5=2\times 5}$, the sum of the first three prime numbers, and the smallest semiprime that is the sum of all the distinct prime numbers from its lower factor through its higher factor.[3]
• ${\displaystyle 10=3+7=5+5}$, the smallest number that can be written as the sum of two prime numbers in two different ways.
• ${\displaystyle 10=1^{2}+3^{2}}$, the sum of the squares of the first two odd numbers.

The factorial of ten is equal to the product of the factorials of the first three odd primes, ${\displaystyle 10!=3!\cdot 5!\cdot 7!}$.[4] Ten is also the first number whose fourth power can be written as a sum of two squares in two different ways (${\displaystyle 80^{2}+60^{2}}$ and ${\displaystyle 96^{2}+28^{2}}$).

Ten has an aliquot sum σ(10) of 8 and is accordingly the first discrete semiprime to be in deficit, with all subsequent discrete semiprimes in deficit.[5] The aliquot sequence for 10 comprises five members (10, 8, 7, 1, 0) with this number being the second composite member of the 7-aliquot tree.[6] 10 is also a Perrin number, preceded in the sequence by (5, 5, 7).[7] In the sequence of triangular numbers, indexed powers of 10 in this sequence generate the following sequence of triangular numbers in decimal: 55 (10th), 5,050 (100th), 500,500 (1,000th), ...[8][lower-alpha 1]

While 55 is the tenth triangular number, it is also the tenth Fibonacci number, and the largest such number to also be a triangular number.[9] Ten is also the first non-trivial decagonal number,[10] the third centered triangular number[11] and tetrahedral number,[12] and the fifth semi-meandric number.[13]

10 is the fourth telephone number, and the number of Young tableaux with four cells.[14] It is the number of ${\displaystyle n}$-queens problem solutions for ${\displaystyle n=5}$.[15]

A ${\displaystyle 10\times 10}$ magic square has a magic constant of 505.[16]

There are ten small Pisot numbers that do not exceed the golden ratio.[17]

According to conjecture, ten is the average sum of the proper divisors of the natural numbers ${\displaystyle \mathbb {N} }$ if the size of the numbers approaches infinity.[18]

The SI prefix for 10 is "deca-".

The meaning "10" is part of the following terms:

• decapoda, an order of crustaceans with ten feet.
• decane, a hydrocarbon with 10 carbon atoms.

Also, the number 10 plays a role in the following:

• The atomic number of neon.
• The number of hydrogen atoms in butane, a hydrocarbon.
• The number of spacetime dimensions in some superstring theories.

The metric system is based on the number 10, so converting units is done by adding or removing zeros (e.g. 1 centimeter = 10 millimeters, 1 decimeter = 10 centimeters, 1 meter = 100 centimeters, 1 dekameter = 10 meters, 1 kilometer = 1,000 meters).

The tetractys

• References in the Bible and the Quran, Judaism, Christianity, and Islam:
  • The Ten Commandments of Exodus[22] and Deuteronomy[23] are considered a cornerstone of Judaism and Christianity.
  • People traditionally tithed one-tenth of their produce. The practice of tithing is still common in Christian churches today, though it is disputed in some circles as to whether or not it is required of Christians.
  • In Deuteronomy 26:12, the Torah commands Jews to give one-tenth of their produce to the poor (Maaser Ani). From this verse and from an earlier verse (Deut. 14:22) there derives a practice for Jews to give one-tenth of all earnings to the poor.[24]
  • Ten Plagues were inflicted on Egypt in Exodus 7–12.
  • Jews observe the annual Ten Days of Repentance beginning on Rosh Hashanah and ending on Yom Kippur.
  • In Jewish liturgy, Ten Martyrs are singled out as a group.
  • There are said to be Ten Lost Tribes of Israel (those other than Judah and Benjamin).
  • There are Ten Sephirot in the Kabbalistic Tree of Life.
  • In Judaism, ten men are the required quorum, called a minyan, for prayer services.
  • In Genesis 18:23-32, Abraham pleads on behalf of Sodom and Gomorrah, asking to save the cities if there are enough righteous people there. He starts at 10 per city, and ends with 10 total in all cities.
  • Interpretations of Genesis in Talmudic and Midrashic teachings suggest that on the first day, God drew forth ten primal elements from the abyss in order to construct all of Creation: Heaven (or Fire), Earth, Chaos, Void, Light, Darkness, Wind (or Spirit), Water, Day, and Night. See also Bereshit (parsha).
  • Jesus tells the Parable of the Ten Virgins in Matthew 25:1–13.

  • Complete the pilgrimage and minor pilgrimage for Allah.¹ But if prevented ˹from proceeding˺, then ˹offer˺ whatever sacrificial animals you can afford. And do not shave your heads until the sacrificial animal reaches its destination. But if any of you is ill or has a scalp ailment ˹requiring shaving˺, then compensate either by fasting, charity, or a sacrificial offering. In times of peace, you may combine the pilgrimage and minor pilgrimage then make the sacrificial offering you can afford. Whoever cannot afford that ˹offering˺, let them fast three days during pilgrimage and seven after returning ˹home˺—completing ten. These offerings are for those who do not live near the Sacred House. And be mindful of Allah, and know that Allah is severe in punishment.

    — Surah al-Baqarah (The Heifer):196[25]

  • Note 1: Ḥajj, the pilgrimage to Mecca, is required at least once in a Muslim’s lifetime if they are physically and financially able to. Umrah, which is a short version of ḥajj, is recommended, but not obligatory.

  • Allah will not call you to account for your thoughtless oaths, but He will hold you accountable for deliberate oaths. The penalty for a broken oath is to feed ten poor people from what you normally feed your own family, or to clothe them, or to free a bondsperson. But if none of this is affordable, then you must fast three days. This is the penalty for breaking your oaths. So be mindful of your oaths. This is how Allah makes things clear to you, so perhaps you will be grateful.

    — Surah al-Maidah (The Table):89[26]

• In Pythagoreanism, the number 10 played an important role and was symbolized by the tetractys.
• In Hinduism, Lord Vishnu appeared on the earth in 10 incarnations, popularly known as Dashaavathar.
• In Sikhism, there are ten human Gurus.

Most countries issue coins and bills with a denomination of 10 (See e.g. 10 dollar note). Of these, the U.S. dime, with the value of ten cents, or one tenth of a dollar, derives its name from the meaning "one-tenth" − see Dime (United States coin)#Denomination history and etymology.

• The interval of a major tenth is an octave plus a major third.
• The interval of a minor tenth is an octave plus a minor third.
• "Ten lords a-leaping" is the gift on the tenth day of Christmas in the carol "The Twelve Days of Christmas".

• Decathlon is a combined event in athletics consisting of ten track and field events.
• In association football, the number 10 is traditionally worn by the team's advanced playmaker. This use has led to "Number 10" becoming a synonym for the player in that particular role, even if they do not wear that number.[27]
• In gridiron football, a team has a limited number of downs to advance the ball ten yards or more from where it was on its last first down; doing this is referred to as gaining another first down.
• In auto racing, driving a car at ten-tenths is driving as fast as possible, on the limit.
• In a regular basketball game, two teams playing against each other have 5 members each, for a total of 10 players on court. Under FIBA, WNBA, and NCAA women's rules, each quarter runs for 10 minutes.
• In blackjack, the Ten, Jack, Queen and King are all worth 10 points.
• In boxing, if the referee counts to 10 whether the boxer is unconscious or not, it will declare a winner by knockout.
• In men's field lacrosse, each team has 10 players on the field at any given time, except in penalty situations.
• Ten-ball is a pool game played with a cue ball and ten numbered balls.
• In most rugby league competitions, the number 10 is worn by one of the two starting props. One exception to this rule is the Super League, which uses static squad numbering.
• In rugby union, the starting fly-half wears the 10 shirt.
• In ten-pin bowling, 10 pins are arranged in a triangular pattern and there are 10 frames per game.

• Ten-codes are commonly used on emergency service radio systems.
• Ten refers to the "meter band" on the radio spectrum between 28 and 29.7 MHz, used by amateur radio.
• ASCII and Unicode code point for line feed.
• In MIDI, Channel 10 is reserved for unpitched percussion instruments.
• In the Rich Text Format specification, all language codes for regional variants of the Spanish language are congruent to 10 mod 256.
• In macOS, the F10 function key tiles all the windows of the current application and grays the windows of other applications.
• The IP addresses in the range 10.0.0.0/8 (meaning the interval between 10.0.0.0 and 10.255.255.255) are reserved for use by private networks by RFC 1918.

• The ESRB recommends video games with an E10+ rating to children aged 10 and up.

10 playing cards of all four suits

• Blake Edwards' 1979 movie 10.
• Series on HBO entitled 1st & Ten which aired between December 1984 and January 1991.
• Series on ESPN and ESPN2 entitled 1st and 10 which launched on ESPN in October 2003 to 2008 and moved to ESPN2 since 2008.
• In astrology, Capricorn is the 10th astrological sign of the Zodiac.
• In Chinese astrology, the 10 Heavenly Stems, refer to a cyclic number system that is used also for time reckoning.
• A 1977 short documentary film Powers of Ten depicts the relative scale of the Universe in factors of ten (orders of magnitude).
• CBS (parent company Paramount Global also owns another entity on this list, Network 10) has a game show called Power of 10, where the player's prize goes up and down by either the previous or next power of ten.
• "Ten Chances" is one of the pricing games on The Price is Right.
• There are ten official inkblots in the Rorschach inkblot test.
• The traditional Snellen chart uses 10 different letters.
• Network 10 is an Australian television network. The Sydney member of the network has the three-letter call-sign TEN and used to broadcast in analogue on VHF Channel 10. Paramount Global owns this network since December 4, 2019.
• Number Ten (also called Ella) is a character in the book series Lorien Legacies. The sixth book, The Fate of Ten, is named after her.
• A Cartoon Network franchise Ben 10, which has a number on its title.

•  Mathematics portal
• List of highways numbered 10