48 (number)

48 (forty-eight) is the natural number following 47 and preceding 49. It is one third of a gross, or four dozens.

In mathematics
Forty-eight is the double factorial of 6, a highly composite number. Like all other multiples of 6, it is a semiperfect number. 48 is the smallest non-trivial 17-gonal number.

48 is the smallest number with exactly ten divisors, and the first multiple of 12 not to be a sum of twin primes.

There are eleven solutions to the equation φ(x) = 48, namely {65, 104, 105, 112, 130, 140, 144, 156, 168, 180, 210}. This is more than any integer below it, making 48 a highly totient number. On the other hand, the totient of 48 is 16, a third of its numeric value, that is also the number of divisors of 168, the seventeenth record for sum-of-divisors of natural numbers where 48 specifically sets the sixteenth such record value, of 124.

Since the greatest prime factor of 482 + 1 = 2305 is 461, which is clearly more than twice 48, 48 is a Størmer number.

48 is a Harshad number in decimal, as it is divisible by 4+8 = 12.

By a classical result of Honsberger, the number of incongruent integer-sided triangles of perimeter $$m$$ is given by the equations $$\tfrac {m^2}{48}$$ for even $$m$$, and $$\tfrac{(m+3)^2}{48}$$ for odd $$m$$.

48 is the order of full octahedral symmetry, which describes three-dimensional mirror symmetries associated with the regular octahedron and cube. 48 is also twice the order of full tetrahedral symmetry (24).

Riemann zeta function
48 is the floor and nearest-integer value of the ninth imaginary part of non-trivial 0|zeroes in the Riemann zeta function (see, Riemann hypothesis). Among the nine first such floor and ceiling values, this is the closest to an integer, differing from 48 by a value of around $$+ \text { } 0.0051508811671597279 \dots$$

Meanwhile, the fifth such ceiling value is 33, which is the smallest of only three numbers to hold a sum-of-divisors of 48 (the others are 35 and 47). The composite index of 48 represents the fifth floor value in this sequence, 32.<ref name="A13629" The smallest floor and ceiling values in the Riemann zeta function are 14 and 15, which are the two smallest numbers (of three total) to hold a sum-of-divisors of 24 (half 48).

In science

 * The atomic number of cadmium.
 * The number of Ptolemaic constellations.
 * The number of symmetries of a cube.

Astronomy

 * Messier object M48, a magnitude 5.5 open cluster in the constellation Hydra.
 * The New General Catalogue object NGC 48, a barred spiral galaxy in the constellation Andromeda.

In religion

 * The prophecies of 48 Jewish prophets and 7 prophetesses were recorded in the Tanakh for posterity.
 * According to the Mishnah, Torah wisdom is acquired via 48 ways (Pirkei Avoth 6:6).
 * In Buddhism, Amitabha Buddha had made 48 great vows and promises to provide ultimate salvation to countless beings through countless eons, with benefits said to be available merely by thinking about his name with Nianfo practice. He is thus hailed as "King of Buddhas" through such skillful compassion and became a popular and formal refuge figure in Pureland Buddhism.

In music

 * Johann Sebastian Bach's Well-Tempered Clavier is informally known as because it consists of a prelude and a fugue in each major and minor key, for a total of forty-eight pieces.
 * "48" is a song by Sunny Day Real Estate.
 * "48" is a song by Tyler, The Creator.
 * "Forty eight" is a song by Truckfighters on their 2007 album, Phi.
 * "48 Hour Parole" is a song by the Hollies.
 * "48 Crash" is a song by Suzi Quatro.
 * Familiar 48 is an alternative pop/rock band formerly known as Bonehead.
 * On Tool's album Ænima, there is a song named "Forty-Six & 2", the sum of which is 48.
 * AKB48 Group is a Japanese female idol group.

In sports

 * 48 is the total number of minutes in a full NBA game.

In other fields
Forty-eight may also refer to:
 * the code for international direct dial phone calls to Poland.
 * the model number of the HP-48 S/SX/G/GX/G+/GII.
 * the 48 Hour Film Project.
 * The First 48, an American crime program, 2004-present.
 * 48 Hours is a television news program on CBS.
 * 48 Hrs., a 1982 film starring Nick Nolte and Eddie Murphy, followed by Another 48 Hrs.
 * Arizona is the 48th state in the United States.
 * The 48 United States (excluding Alaska and Hawaii) are also referred to as the "Lower 48" or the "48 contiguous states".
 * '48 is an alternate history novel by James Herbert.
 * The 48 Laws of Power is a book by Robert Greene.
 * The number 48 in ASCII is what you add to any single digit integer to convert to its ASCII value.
 * MAC address uses 48-bit (6-byte).
 * The number of the French department Lozère.
 * a model of Harley-Davidson in the Sportster line.
 * In Chinese numerology, 48 is an auspicious number meaning 'determined to prosper', or simply 'prosperity', which is good for business.
 * 48 is the number of mountain peaks in New Hampshire over 4000 feet above sea level, as defined by the Appalachian Mountain Club.
 * '48 is a slang term in Palestinian Arabic for parts of Israel or Palestine not under the control of the State of Palestine. Arab people from those parts are colloquially known as 48-Arabs (عرب ٤٨).