48 (number)

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

← 47 48 49 →
← 40 41 42 43 44 45 46 47 48 49 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	forty-eight
Ordinal	48th (forty-eighth)
Factorization	24 × 3
Divisors	1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Greek numeral	ΜΗ´
Roman numeral	XLVIII
Binary	1100002
Ternary	12103
Senary	1206
Octal	608
Duodecimal	4012
Hexadecimal	3016

In mathematics

Forty-eight is the double factorial of 6,^[1][2] a highly composite number.^[3] Like all other multiples of 6, it is a semiperfect number.^[4] 48 is the smallest non-trivial 17-gonal number.^[5]

48 is the smallest number with exactly ten divisors,^[6] and the first multiple of 12 not to be a sum of twin primes.

There are 11 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.^[7] On the other hand, the totient of 48 is 16,^[8] a third of its numeric value, that is also the number of divisors of 168,^[9] the seventeenth record for sum-of-divisors of natural numbers where 48 specifically sets the sixteenth such record value, of 124.^[10]

Since the greatest prime factor of 48² + 1 = 2305 is 461, which is clearly more than twice 48, 48 is a Størmer number.^[11]

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

By a classical result of Honsberger, the number of incongruent integer-sided triangles of perimeter ${\displaystyle m}$ is given by the equations ${\displaystyle {\tfrac {m^{2}}{48}}}$ for even ${\displaystyle m}$, and ${\displaystyle {\tfrac {(m+3)^{2}}{48}}}$ for odd ${\displaystyle m}$.^[13]

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 zeroes in the Riemann zeta function (see, Riemann hypothesis).^[14][15] Among the nine first such floor and ceiling values, this is the closest to an integer, differing from 48 by a value of around ${\displaystyle +{\text{ }}0.0051508811671597279\dots }$^[16]

Meanwhile, the fifth such ceiling value is 33,^[17] which is the smallest of only three numbers to hold a sum-of-divisors of 48 (the others are 35 and 47).^[18] The composite index of 48 represents the fifth floor value in this sequence, 32.^[19][14] 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^[20] 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 The Forty-Eight 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.^[21]
• 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 (Arabic: عرب ٤٨, romanized: ‘Arab Thamāniya wa-Arba‘īn).

See also

• List of highways numbered 48

References

1. Sloane, N. J. A. (ed.). "Sequence A006882 (Double factorials n!!: a(n) = n*a(n-2) for n > 1, a(0) = a(1) = 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. Sloane, N. J. A. (ed.). "Sequence A000165 (Double factorial of even numbers: (2n)!! = 2^n*n!)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. Sloane, N. J. A. (ed.). "Sequence A002182 (Highly composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
4. Sloane, N. J. A. (ed.). "Sequence A005835 (Pseudoperfect (or semiperfect) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. Sloane, N. J. A. (ed.). "Sequence A051869 (17-gonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
6. Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
7. Sloane, N. J. A. (ed.). "Sequence A097942 (Highly totient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
8. Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-13.
9. Sloane, N. J. A. (ed.). "Sequence A000005 (d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-13.
10. Sloane, N. J. A. (ed.). "Sequence A034885 (Record values of sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-13.
11. Sloane, N. J. A. (ed.). "Sequence A005528 (Størmer numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
12. Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
13. East, James; Niles, Ron (2019). "Integer polygons of given perimeter". Bulletin of the Australian Mathematical Society. 100 (1). Canberra: Australian Mathematical Society: 131–147. arXiv:1710.11245. doi:10.1017/S0004972718001612. MR 3977311. S2CID 119729735. Zbl 1420.52014.
14. Sloane, N. J. A. (ed.). "Sequence A013629 (Floor of imaginary parts of nontrivial zeros of Riemann zeta function)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
15. Sloane, N. J. A. (ed.). "Sequence A002410 (Nearest integer to imaginary part of n-th zero of Riemann zeta function)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
16. Odlyzko, Andrew. "The first 100 (non trivial) zeros of the Riemann Zeta function [AT&T Labs]". Andrew Odlyzko: Home Page. UMN CSE. Retrieved 2024-01-16.
17. Sloane, N. J. A. (ed.). "Sequence A092783 (Ceiling of imaginary parts of zeros of Riemann zeta function)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-05.
18. Sloane, N. J. A. (ed.). "Sequence A000203 (The sum of divisors of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-05.
19. Sloane, N. J. A. (ed.). "Sequence A02808 (The composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-05.
20. "How many prophets were there and who were they? - philosophy prophecy history prophets torah the bible the prophets". Archived from the original on 2007-08-10. Retrieved 2006-07-17.
21. "Chinese Numerology: Lucky and Unlucky Numbers Meaning". My Today's Horoscope. 2019-11-09. Retrieved 2021-09-20.