Jump to content

54 (number)

From Wikipedia, the free encyclopedia
This article is about the number. For the years, see 54 BC and AD 54. For other uses, see 54.
← 53 54 55 →
Cardinalfifty-four
Ordinal54th
(fifty-fourth)
Factorization2 × 33
Divisors1, 2, 3, 6, 9, 18, 27, 54
Greek numeralΝΔ´
Roman numeralLIV, liv
Binary1101102
Ternary20003
Senary1306
Octal668
Duodecimal4612
Hexadecimal3616
Eastern Arabic, Kurdish, Persian, Sindhi٥٤
Assamese & Bengali৫৪
Chinese numeral,
Japanese numeral		五十四
Devanāgarī५४
Ge'ez፶፬
Georgianნდ
Hebrewנ"ד
Kannada೫೪
Khmer៥៤
ArmenianԾԴ
Malayalam൫൰൪
Meitei꯵꯴
Thai๕๔
Telugu౫౪
Babylonian numeral𒐐𒐘
Egyptian hieroglyph𓎊𓏽
Mayan numeral𝋢𝋮
Urdu numerals۵۴
Tibetan numerals༥༤
Korean numerals오십사, 쉰넷
Financial kanji/hanja五拾四, 伍拾肆
Look up fifty-four in Wiktionary, the free dictionary.

54 (fifty-four) is the natural number and positive integer following 53 and preceding 55. As a multiple of 2 but not of 4, 54 is an oddly even number and a composite number.

In mathematics

[edit]

Number theory

[edit]
54 as the sum of three positive squares

54 is an abundant number[1] because the sum of its proper divisors (66),[2] which excludes 54 as a divisor, is greater than itself. Like all multiples of 6,[3] 54 is equal to some of its proper divisors summed together,[a] so it is also a semiperfect number.[4] These proper divisors can be summed in various ways to express all positive integers smaller than 54, so 54 is a practical number as well.[5] Additionally, as an integer for which the arithmetic mean of all its positive divisors (including itself) is also an integer, 54 is an arithmetic number.[6]

54 can be constructed mathematically in a variety of ways. It is the smallest number that can be expressed as the sum of three positive squares in more than two different ways.[b][7] It can also be expressed as twice the third power of 3,[c] so it is a Leyland number.[8] 54 can be expressed as the sum of two-or-more consecutive integers in three ways,[d] so it has a politeness of 3.[9] Additionally, fifty-four objects can be positioned to construct the vertices of a polygon with nineteen sides, so 54 is an enneadecagonal number and the first 19-gonal number after 19 itself.[e][10]

Trigonometry

[edit]

If the complementary angle of a triangle's corner is 54 degrees, the sine of that angle is half the golden ratio.[11][12] This is because the corresponding interior angle is equal to π/5 radians (or 36 degrees).[f] If that triangle is isoceles, the relationship with the golden ratio makes it a golden triangle. The golden triangle is most readily found as the spikes on a regular pentagram.

If, instead, 54 is the length of a triangle's side and all the sides lengths are rational numbers, the 54 side cannot be the hypotenuse. Using the Pythagorean theorem, there is no way to construct 542 as the sum of two other square rational numbers. Therefore, 54 is a nonhypotenuse number.[13]

However, 54 can be expressed as the area of a triangle with three rational side lengths.[g] Therefore, it is a congruent number.[14] One of these combinations of three rational side lengths is composed of integers: 9:12:15, which is a 3:4:5 right triangle that is Pythagorean, Heronian, and a Brahmagupta triangle.

Smoothness and Babylonian mathematics

[edit]

The prime factorization of 54 contains four numbers,[h] so it is 4-almost prime.[15] Because the largest prime in 54's prime factorization is not greater than 3, it is a 3-smooth number.[16] By extension, 54 is also k-smooth for any prime number k larger than 3; the most relevant of these is the fact that 54 is a regular number (5-smooth).[17]

As a regular number, 54 is a divisor of many powers of 60,[i] which is an important property in Assyro-Babylonian mathematics because it uses a sexagesimal (base-60) number system. In base-60, the reciprocal of a regular number has a finite representation, and that simplified multiplication and division.[18] For instance, 54 is a divisor of 603, and 603/54 = 4000. In base-60, division by 54 can be accomplished by multiplying by 4000 and shifting three places. In base-60, 4000 can be written as 1:6:40.[j] Thus, 1/54 can be written as 1/601 + 6/602 + 40/603, also denoted 1:6:40 as Babylonian notational conventions did not specify the power of the starting digit. Conversely 1/4000 = 54/603, so division by 1:6:40 = 4000 can be accomplished by instead multiplying by 54 and shifting three base-60 places.

Recreational mathematics

[edit]

54 is divisible by the sum of its digits in twenty-one bases, meaning it is a Harshad number in those bases.[19] For example, in base 10, the sum of 54's digits (5 and 4) is 9, which is a divisor of 54. Therefore, 54 is a Harshad number in base 10.[20]

List of basic calculations

[edit]
Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
54 × x 54 108 162 216 270 324 378 432 486 540 594 648 702 756 810 918 972 1026 1080 1134
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
54 ÷ x 54 27 18 13.5 10.8 9 7.714285 6.75 6 5.4 4.90 4.5 4.153846 3.857142 3.6
x ÷ 54 0.01851 0.037 0.05 0.074 0.0925 0.1 0.1296 0.148 0.16 0.185 0.2037 0.2 0.2407 0.259 0.27
Exponentiation 1 2 3 4 5
54x 54 2916 157464 8503056 459165024
x54 1 18014398509481984 58149737003040059690390169 324518553658426726783156020576256 55511151231257827021181583404541015625
54 7.34846... 3.77976... 2.71080... 2.22064...

Because 54 is a multiple of 2 but not a square number, its square root is irrational.[21]

In literature

[edit]

In The Hitchhiker's Guide to the Galaxy by Douglas Adams, the "Answer to the Ultimate Question of Life, the Universe, and Everything" famously was 42.[22] Eventually, one character's unsuccessful attempt to divine the Ultimate Question elicited "What do you get if you multiply six by nine?"[23] The mathematical answer was 54, not 42. Some readers who were trying to find a deeper meaning in the passage soon noticed a certain veracity when using base 13: the base-10 expression 5410 can be encoded as the base-13 expression 4213.[24] Adams said this was a coincidence.[25]

See also

[edit]
  • 45 – 54 reversed

Explanatory footnotes

[edit]
  1. ^ 54 can be expressed as: 9 + 18 + 27 = 54.
  2. ^ 54 can be expressed as:
    • 72 + 22 + 12 = 54
    • 62 + 32 + 32 = 54
    • 52 + 52 + 22 = 54
  3. ^ 54 can be expressed as: 33 + 33 = 54.
  4. ^ 54 can be expressed as:
    • 17 + 18 + 19 = 54
    • 12 + 13 + 14 + 15 = 54
    • 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 54
  5. ^ To show this, make an inner shell of nineteen points at the regular polygon's vertices. By convention, choose one point of the inner shell that will be shared with an outer shell. Then, make that outer shell by placing thirty-five points (plus the one shared point) in segments two-points-long to represent the nineteen sides of a second regular polygon. 19 + 35 = 54.
  6. ^ There are various ways to prove this, but the algebraic method will eventually show that cos(π/5) = 2*sqrt(5)/4 = phi/2.
  7. ^ AreaΔ = 1/2 base × height. Therefore a triangle with a base of 9 and a height of 12 has an area of 54. By the Pythagoean theorem, the hypotenuse of that triangle is 15.
  8. ^ 54 can be expressed as: 2 × 33 = 2 × 3 × 3 × 3 = 54.
  9. ^ 603 and its multiples are divisible by 54.
  10. ^ 1:6:40 = 1×602 + 6×601 + 40×600 = 4000. Of course, this would have been written in actual Babylonian numerals.

References

[edit]
Genji-mon, the traditional symbols that represent the 54 chapters of The Tale of Genji
  1. ^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A001065". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  3. ^ Zachariou, Andreas; Zachariou, Eleni (1972). "Perfect, Semiperfect and Ore Numbers". Bull. Soc. Math. Grèce. Nouvelle Série. 13: 12–22. MR 0360455. Zbl 0266.10012.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A005835 (Pseudoperfect (or semiperfect) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A005153 (Practical numbers: positive integers m such that every k <= sigma(m) is a sum of distinct divisors of m. Also called panarithmic numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. ^ Sloane, N. J. A. (ed.). "Sequence A003601 (Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  7. ^ Sloane, N. J. A. (ed.), "Sequence A025331 (Numbers that are the sum of 3 nonzero squares in 3 or more ways.)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation; Sloane, N. J. A. (ed.). "Sequence A025323 (Numbers that are the sum of 3 nonzero squares in exactly 3 ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  8. ^ "Sloane's A076980 : Leyland numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A138591 (Sums of two or more consecutive nonnegative integers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  10. ^ "Sloane's A051871 : 19-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  11. ^ Khan, Sameen Ahmed (2020-10-11). "Trigonometric Ratios Using Geometric Methods". Advances in Mathematics: Scientific Journal. 9 (10): 8698. doi:10.37418/amsj.9.10.94. ISSN 1857-8365.
  12. ^ Sloane, N. J. A. (ed.). "Sequence A019863 (Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  13. ^ Sloane, N. J. A. (ed.). "Sequence A004144 (Nonhypotenuse numbers (indices of positive squares that are not the sums of 2 distinct nonzero squares).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  14. ^ Sloane, N. J. A. (ed.). "Sequence A003273 (Congruent numbers: positive integers k for which there exists a right triangle having area k and rational sides.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  15. ^ Sloane, N. J. A. (ed.). "Sequence A014613". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  16. ^ Sloane, N. J. A. (ed.). "Sequence A003586 (3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  17. ^ Sloane, N. J. A. (ed.). "Sequence A051037 (5-smooth numbers, i.e., numbers whose prime divisors are all <= 5.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  18. ^ Aaboe, Asger (1965), "Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers)", Journal of Cuneiform Studies, 19 (3), The American Schools of Oriental Research: 79–86, doi:10.2307/1359089, JSTOR 1359089, MR 0191779, S2CID 164195082.
  19. ^ Sloane, N. J. A. (ed.). "Sequence A080221 (n is Harshad (divisible by the sum of its digits) in a(n) bases from 1 to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  20. ^ Sloane, N. J. A. (ed.), "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation (includes only base 10 Harshad numbers).
  21. ^ Jackson, Terence (2011-07-01). "95.42 Irrational square roots of natural numbers — a geometrical approach". The Mathematical Gazette. 95 (533): 327–330. doi:10.1017/S0025557200003193. ISSN 0025-5572. S2CID 123995083.
  22. ^ Adams, Douglas (1979). The Hitchhiker's Guide to the Galaxy. p. 179-80.
  23. ^ Adams, Douglas (1980). The Restaurant at the End of the Universe. p. 181-84.
  24. ^ Adams, Douglas (1985). Perkins, Geoffrey (ed.). The Original Hitchhiker Radio Scripts. London: Pan Books. p. 128. ISBN 0-330-29288-9.
  25. ^ Diaz, Jesus. "Today Is 101010: The Ultimate Answer to the Ultimate Question". io9. Archived from the original on 26 May 2017. Retrieved 8 May 2017.
Retrieved from "https://en.wikipedia.org/w/index.php?title=54_(number)&oldid=1268214742"