54 (number)

This article is about the number. For the years, see 54 BC and AD 54. For other uses, see 54.

← 53 54 55 →
← 50 51 52 53 54 55 56 57 58 59 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	fifty-four
Ordinal	54th (fifty-fourth)
Factorization	2 × 33
Divisors	1, 2, 3, 6, 9, 18, 27, 54
Greek numeral	ΝΔ´
Roman numeral	LIV, liv
Binary	1101102
Ternary	20003
Senary	1306
Octal	668
Duodecimal	4612
Hexadecimal	3616
Eastern Arabic, Kurdish, Persian, Sindhi	٥٤
Assamese & Bengali	৫৪
Chinese numeral, Japanese numeral	五十四
Devanāgarī	५४
Ge'ez	፶፬
Georgian	ნდ
Hebrew	נ"ד
Kannada	೫೪
Khmer	៥៤
Armenian	ԾԴ
Malayalam	൫൰൪
Meitei	꯵꯴
Thai	๕๔
Telugu	౫౪
Babylonian numeral	𒐐𒐘
Egyptian hieroglyph	𓎊𓏽
Mayan numeral	𝋢𝋮
Urdu numerals	۵۴
Tibetan numerals	༥༤
Financial kanji/hanja	五拾四, 伍拾肆
Morse code	........._
NATO phonetic alphabet	FIFE FOW-ER
ASCII value	6

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.

54 is related to the golden ratio through trigonometry: the sine of a 54 degree angle is half of the golden ratio. Also, 54 is a regular number, and its even division of powers of 60 was useful to ancient mathematicians who used the Assyro-Babylonian mathematics system.

It is also an abundant number ^[1] , since the sum of its proper divisors (66) ^[2] is greater than itself.

In mathematics

Number theory

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]

Trigonometry and the golden ratio

If the complementary angle of a triangle's corner is 54 degrees, the sine of that angle is half the golden ratio. ^{[7] [8]} This is because the corresponding interior angle is equal to π/5 radians (or 36 degrees). ^[b] 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. ^[9]

However, 54 can be expressed as the area of a triangle with three rational side lengths. ^[c] Therefore, it is a congruent number. ^[10] 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 a Pythagorean, a Heronian, and a Brahmagupta triangle.

Regular number used in Assyro-Babylonian mathematics

As a regular number, 54 is a divisor of many powers of 60. ^[d] This is an important property in Assyro-Babylonian mathematics because that system uses a sexagesimal (base-60) number system. In base 60, the reciprocal of a regular number has a finite representation. Babylonian computers kept tables of these reciprocals to make their work more efficient. Using regular numbers simplifies multiplication and division in base 60 because dividing a by b can be done by multiplying a by b's reciprocal when b is a regular number. ^{[11] [12]}

For instance, division by 54 can be achieved in the Assyro-Babylonian system by multiplying by 4000 because 60³ ÷ 54 = 60³ × (1/54) = 4000. In base 60, 4000 can be written as 1:6:40. ^[e] Because the Assyro-Babylonian system does not have a symbol separating the fractional and integer parts of a number ^[13] and does not have the concept of 0 as a number, ^[14] it does not specify the power of the starting digit. Accordingly, 1/54 can also be written as 1:6:40. ^{[f] [13]} Therefore, the result of multiplication by 1:6:40 (4000) has the same Assyro-Babylonian representation as the result of multiplication by 1:6:40 (1/54). To convert from the former to the latter, the result's representation is interpreted as a number shifted three base-60 places to the right, reducing it by a factor of 60³. ^[g]

Graph theory

The Ellingham–Horton 54-graph

The second Ellingham–Horton graph was published by Mark N. Ellingham and Joseph D. Horton in 1983; it is of order 54. ^[15] These graphs provided further counterexamples to the conjecture of W. T. Tutte that every cubic 3-connected bipartite graph is Hamiltonian. ^[16] Horton disproved the conjecture some years earlier with the Horton graph, but that was larger at 92 vertices. ^[17] The smallest known counter-example is now 50 vertices. ^[18]

In literature

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. ^[19] Eventually, one character's unsuccessful attempt to divine the Ultimate Question elicited "What do you get if you multiply six by nine?" ^[20] The mathematical answer was 54, not 42. Some readers who were trying to find a deeper meaning in the passage soon noticed the fact was true in base 13: the base-10 expression 54₁₀ can be encoded as the base-13 expression 6₁₃ × 9₁₃ = 42₁₃. ^[21] Adams said this was a coincidence. ^[22]

List of basic calculations

Multiplication	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
54 × x	54	108	162	216	270	324	378	432	486	540	594	648	702	756	810

Division	1	2	3	4	5	6	7	8	9	10
54 ÷ x	54	27	18	13.5	10.8	9	7.714285	6.75	6	5.4
x ÷ 54	0.01851	0.037	0.05	0.074	0.0925	0.1	0.1296	0.148	0.16	0.185

Exponentiation	1	2	3
54x	54	2916	157464
x54	1	18014398509481984	58149737003040059690390169
${\displaystyle {\sqrt[{x}]{54}}}$	54	7.34846... [h]	3.77976...

Explanatory footnotes

Genji-mon, the traditional symbols that represent the fifty-four chapters of The Tale of Genji

1. 54 can be expressed as: 9 + 18 + 27 = 54.
2. There are various ways to prove this, but the algebraic method will eventually show that ${\textstyle \sin {54}=\cos {\frac {\pi }{5}}={\frac {2{\sqrt {5}}}{4}}={\frac {\phi }{2}}}$.
3. Area_Δ = 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.
4. 60³ and its multiples are divisible by 54.
5. 1:6:40 = 1×60² + 6×60¹ + 40×60⁰ = 4000. This is the number written in Babylonian numerals: 𒐕𒐚𒐏.
6. 1:6:40 = 1×60^-1 + 6×60^-2 + 40×60^-3 = 1/54. This is the number written in Babylonian numerals: 𒐕𒐚𒐏.
7. For example, 6534 ÷ 54 = 121. The Assyro-Babylonian method is to calculate 6534 × 4000 = 26136000. This result can be written in Babylonian numerals as 𒐖𒐕 (2:1), meaning 2×60⁴ + 1×60³. To complete the division by 54, one must divide by 60³. Shifting the numeral three base-60 digits to the right divides the number by 60³, so 𒐖𒐕 (2:1) is already the answer: 2×60¹ + 1×60⁰ = 121.
8. Because 54 is a multiple of 2 but not a square number, its square root is irrational. ^[23]

References

1. Sloane, N. J. A. (ed.). "SequenceA005101(Abundant numbers (sum of divisors of m exceeds 2m))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
2. Sloane, N. J. A. (ed.). "SequenceA001065(Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". 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.). "SequenceA005835(Pseudoperfect (or semiperfect) numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
5. Sloane, N. J. A. (ed.). "SequenceA005153(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.). "SequenceA003601(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.
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8. Sloane, N. J. A. (ed.). "SequenceA019863(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.
9. Sloane, N. J. A. (ed.). "SequenceA004144(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.
10. Sloane, N. J. A. (ed.). "SequenceA003273(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.
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15. Ellingham, M. N.; Horton, J. D. (1983), "Non-Hamiltonian 3-connected cubic bipartite graphs", Journal of Combinatorial Theory, Series B, 34 (3): 350–353, doi: 10.1016/0095-8956(83)90046-1 .
16. Tutte, W. T. (1971), "On the 2-factors of bicubic graphs", Discrete Mathematics, 1 (2): 203–208, doi: 10.1016/0012-365X(71)90027-6 .
17. "Horton Graph". Archived from the original on 2016-03-04. Retrieved 2022-02-19.
18. Georges, J. P. (1989), "Non-hamiltonian bicubic graphs", Journal of Combinatorial Theory, Series B, 46 (1): 121–124, doi:10.1016/0095-8956(89)90012-9 .
19. Adams, Douglas (1979). The Hitchhiker's Guide to the Galaxy. p. 179-80.
20. Adams, Douglas (1980). The Restaurant at the End of the Universe. p. 181-84.
21. Adams, Douglas (1985). Perkins, Geoffrey (ed.). The Original Hitchhiker Radio Scripts. London: Pan Books. p. 128. ISBN   0-330-29288-9.
22. 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.
23. 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.