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
Korean numerals	오십사, 쉰넷
Financial kanji/hanja	五拾四, 伍拾肆

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

Number theory

54 as the sum of three positive squares

54 is an abundant number because the sum of its proper divisors (66), which excludes 54 as a divisor, is greater than itself. Like all multiples of 6, 54 is equal to some of its proper divisors summed together, so it is also a semiperfect number. 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. 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.

The prime factorization of 54 contains four numbers, so it is 4-almost prime. Because the largest prime in 54's prime factorization is not greater than 3, this number is a 3-smooth number. By extension, it is also k-smooth for any prime number k larger than three; the most commonly studied of these are the regular numbers (5-smooth) and the humble numbers (7-smooth).

As a regular number, 54 is a divisor of many powers of 60, which is an important property in Assyro-Babylonian mathematics because they used a sexagesimal (base-60) number system. In base-60, the reciprocal of a regular number has a finite representation. If n divides $60^{k}$, then the sexagesimal representation of $1/n$ is just that for $60^{k}/n$, shifted by some number of places. This allows for easy division by these numbers: to divide by $n$, multiply by $1/n$, then shift.

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 sexagesimal, 4000 = 1×3600 + 6×60 + 40×1, or (as listed by Joyce) 1:6:40. Thus, 1/54, in sexagesimal, is 1/60 + 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 sexagesimal places.

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. It can also be expressed as twice the third power of three, so it is a Leyland number. 54 can be expressed as the sum of two-or-more consecutive integers in three ways, so it has a politeness of 3. Additionally, 54 objects can be positioned to construct the vertices of a 19-sided polygon, so 54 is an enneadecagonal number and the first 19-gonal number after 19 itself.

Trigonometry

If the complementary angle of a triangle's corner is 54 degrees, the sine of that angle is half the golden ratio. This is because the corresponding interior angle is equal to π/5 radians (or 36 degrees). 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.

However, 54 can be expressed as the area of a triangle with three rational side lengths. Therefore, it is a congruent number.

Recreational mathematics

54 is divisible by the sum of its digits in 21 bases, meaning it is a Harshad number in those bases. For example, in base 10, the sum of 54's digits (5 and 4), is 9, which is a divisor of 54, so 54 is a Harshad number in base 10.

List of basic calculations

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
${\sqrt[{x}]{54}}$	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.

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. Eventually, one character's attempt to divine the Ultimate Question elicited "What do you get if you multiply six by nine?" The story did not present this as the true Ultimate Question, and 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 decimal expression 54 is encoded as 42 in base-13. Adams said this was a coincidence.

See also

• 45 (number) – 54 reversed

References

Genji-mon, the traditional symbols that represent the 54 chapters of The Tale of Genji