5040 (number)
Encyclopedia
5040 is a factorial
(7!), a highly composite number
, a superior highly composite number
, a colossally abundant number
, and the number
of permutation
s of 4 items out of 10 choices (10 × 9 × 8 × 7 = 5040).
mentions in his Laws
that 5040 is a convenient number to use for dividing
many things (including both the citizens and the land of a state) into lesser parts. He remarks that this number can be divided by all the (natural) numbers from 1 to 12
with the single exception of 11
. He rectifies this "defect" by suggesting that two families could be subtracted from the citizen body to produce the number 5038, which is divisible by 11. Plato also took notice of the fact that 5040 can be divided by 12 twice over. Indeed, Plato's repeated insistence on the use of 5040 for various state purposes is so evident that it is written, "Plato, writing under Pythagorean
influences, seems really to have supposed that the well-being of the city depended almost as much on the number 5040 as on justice
and moderation
."
is the divisor function
and
is the Euler-Mascheroni constant
, then 5040 is the largest known number for which this inequality holds:
.
This is somewhat unusual, since in the limit
we have:
Guy Robin showed in 1984 that the inequality fails for all larger numbers if and only if
the Riemann hypothesis
is true.
, not only because of the Plato connection, but because using round figures, the sum of the radii
of both the Earth
and Moon
(in miles) is 3960 + 1080 = 5040. Incidentally, the sum of their diameter
s is also the number of minutes in a week (7 days × 24 hours × 60 minutes = 10,080).
▪ The ratio
of the radius
of the moon and the radius of the earth is 1080/3960, which simplifies to 3/11. This ratio can also be expressed as (4 - π)/π, when using 22/7 as the value of π. This means that the sizes of the earth and the moon are related by a simple function of π.
▪ Given that the radius of the moon is 3/11 that of the earth, the sum of their radii can be broken into 3/14 (for the radius of the moon) and 11/14 (for the radius of the earth). Further, the sum of their radii in miles is 5040, which when divided by 14 is 360 (the number of degrees in a circle). This would not happen for another pair of objects with radii in the same ratio - it only happens when the sum of their radii is 5040.
▪ 5040 has exactly 60 divisors, counting itself and 1.
▪ In a vigesimal
system, 5040 is represented as 12 groups of 20's and 12 groups of 20-squared's (12 • 20 = 240, 12 • 202 = 4800, and 240 + 4800 = 5040).
▪ 5040 is the sum of 42 consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 +163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229).
of both 1 × 2 × 3 × 4 × 5 × 6 × 7 and 7 × 8 × 9 × 10. Additionally, 7920 is the product of 8 × 9 × 10 × 11.
▪ The ratio
7920/5040, which is (11!/7!)/7!, or equivalently 11!/(7!)², also equals 11/7. The numbers 11 and 7 are the only pair of natural numbers (along with 3 and 2) for which this occurs: (n!/m!)/m! = n/m; or equivalently: n!/(m!)² = n/m.
▪ The ratio 7920/5040 simplifies to 11/7, which is π/2 (when using 22/7 for π). Similarly, 5040/7920 is 7/11, which is 2/π (when using 22/7 for π).
▪ Given a circle
with radius
5040 units, a quarter of that circle's circumference
is 7920 units (when using 22/7 for π).
▪ On the 27th of August, 2002, a crop formation was discovered at Crooked Soley in Wiltshire
, England
that had an outer ring split into 1,296 curved, diamond-like sections. The crops in 504 of those sections were still standing while the crops in the other 792 were leaning virtually horizontally.
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
(7!), a highly composite number
Highly composite number
A highly composite number is a positive integer with more divisors than any positive integer smaller than itself.The initial or smallest twenty-one highly composite numbers are listed in the table at right....
, a superior highly composite number
Superior highly composite number
In mathematics, a superior highly composite number is a certain kind of natural number with the following properties. A natural number n is called superior highly composite if there is an ε > 0 such that for all natural numbers k ≥ 1,...
, a colossally abundant number
Colossally abundant number
In mathematics, a colossally abundant number is a natural number that, in some rigorous sense, has a lot of divisors...
, and the number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
of permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
s of 4 items out of 10 choices (10 × 9 × 8 × 7 = 5040).
| 5040 | |
|---|---|
| Cardinal Cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite... |
5040 Five thousand [and] forty |
| Ordinal Ordinal number In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals... |
5040th Five thousand [and] fortieth |
| Roman numeral | XL |
| Factorization Factorization In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original... |  |
| Divisors | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520, 5040 |
| Binary Binary numeral system The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2... |
1001110110000 |
| Duodecimal Duodecimal The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written as 'A', 'T' or 'X', and the number eleven as 'B' or 'E'... |
2B00 |
| Hexadecimal Hexadecimal In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen... |
13B0 |
Philosophy
PlatoPlato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
mentions in his Laws
Laws (dialogue)
The Laws is Plato's last and longest dialogue. The question asked at the beginning is not "What is law?" as one would expect. That is the question of the Minos...
that 5040 is a convenient number to use for dividing
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
many things (including both the citizens and the land of a state) into lesser parts. He remarks that this number can be divided by all the (natural) numbers from 1 to 12
12 (number)
12 is the natural number following 11 and preceding 13.The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten...
with the single exception of 11
11 (number)
11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...
. He rectifies this "defect" by suggesting that two families could be subtracted from the citizen body to produce the number 5038, which is divisible by 11. Plato also took notice of the fact that 5040 can be divided by 12 twice over. Indeed, Plato's repeated insistence on the use of 5040 for various state purposes is so evident that it is written, "Plato, writing under Pythagorean
Pythagoreanism
Pythagoreanism was the system of esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics. Pythagoreanism originated in the 5th century BCE and greatly influenced Platonism...
influences, seems really to have supposed that the well-being of the city depended almost as much on the number 5040 as on justice
Justice
Justice is a concept of moral rightness based on ethics, rationality, law, natural law, religion, or equity, along with the punishment of the breach of said ethics; justice is the act of being just and/or fair.-Concept of justice:...
and moderation
Moderation
Moderation is the process of eliminating or lessening extremes. It is used to ensure normality throughout the medium on which it is being conducted...
."
Number Theory
If is the divisor functionDivisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
and
is the Euler-Mascheroni constantEuler-Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
, then 5040 is the largest known number for which this inequality holds:
.This is somewhat unusual, since in the limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
we have:
Guy Robin showed in 1984 that the inequality fails for all larger numbers if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
is true.
Other
▪ 5040 is considered an important number in some systems of numerologyNumerology
Numerology is any study of the purported mystical relationship between a count or measurement and life. It has many systems and traditions and beliefs...
, not only because of the Plato connection, but because using round figures, the sum of the radii
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
of both the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
and Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
(in miles) is 3960 + 1080 = 5040. Incidentally, the sum of their diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
s is also the number of minutes in a week (7 days × 24 hours × 60 minutes = 10,080).
▪ The ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
of the radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
of the moon and the radius of the earth is 1080/3960, which simplifies to 3/11. This ratio can also be expressed as (4 - π)/π, when using 22/7 as the value of π. This means that the sizes of the earth and the moon are related by a simple function of π.
▪ Given that the radius of the moon is 3/11 that of the earth, the sum of their radii can be broken into 3/14 (for the radius of the moon) and 11/14 (for the radius of the earth). Further, the sum of their radii in miles is 5040, which when divided by 14 is 360 (the number of degrees in a circle). This would not happen for another pair of objects with radii in the same ratio - it only happens when the sum of their radii is 5040.
▪ 5040 has exactly 60 divisors, counting itself and 1.
▪ In a vigesimal
Vigesimal
The vigesimal or base 20 numeral system is based on twenty .- Places :...
system, 5040 is represented as 12 groups of 20's and 12 groups of 20-squared's (12 • 20 = 240, 12 • 202 = 4800, and 240 + 4800 = 5040).
▪ 5040 is the sum of 42 consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 +163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229).
5040 and 7920
▪ As previously stated, 5040 is the productProduct (mathematics)
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...
of both 1 × 2 × 3 × 4 × 5 × 6 × 7 and 7 × 8 × 9 × 10. Additionally, 7920 is the product of 8 × 9 × 10 × 11.
▪ The ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
7920/5040, which is (11!/7!)/7!, or equivalently 11!/(7!)², also equals 11/7. The numbers 11 and 7 are the only pair of natural numbers (along with 3 and 2) for which this occurs: (n!/m!)/m! = n/m; or equivalently: n!/(m!)² = n/m.
▪ The ratio 7920/5040 simplifies to 11/7, which is π/2 (when using 22/7 for π). Similarly, 5040/7920 is 7/11, which is 2/π (when using 22/7 for π).
▪ Given a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
with radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
5040 units, a quarter of that circle's circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....
is 7920 units (when using 22/7 for π).
▪ On the 27th of August, 2002, a crop formation was discovered at Crooked Soley in Wiltshire
Wiltshire
Wiltshire is a ceremonial county in South West England. It is landlocked and borders the counties of Dorset, Somerset, Hampshire, Gloucestershire, Oxfordshire and Berkshire. It contains the unitary authority of Swindon and covers...
, England
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
that had an outer ring split into 1,296 curved, diamond-like sections. The crops in 504 of those sections were still standing while the crops in the other 792 were leaning virtually horizontally.
External links
- MathworldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...
article on Plato's numbers