1728 is the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

following 1727 and preceding

1729 Events January–March * January 8 – Frederick, the eldest son of King George II of Great Britain is made Prince of Wales at the age of 21, a few months after he comes to Britain for the first time after growing up in Hano ...

. 1728 is a

dozen A dozen (commonly abbreviated doz or dz) is a grouping of twelve. The dozen may be one of the earliest primitive integer groupings, perhaps because there are approximately a dozen cycles of the Moon, or months, in a cycle of the Sun, or year ...

gross, one great gross (or grand gross). It is the number of cubic

inch Measuring tape with inches The inch (symbol: in or ″) is a unit of length in the British imperial and the United States customary systems of measurement. It is equal to yard or of a foot. Derived from the Roman uncia ("twelfth ...

es in a cubic

foot The foot ( : feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb which bears weight and allows locomotion. In many animals with feet, the foot is a separate organ at the terminal part of the leg mad ...

. It is also the number of daily chants of the

Hare Krishna mantra The Hare Krishna mantra, also referred to reverentially as the ("Great Mantra"), is a 16-word Vaishnava mantra which is mentioned in the Kali-Santarana Upanishad and which from the 15th century rose to importance in the Bhakti movement follo ...

by a Hare Krishna devotee. The number comes from 16 rounds on a 108

japamala A japamala, , or simply mala ( sa, माला; , meaning 'garland') is a loop of prayer beads commonly used in Indian religions such as Hinduism, Jainism, Sikhism, and Buddhism for counting recitations when performing ''japa'' (reciting a ...

bead.

In mathematics

1728 is the

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

of 12 and, as such, is important in the

duodecimal The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wr ...

number system, in which it is represented as "1000". * 1728 = 12³ * 1728 = 3³ × 4³ * 1728 = 2³ × 6³ * 1728 = 6³ + 8³ + 10³ * 1728 = 24² + 24² + 24² * 1728 = 289³ + 287³ + (−288)³ + (−288)³ * 28

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

s: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864, 1728 1728 is the number of

directed Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''D ...

open knight's tours on a chessboard. 1728 occurs in the algebraic formula for the

j-invariant In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is hol ...

of an

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...

, as a function over a

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

variable on the

upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...

\,\mathcal : \

, :

j(\tau) = 1728 \frac = 1728 \frac.

Inputting a value of

2i

for

\tau

, where

i

is the

imaginary number An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . F ...

, yields another cubic

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

: :

j(2i) = 1728 \frac = 66^3.

1728 is one less than the first Hardy–Ramanujan or taxicab number,

References

{{Integers, 1000 Integers

In mathematics

See Also

References