Cube (algebra)

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y = x3 for values of 0 ≤ x ≤ 25.

In arithmetic and algebra, the cube of a number n is its third power: the result of the number multiplied by itself twice:

n3 = n × n × n.

It is also the number multiplied by its square:

n3 = n × n2.

This is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n. It determines the side of the cube of a given volume. It is also n raised to the one-third power.

Both cube and cube root are odd functions:

(−n)3 = −(n3).

The cube of a number or any other mathematical expression is denoted by a superscript 3, for example 23 = 8 or (x + 1)3.

In integers[edit]

A cube number, or a perfect cube, or sometimes just a cube is a number which is the cube of an integer. The positive perfect cubes up to 603 are (sequence A000578 in OEIS):

13 = 1 113 = 1331 213 = 9261 313 = 29791 413 = 68921 513 = 132651
23 = 8 123 = 1728 223 = 10648 323 = 32768 423 = 74088 523 = 140608
33 = 27 133 = 2197 233 = 12167 333 = 35937 433 = 79507 533 = 148877
43 = 64 143 = 2744 243 = 13824 343 = 39304 443 = 85184 543 = 157464
53 = 125 153 = 3375 253 = 15625 353 = 42875 453 = 91125 553 = 166375
63 = 216 163 = 4096 263 = 17576 363 = 46656 463 = 97336 563 = 175616
73 = 343 173 = 4913 273 = 19683 373 = 50653 473 = 103823 573 = 185193
83 = 512 183 = 5832 283 = 21952 383 = 54872 483 = 110592 583 = 195112
93 = 729 193 = 6859 293 = 24389 393 = 59319 493 = 117649 593 = 205379
103 = 1000 203 = 8000 303 = 27000 403 = 64000 503 = 125000 603 = 216000

Geometrically speaking, a positive number m is a perfect cube if and only if one can arrange m solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one larger one with the appearance of a Rubik's Cube, since 3 × 3 × 3 = 27.

The pattern between every perfect cube from negative infinity to positive infinity is as follows,

n3 = (n − 1)3 + 3(n − 1)n + 1.

or

n3 = (n + 1)3 − 3(n + 1)n − 1.

There is no smallest perfect cube, since negative integers are included. For example, (−4) × (−4) × (−4) = −64.

Base ten[edit]

Unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25, 75 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6 and e8 can be the last two digits of a perfect cube (where o stands for any odd digit and e for any even digit). Some cube numbers are also square numbers, for example 64 is a square number (8 × 8) and a cube number (4 × 4 × 4); this happens if and only if the number is a perfect sixth power.

It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root 1, 8 or 9. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3:

  • If the number is divisible by 3, its cube has digital root 9;
  • If it has a remainder of 1 when divided by 3, its cube has digital root 1;
  • If it has a remainder of 2 when divided by 3, its cube has digital root 8.

Waring's problem for cubes[edit]

Main article: Waring's problem

Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes:

23 = 23 + 23 + 13 + 13 + 13 + 13 + 13 + 13 + 13.

Fermat's last theorem for cubes[edit]

Main article: Fermat's last theorem

The equation x3 + y3 = z3 has no non-trivial (i.e. xyz ≠ 0) solutions in integers. In fact, it has none in Eisenstein integers.[1]

Both of these statements are also true for the equation[2] x3 + y3 = 3z3.

Sum of first n cubes[edit]

The sum of the first n cubes is the nth triangle number squared:

1^3+2^3+\dots+n^3 = (1+2+\dots+n)^2=\left(\frac{n(n+1)}{2}\right)^2.

For example, the sum of the first 5 cubes is the square of the 5th triangular number,

1^3+2^3+3^3+4^3+5^3 = 15^2 \,

A similar result can be given for the sum of the first y odd cubes,

1^3+3^3+\dots+(2y-1)^3 = (xy)^2

but x, y must satisfy the negative Pell equation x^2-2y^2 = -1. For example, for y = 5 and 29, then,

1^3+3^3+\dots+9^3 = (7\cdot 5)^2 \,
1^3+3^3+\dots+57^3 = (41\cdot 29)^2

and so on. Also, every even perfect number, except the first one, is the sum of the first 2(p−1)/2 odd cubes,

28 = 2^2(2^3-1) = 1^3+3^3
496 = 2^4(2^5-1) = 1^3+3^3+5^3+7^3
8128 = 2^6(2^7-1) = 1^3+3^3+5^3+7^3+9^3+11^3+13^3+15^3

Sum of cubes in arithmetic progression[edit]

There are examples of cubes in arithmetic progression whose sum is a cube,

3^3+4^3+5^3 = 6^3
11^3+12^3+13^3+14^3 = 20^3
31^3+33^3+35^3+37^3+39^3+41^3 = 66^3

with the first one also known as Plato's number. The formula F for finding the sum of an n number of cubes in arithmetic progression with common difference d and initial cube a3,

F(d,a,n) = a^3+(a+d)^3+(a+2d)^3+\cdots+(a+dn-d)^3

is given by

F(d,a,n) = (n/4)(2a-d+dn)(2a^2-2ad+2adn-d^2n+d^2n^2)

A parametric solution to

F(d,a,n) = y^3

is known for the special case of d = 1, or consecutive cubes, but only sporadic solutions are known for integer d > 1, such as d = 2, 3, 5, 7, 11, 13, 37, 39, etc.[3]

Cubes as sums of successive odd integers[edit]

In the sequence of odd integers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ..., the first one is a cube (1 = 13); the sum of the next two is the next cube (3+5 = 23); the sum of the next three is the next cube (7+9+11 = 33); and so forth.

In rational numbers[edit]

Every positive rational number is the sum of three positive rational cubes,[4] and there are rationals that are not the sum of two rational cubes.[5]

In real numbers, other fields, and rings[edit]

Further information: cubic function
x³ plotted on a Cartesian plane

In real numbers, the cube function preserves the order: larger numbers have larger cubes. In other words, cubes (strictly) monotonically increase. Also, its codomain is the entire real line: the function xx3 : RR is a surjection (takes all possible values). Only three numbers equal to the own cubes: −1, 0, and 1. If −1 < x < 0 or 1 < x, then x3 > x. If x < −1 or 0 < x < 1, then x3 < x. All aforementioned properties pertain also to any higher odd power (x5, x7, …) of real numbers. Equalities and inequalities are also true in any ordered ring.

Volumes of similar Euclidean solids are related as cubes of their linear sizes.

In complex numbers, the cube of a purely imaginary number is also purely imaginary. For example, i3 = −i.

The derivative of x3 equals to 3x2.

Cubes occasionally have the surjective property in other fields, such as in Fp for such prime p that p ≠ 1 (mod 3),[6] but not necessarily: see the counterexample with rationals above. Also in F7 only three elements 0, ±1 are perfect cubes, of seven total. −1, 0, and 1 are perfect cubes anywhere and the only elements of a field equal to the own cubes: x3x = x(x − 1)(x + 1).

History[edit]

Determination of the cubes of large numbers was very common in many ancient civilizations. Mesopotamian mathematicians created cuneiform tablets with tables for calculating cubes and cube roots by the Old Babylonian period (20th to 16th centuries BC).[7][8] Cubic equations were known to the ancient Greek mathematician Diophantus.[9] Hero of Alexandria devised a method for calculating cube roots in the 1st century CE.[10] Methods for solving cubic equations and extracting cube roots appear in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the 2nd century BCE and commented on by Liu Hui in the 3rd century CE.[11] The Indian mathematician Aryabhata wrote an explanation of cubes in his work Aryabhatiya. In 2010 Alberto Zanoni found a new algorithm[12] to compute the cube of a long integer in a certain range, faster than squaring-and-multiplying.

Notes[edit]

  1. ^ Hardy & Wright, Thm. 227
  2. ^ Hardy & Wright, Thm. 232
  3. ^ "A Collection of Algebraic Identities". 
  4. ^ Hardy & Wright, Thm. 234
  5. ^ Hardy & Wright, Thm. 233
  6. ^ The multiplicative group of Fp is cyclic of order p − 1, and if it is not divisible by 3, then cubes define a group automorphism.
  7. ^ Cooke, Roger (8 November 2012). The History of Mathematics. John Wiley & Sons. p. 63. ISBN 978-1-118-46029-0. 
  8. ^ Nemet-Nejat, Karen Rhea (1998). Daily Life in Ancient Mesopotamia. Greenwood Publishing Group. p. 306. ISBN 978-0-313-29497-6. 
  9. ^ Van der Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich 1983 ISBN 0-387-12159-5
  10. ^ Smyly, J. Gilbart (1920). "Heron's Formula for Cube Root". Hermathena (Trinity College Dublin) 19 (42): 64–67. 
  11. ^ Crossley, John; W.-C. Lun, Anthony (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford University Press. pp. 176, 213. ISBN 978-0-19-853936-0. 
  12. ^ http://www.springerlink.com/content/q1k57pr4853g1513/

See also[edit]

References[edit]

External links[edit]