Cube (algebra)
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In arithmetic and algebra, the cube of a number Template:Mvar is its third power: the result of the number multiplied by itself twice:
- n^{3} = n × n × n.
It is also the number multiplied by its square:
- n^{3} = n × n^{2}.
This is also the volume formula for a geometric cube with sides of length Template:Mvar, giving rise to the name. The inverse operation of finding a number whose cube is Template:Mvar is called extracting the cube root of Template:Mvar. It determines the side of the cube of a given volume. It is also Template:Mvar raised to the one-third power.
Both cube and cube root are odd functions:
- (−n)^{3} = −(n^{3}).
The cube of a number or any other mathematical expression is denoted by a superscript 3, for example 2^{3} = 8 or (x + 1)^{3}.
In integers
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 60^{3} are (sequence A000578 in OEIS):
1^{3} = 1 | 11^{3} = 1331 | 21^{3} = 9261 | 31^{3} = 29791 | 41^{3} = 68921 | 51^{3} = 132651 |
2^{3} = 8 | 12^{3} = 1728 | 22^{3} = 10648 | 32^{3} = 32768 | 42^{3} = 74088 | 52^{3} = 140608 |
3^{3} = 27 | 13^{3} = 2197 | 23^{3} = 12167 | 33^{3} = 35937 | 43^{3} = 79507 | 53^{3} = 148877 |
4^{3} = 64 | 14^{3} = 2744 | 24^{3} = 13824 | 34^{3} = 39304 | 44^{3} = 85184 | 54^{3} = 157464 |
5^{3} = 125 | 15^{3} = 3375 | 25^{3} = 15625 | 35^{3} = 42875 | 45^{3} = 91125 | 55^{3} = 166375 |
6^{3} = 216 | 16^{3} = 4096 | 26^{3} = 17576 | 36^{3} = 46656 | 46^{3} = 97336 | 56^{3} = 175616 |
7^{3} = 343 | 17^{3} = 4913 | 27^{3} = 19683 | 37^{3} = 50653 | 47^{3} = 103823 | 57^{3} = 185193 |
8^{3} = 512 | 18^{3} = 5832 | 28^{3} = 21952 | 38^{3} = 54872 | 48^{3} = 110592 | 58^{3} = 195112 |
9^{3} = 729 | 19^{3} = 6859 | 29^{3} = 24389 | 39^{3} = 59319 | 49^{3} = 117649 | 59^{3} = 205379 |
10^{3} = 1000 | 20^{3} = 8000 | 30^{3} = 27000 | 40^{3} = 64000 | 50^{3} = 125000 | 60^{3} = 216000 |
Geometrically speaking, a positive number Template:Mvar is a perfect cube if and only if one can arrange Template:Mvar 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,
- n^{3} = (n − 1)^{3} + 3(n − 1)n + 1.
or
- n^{3} = (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
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
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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 = 2^{3} + 2^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3}.
Fermat's last theorem for cubes
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The equation x^{3} + y^{3} = z^{3} 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]} x^{3} + y^{3} = 3z^{3}.
Sum of first n cubes
The sum of the first Template:Mvar cubes is the Template:Mvarth triangle number squared:
For example, the sum of the first 5 cubes is the square of the 5th triangular number,
A similar result can be given for the sum of the first Template:Mvar odd cubes,
but Template:Mvar, Template:Mvar must satisfy the negative Pell equation . For example, for y = 5 and 29, then,
and so on. Also, every even perfect number, except the first one, is the sum of the first 2^{(p−1)/2} odd cubes,
Sum of cubes of numbers in arithmetic progression
There are examples of cubes of numbers in arithmetic progression whose sum is a cube:
with the first one also known as Plato's number. The formula Template:Mvar for finding the sum of Template:Mvar cubes of numbers in arithmetic progression with common difference Template:Mvar and initial cube a^{3},
is given by
A parametric solution to
is known for the special case of d = 1, or consecutive cubes, but only sporadic solutions are known for integer d > 1, such as Template:Mvar = 2, 3, 5, 7, 11, 13, 37, 39, etc.^{[3]}
Cubes as sums of successive odd integers
In the sequence of odd integers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ..., the first one is a cube (1 = 1^{3}); the sum of the next two is the next cube (3+5 = 2^{3}); the sum of the next three is the next cube (7+9+11 = 3^{3}); and so forth.
In rational numbers
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
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 x ↦ x^{3} : R → R 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 x^{3} > x. If x < −1 or 0 < x < 1, then x^{3} < x. All aforementioned properties pertain also to any higher odd power (x^{5}, x^{7}, …) 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, i^{3} = −i.
The derivative of x^{3} equals to 3x^{2}.
Cubes occasionally have the surjective property in other fields, such as in F_{p} for such prime Template:Mvar that p ≠ 1 (mod 3),^{[6]} but not necessarily: see the counterexample with rationals above. Also in F_{7} 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: x^{3} − x = x(x − 1)(x + 1).
History
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
- ↑ Hardy & Wright, Thm. 227
- ↑ Hardy & Wright, Thm. 232
- ↑ Template:Cite web
- ↑ Hardy & Wright, Thm. 234
- ↑ Hardy & Wright, Thm. 233
- ↑ The multiplicative group of F_{p} is cyclic of order p − 1, and if it is not divisible by 3, then cubes define a group automorphism.
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ Van der Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich 1983 ISBN 0-387-12159-5
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ http://www.springerlink.com/content/q1k57pr4853g1513/
See also
- Perfect power
- Euler's sum of powers conjecture
- Taxicab number
- Cabtaxi number
- Doubling the cube
- Kepler's laws of planetary motion#Third law
- Monkey saddle
References
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