Санның кубы және куб түбірі

Number raised to the third power

In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or (x + 1)^(3). The cube is also the number multiplied by its square:
size=120%. The cube function is the function x ↦ x^(3) (often denoted 1=y = x^(3)) that maps a number to its cube. It is an odd function, as
size=120%. The volume of a geometric cube is the cube of its side length, giving rise to the name. The inverse operation that consists 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. The graph of the cube function is known as the cubic parabola. Because the cube function is an odd function, this curve has a center of symmetry at the origin, but no axis of symmetry.

It is conjectured that every integer (positive or negative) not congruent to ±4 modulo 9 can be written as a sum of three (positive or negative) cubes with infinitely many ways. For example, Integers congruent to ±4 modulo 9 are excluded because they cannot be written as the sum of three cubes. The smallest such integer for which such a sum is not known is 114. In September 2019, the previous smallest such integer with no known 3 cube sum, 42, was found to satisfy this equation:

One solution to is given in the table below for n ≤ 78, and n not congruent to 4 or 5 modulo 9. The selected solution is the one that is primitive (1=gcd(x, y, z) = 1), is not of the form or (since they are infinite families of solutions), satisfies , and has minimal values for and (tested in this order). Only primitive solutions are selected since the non primitive ones can be trivially deduced from solutions for a smaller value of n. For example, for 1=n = 24, the solution results from the solution by multiplying everything by Therefore, this is another solution that is selected. Similarly, for 1=n = 48, the solution 1=(x, y, z) = ( 2, 2, 4) is excluded, and this is the solution 1=(x, y, z) = ( 23, 26, 31) that is selected.

The equation 1=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. Both of these statements are also true for the equation 1=x^(3) + y^(3) = 3z^(3).

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

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.

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

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 are equal to their own cubes: , , and 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, 1=[[imaginary unit. The derivative of x^(3) equals 3x^(2). Cubes occasionally have the surjective property in other fields, such as in [[finite field for such prime p that p ≠ 1 (mod 3), 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: 1=x^(3) − x = [[factorization of polynomials.

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). Cubic equations were known to the ancient Greek mathematician Diophantus. Hero of Alexandria devised a method for calculating cube roots in the 1st century CE. 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.