"it is possible to find sequences of 64 consecutive integers such that each inner member shares a factor with either the first or the last member, 64 is an Erdős–Woods number. In base 10, no integer added up to its own digits yields 64, hence it is a self number. 64 is a superperfect number—a number such that σ(σ(""n"")) = 2""n"". 64 is the index of Graham's number in the rapidly growing sequence 3↑↑↑↑3, 3 ↑ 3,… Sixty-four is: 64 (number) 64 (sixty-four) is the natural number following 63 and preceding 65. Sixty-four is the square of 8, the cube"
"Cube root In mathematics, a cube root of a number ""x"" is a number ""y"" such that ""y"" = ""x"". All real numbers (except zero) have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 2 = 8, while the other cube roots of 8 are −1 + ""i"" and −1 − ""i"". The three cube roots of −27""i"" are The cube root operation is not distributive with addition or subtraction. In"
"the cube root of numbers having many digits in the ""Aryabhatiya"" (section 2.5). Cube root In mathematics, a cube root of a number ""x"" is a number ""y"" such that ""y"" = ""x"". All real numbers (except zero) have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 2 = 8, while the other cube roots of 8 are −1 + ""i"" and −1 − ""i"". The three cube roots of −27""i"""
"cube root of 29791 is 31. Another example: This process can be extended to find cube roots that are 3 digits long, by using arithmetic modulo 11. These types of tricks can be used in any root where the order of the root is coprime with 10; thus it fails to work in square root, since the power, 2, divides into 10. 3 does not divide 10, thus cube roots work. To approximate a common log (to at least one decimal point accuracy), a few log rules, and the memorization of a few logs is required. One must know: From"
"some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the ""principal cube root"", denoted with the radical sign . The cube root operation is associative with exponentiation and distributive with multiplication and division if considering only real numbers, but not always if considering complex numbers: for example, the cube of any cube root of 8 is 8, but the three cube roots of 8 are 8, −4 + 4""i"", and −4 − 4""i"". The cube roots"
"is a pattern in the rightmost digit: adding and subtracting with 1 or 3. Starting from zero: There are two steps to extracting the cube root from the cube of a two digit number. Say you are asked to extract the cube root of 29791. Begin by determining the one's place (units) of the two digit number. You know it must be one, since the cube ends in 1, as seen above. Note that every digit corresponds to itself except for 2, 3, 7 and 8, which are just subtracted from ten to obtain the corresponding digit. The second step"
"root is coprime with 10 are somewhat easier since the digits are scrambled in consistent ways, as we shall see in the next section. An easy task for the beginner is extracting cube roots from the cubes of 2 digit numbers. For example, given 74088, determine what two digit number, when multiplied by itself once and then multiplied by the number again, yields 74088. One who knows the method will quickly know the answer is 42, as 42 = 74088. Before learning the procedure, it is required that the performer memorize the cubes of the numbers 1-10: Observe that there"
"is to determine the first digit of the two digit cube root by looking at the magnitude of the given cube. To do this, remove the last three digits of the given cube (29791 → 29) and find the greatest cube it is greater than (this is where knowing the cubes of numbers 1-10 is needed). Here, 29 is greater than 1 cubed, greater than 2 cubed, greater than 3 cubed, but not greater than 4 cubed. The greatest cube it is greater than is 3, so the first digit of the two digit cube must be 3. Therefore, the"
"this question in ""Scientific American"", Kay R. Pechenick DeVicci and several other readers showed that the answer for the (3,4) case is the square root of the smaller of two real roots of the polynomial formula_7, which works out to approximately 1.007435. For formula_8, the optimal side length of the largest square in an formula_5-dimensional hypercube is either formula_10 or formula_11, depending on whether formula_5 is even or odd respectively. Prince Rupert's cube In geometry, Prince Rupert's cube (named after Prince Rupert of the Rhine) is the largest cube that can pass through a hole cut through a unit cube,"
"the cube roots being the real cube roots. In the case of three real roots, the square root expression is an imaginary number; here any real root is expressed by defining the first cube root to be any specific complex cube root of the complex radicand, and by defining the second cube root to be the complex conjugate of the first one. The nested radicals in this solution cannot in general be simplified unless the cubic equation has at least one rational solution. Indeed, if the cubic has three irrational but real solutions, we have the ""casus irreducibilis"", in which"
"however, easy to show that most numbers are not perfect cubes because ""all"" perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only −1, 1 and 0. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3: 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: The equation has"
"which can be simplified by noting that 5/10 = 3/6 = 1/2: formula_75 The square root can also be expressed by a periodic continued fraction, but the above form converges more quickly with the proper ""x"" and ""y"". The cube root of two (2 or ≈ 1.259921...): (A) ""Standard notation"" of ""x"" = 1, ""y"" = 1, and 2""z - y"" = 3: formula_76 (B) Rapid convergence with ""x"" = 5, ""y"" = 3 and 2""z - y"" = 253: formula_77 Pogson's ratio (100 or ≈ 2.511886...), with ""x"" = 5, ""y"" = 75 and 2""z - y"" = 6325:"
"formula_57 is a Legendre symbol. From formula_58 follows that formula_59. Thus ""c"" is a quadratic residue modulo ""p"". Hence formula_60 and therefore The relation formula_55 is not a requirement because square roots modulo other primes can be computed too. E.g., Rabin proposes to find the square roots modulo primes by using a special case of Berlekamp's algorithm. Decoding produces three false results in addition to the correct one, so that the correct result must be guessed. This is the major disadvantage of the Rabin cryptosystem and one of the factors which have prevented it from finding widespread practical use. If"
"Square root In mathematics, a square root of a number ""a"" is a number ""y"" such that ; in other words, a number ""y"" whose ""square"" (the result of multiplying the number by itself, or ) is ""a"". For example, 4 and −4 are square roots of 16 because . Every nonnegative real number ""a"" has a unique nonnegative square root, called the ""principal square root"", which is denoted by , where √ is called the ""radical sign"" or ""radix"". For example, the principal square root of 9 is 3, which is denoted by = 3, because and 3 is"
"Square root of 2 The square root of 2, or the (1/2)th power of 2, written in mathematics as or , is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. As a good rational approximation for"
"Snub cube In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images (or ""enantiomorphs"") of each other. The union of both forms is a compound of two snub cubes, and the convex hull of both sets of vertices is a truncated cuboctahedron. Kepler first named it in Latin as cubus simus in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived"
"real number is a positive real number, negative numbers do not have real square roots. However, for every negative real number there are two imaginary square roots. For example, the square roots of −25 are 5""i"" and −5""i"", where ""i"" represents a number whose square is . A cube root of a number ""x"" is a number ""r"" whose cube is ""x"": Every real number ""x"" has exactly one real cube root, written formula_4. For example, Every real number has two additional complex cube roots. Expressing the degree of an ""n""th root in its exponent form, as in formula_37, makes"
"solution expressions involve the real cube root of a real number, while if all three solutions are real numbers then they may be expressed in terms of the complex cube root of a complex number. Quartic equations can also be solved in terms of cube roots and square roots. The calculation of cube roots can be to traced back to Babylonian mathematicians from as early as 1800 BCE. In the fourth century BCE Plato posed the problem of doubling the cube, which required a compass-and-straightedge construction of the edge of a cube with twice the volume of a given cube;"
"it is a nonzero quadratic residue mod ""p"". Note that the quadratic reciprocity law allows one to easily test whether ""a"" is a nonzero quadratic residue mod ""p"", thus we get a practical way to determine which ""p""-adic numbers (for ""p"" odd) have a ""p""-adic square root, and it can be extended to cover the case ""p""=2 using the more general version of Hensel's lemma (an example with 2-adic square roots of 17 is given later). To make the discussion above more explicit, let us find a ""square root of 2"" (the solution to formula_67) in the 7-adic integers. Modulo"
"number of sides). This is because the sum of the angles of any ""n""-gon is 180° × (""n"" – 2) and so the measure of each angle of any regular ""n""-gon is 180° × (""n"" – 2) ÷ ""n"". Thus for example the entry ""45°: square"" means that, with ""n"" = 4, 180° ÷ ""n"" = 45°, and the number of degrees in each angle of a square is (""n"" – 2) × 45° = 90°. For cube roots of non-real numbers that appear in this table, one has to take the principal value, that is the cube root with"
"or the other of the two complex cube roots of 1. For complex numbers, the principal cube root is usually defined as the cube root that has the greatest real part, or, equivalently, the cube root whose argument has the least absolute value. It is related to the principal value of the natural logarithm by the formula If we write ""x"" as where ""r"" is a non-negative real number and ""θ"" lies in the range then the principal complex cube root is This means that in polar coordinates, we are taking the cube root of the radius and dividing the"
"Snub 24-cell In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. Topologically, under its highest symmetry, [3,4,3], as an alternation of a truncated 24-cell, it contains 24 pyritohedra (an icosahedron with T symmetry), 24 regular tetrahedra, and 96 triangular pyramids. It is one of three semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He"
"nested cube roots such as can represent rational numbers as well. Again, by realizing that the whole expression appears inside itself, we are left with the equation If we solve this equation, we find that ""x"" = 2. More generally, we find that is the positive real root of the equation ""x"" − ""x"" − ""n"" = 0 for all ""n"" > 0. For ""n"" = 1, this root is the plastic number ""ρ"", approximately equal to 1.3247. The same procedure also works to get as the real root of the equation ""x"" + ""x"" − ""n"" = 0 for"
"are real or complex, the roots are The modulus of the roots of a quadratic formula_27 can be no greater than formula_28 where formula_29 is the golden ratio formula_30 The square root of a univariate quadratic function gives rise to one of the four conic sections, almost always either to an ellipse or to a hyperbola. If formula_31 then the equation formula_32 describes a hyperbola, as can be seen by squaring both sides. The directions of the axes of the hyperbola are determined by the ordinate of the minimum point of the corresponding parabola formula_33. If the ordinate is negative,"
"has three roots, and can be written alternatively as: which can be expanded as: formula_52 is a volume of an unstable state between formula_33 and formula_34. The cubic equations are identical. Therefore, from the linear terms in these equations, formula_55 can be solved: From the quadratic terms, ""B"" can be solved: And from the cubic terms, ""C"" can be solved: Since formula_33, formula_34 and formula_31 were tabulated for many fluids with formula_32 as a parameter, it is a simple matter to compute ""B"" and ""C"" in the saturation region of these fluids. The results are generally in agreement with those"
"a sum of 14 fourth powers (Vaughan in 1985 and 1989 reduced the 14 successively to 13 and 12). The exact value of ""G""(""k"") is unknown for any other ""k"", but there exist bounds. The number ""G""(""k"") is greater than or equal to In the absence of congruence restrictions, a density argument suggests that ""G""(""k"") should equal . G(3) is at least four (since cubes are congruent to 0, 1 or −1 mod 9); for numbers less than 1.3, 1290740 is the last to require six cubes, and the number of numbers between N and 2N requiring five cubes drops"
"example: In the case of the fifth root of 34 above (after dividing out selected common factors): Building on the digit-by-digit calculation of a square root, it can be seen that the formula used there, formula_64, or formula_65, follows a pattern involving Pascal's triangle. For the ""n""th root of a number formula_66 is defined as the value of element formula_67 in row formula_68 of Pascal's Triangle such that formula_69, we can rewrite the expression as formula_70. For convenience, call the result of this expression formula_71. Using this more general expression, any positive principal root can be computed, digit-by-digit, as follows."
"Cube (algebra) In arithmetic and algebra, the cube of a number is its third power: the result of the number multiplied by itself twice: It is also the number multiplied by its square: This is also the volume formula for a geometric cube with sides of length , giving rise to the name. The inverse operation of finding a number whose cube is is called extracting the cube root of . It determines the side of the cube of a given volume. It is also raised to the one-third power. Both cube and cube root are odd functions: The cube"
"complex conjugate roots. See Discriminant:Nature of the roots for details. The cubic was first partly solved by the 15–16th-century Italian mathematician Scipione del Ferro, who did not however publish his results; this method, though, only solved one type of cubic equation. This solution was then rediscovered independently in 1535 by Niccolò Fontana Tartaglia, who shared it with Gerolamo Cardano, asking him to not publish it. Cardano then extended this to numerous other cases, using similar arguments; see more details at Cardano's method. After the discovery of Ferro's work, he felt that Tartaglia's method was no longer secret, and thus he"
"regular 6-cube or 6-orthoplex. 6-cube In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol {4,3}, being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the ""4-cube"") with ""hex"" for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets. It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can"
"from the depressed cubic (), , we make the substitution , known as Vieta's substitution. This results in the equation Multiplying by , it becomes a sextic equation in , which is in fact a quadratic equation in : The quadratic formula allows equation () to be solved for . If , and are the three cube roots of one of the solutions in , then the roots of the original depressed cubic are , , and . Another way of expressing the roots is to take ; then the roots of the original depressed cubic are , , and,"
"of a number ""x"" are the numbers ""y"" which satisfy the equation For any real number ""x"", there is ""one"" real number ""y"" such that ""y"" = ""x"". The cube function is increasing, so does not give the same result for two different inputs, plus it covers all real numbers. In other words, it is a bijection, or one-to-one. Then we can define an inverse function that is also one-to-one. For real numbers, we can define a unique cube root of all real numbers. If this definition is used, the cube root of a negative number is a negative number."
"If ""x"" and ""y"" are allowed to be complex, then there are three solutions (if ""x"" is non-zero) and so ""x"" has three cube roots. A real number has one real cube root and two further cube roots which form a complex conjugate pair. For instance, the cube roots of 1 are: The last two of these roots lead to a relationship between all roots of any real or complex number. If a number is one cube root of a particular real or complex number, the other two cube roots can be found by multiplying that cube root by one"