"columns. Other equipment in the board game are track pieces, discs, clips, z shaped platform pieces, cannons, blocks, a ball flipper, a beam and card pieces,and a 7 cm x 7 cm x 7 cm plastic cube for playing a series of physical games. The cube is used for a container and to connect onto the cube platform for the electronic games. A reducer also comes with the cube, allowing it to reduce the size of the open top of the cube. This is, in almost all games involving the cube, removed in a simplify. Many well known and classic"
"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"
"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"
"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"""
"Square root of 3 The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property. It is denoted by . The square root of 3 is an irrational number. It is also known as Theodorus' constant, named after Theodorus of Cyrene, who proved its irrationality. The first sixty digits of its decimal expansion are: As of December 2013, its numerical value in decimal has been computed to at least ten"
"If and are non-real, then Since this is the product of a non-zero real number by , its square is a real number less than and therefore . Finally, if the numbers , , and are three distinct real numbers, then the product is a non-zero real number, and so . Viète's trigonometric expression of the roots in the three-real-roots case lends itself to a geometric interpretation in terms of a circle. When the cubic is written in depressed form (), , as shown above, the solution can be expressed as Here formula_32 is an angle in the unit circle;"
"The circle around formula_228 with radius formula_230 intersects the parabola in formula_231. The perpendicular from formula_231 onto the ""x""-axis intersects the unit circle in formula_233 and formula_234 is exactly one third of formula_222. The correctness of this construction can be seen by showing that the ""x""-coordinate of formula_140 is formula_237. Solving the equation system given by the circle around formula_228 and the parabola leads to the cubic equation formula_239. The triple angle formula formula_240 then shows that formula_237 is indeed a solution of that cubic equation. This trisection goes back to René Descartes who described it in his book ""La"
"and two complex roots, the three roots can be represented as points in the complex plane, as can the two roots of the cubic's derivative. There is an interesting geometrical relationship among all these roots. The points in the complex plane representing the three roots serve as the vertices of an isosceles triangle. (The triangle is isosceles because one root is on the horizontal (real) axis and the other two roots, being complex conjugates, appear symmetrically above and below the real axis.) Marden's theorem says that the points representing the roots of the derivative of the cubic are the foci"
"origin to the intersection of the vertical line and the -axis. A simple modern proof of the method is the following: multiplying the equation by and regrouping the terms gives The left-hand side is the value of on the parabola. The equation of the circle being , the right hand side is the value of on the circle. A cubic equation with real coefficients can be solved geometrically using compass, straightedge, and an angle trisector if and only if it has three real roots. Every cubic equation (), , with real coefficients and , has three solutions (some of which"
"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"
"all three real solutions are written in terms of cube roots of complex numbers. On the other hand, consider the equation which has the rational solutions 1, 2, and —3. The general solution formula given above gives the solutions For any given choice of cube root and its conjugate, this contains nested radicals involving complex numbers, yet it is reducible (even though not obviously so) to one of the solutions 1, 2, or –3. Under certain conditions infinitely nested square roots such as represent rational numbers. This rational number can be found by realizing that ""x"" also appears under the"
"not real in general. If both and are equal to , then the equation has a single root (which is a triple root): If and , then there are both a double root, and a simple root, Dividing by and substituting for we get the equation where The left hand side of equation () is a monic trinomial called a depressed cubic, because the quadratic term has coefficient 0. Any formula for the roots of a depressed cubic may be transformed into a formula for the roots of equation () by substituting the above values for and and using the"
"be −2, , or . This is related with the concept of monodromy: if one follows by continuity the function ""cube root"" along a closed path around zero, after a turn the value of the cube root is multiplied (or divided) by formula_10 Cube roots arise in the problem of finding an angle whose measure is one third that of a given angle (angle trisection) and in the problem of finding the edge of a cube whose volume is twice that of a cube with a given edge (doubling the cube). In 1837 Pierre Wantzel proved that neither of these"
"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;"
"this problem, and in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123. The cube has three classes of symmetry, which can be represented by vertex-transitive coloring the faces. The highest octahedral symmetry O has all the faces the same color. The dihedral symmetry D comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D is also a prismatic symmetry, with"
"relation . Therefore, only equation () is considered in the following. When a cubic equation has three real roots, the formulas expressing these roots in terms of radicals involve complex numbers. It has been proved that when none of the three real roots is rational—the ""casus irreducibilis""— one cannot express the roots in terms of real radicals. Nevertheless, purely real expressions of the solutions may be obtained using hypergeometric functions, or more elementarily in terms of trigonometric functions, specifically in terms of the cosine and arccosine functions. The formulas which follow, due to François Viète, are true in general (except"
"+ 41 + 43 + 47 + 53), sum of the first four powers of 4 (4 + 4 + 4 + 4), divisible by the number of primes below it, nontotient, noncototient. 341 = 11 × 31, sum of seven consecutive primes (37 + 41 + 43 + 47 + 53 + 59 + 61), octagonal number, centered cube number, super-Poulet number. 341 is the smallest Fermat pseudoprime; it is the ""least"" ""composite"" ""odd"" modulus ""m"" greater than the base ""b"", that satisfies the ""Fermat"" property """"b"" − 1 is divisible by ""m"""", for bases up to 128 of"
"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,"
"(or other trigonometric function) of an arbitrary angle, the cosine of one-third of that angle is one of the roots of a cubic. The solution of the general quartic equation relies on the solution of its resolvent cubic. The eigenvalues of a 3×3 matrix are the roots of a cubic polynomial which is the characteristic polynomial of the matrix. The characteristic equation of a third-order linear difference equation or differential equation is a cubic equation. In analytical chemistry, the Charlot equation, which can be used to find the pH of buffer solutions, can be solved using a cubic equation. In"
"locally observed frequency. The fundamental equation of the orbit is easier to solve if it is expressed in terms of the inverse radius ""u"" = 1/""r"" The right-hand side of this equation is a cubic polynomial, which has three roots, denoted here as ""u"", ""u"", and ""u"" The sum of the three roots equals the coefficient of the ""u"" term A cubic polynomial with real coefficients can either have three real roots, or one real root and two complex conjugate roots. If all three roots are real numbers, the roots are labeled so that ""u"" < ""u"" < ""u"". If"
"41 + 43 + 47), Motzkin number, self number. A Lucas and Fibonacci pseudoprime. ""See 323 (disambiguation)"" 324 = 2 × 3 = 18. 324 is the sum of four consecutive primes (73 + 79 + 83 + 89), totient sum of the first 32 integers, untouchable number, and a Harshad number. 325 = 5 × 13. 325 is a triangular number, hexagonal number, nonagonal number, centered nonagonal number. 325 is the smallest number to be the sum of two squares in 3 different ways: 1 + 18, 6 + 17 and 10 + 15. 325 is also the smallest"
"Cubic function In algebra, a cubic function is a function of the form in which is nonzero. Setting produces a cubic equation of the form The solutions of this equation are called roots of the polynomial . If all of the coefficients , , , and of the cubic equation are real numbers, then it has at least one real root (this is true for all odd degree polynomials). All of the roots of the cubic equation can be found algebraically. (This is also true of quadratic (second degree) or quartic (fourth degree) equations, but not of higher-degree equations, by"
"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,"
"roots and cubic roots using ""Yang Hui's Triangle"" was invented by mathematician Jia Xian who expounded it around 1100 AD, about 500 years before Pascal. In his book (now lost) known as ""Rújī Shìsuǒ"" (如積釋鎖) or ""Piling-up Powers and Unlocking Coefficients"", which is known through his contemporary mathematician Liu Ruxie (劉汝諧). Jia described the method used as 'li cheng shi suo' (the tabulation system for unlocking binomial coefficients). It appeared again in a publication of Zhu Shijie's book ""Jade Mirror of the Four Unknowns"" (四元玉鑒) of 1303 AD. Around 1275 AD, Yang finally had two published mathematical books, which were"
"equation in sum to zero. If a cubic is plotted in the Cartesian plane, the real root can be seen graphically as the horizontal intercept of the curve. But further, if the complex conjugate roots are written as then is the abscissa (the positive or negative horizontal distance from the origin) of the tangency point of a line that is tangent to the cubic curve and intersects the horizontal axis at the same place as does the cubic curve; and is the square root of the tangent of the angle between this line and the horizontal axis. With one real"
"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"
"be represented in the form In this representation, and the trigonometric functions are defined as In the case that , that is, the unit vector. This leads to the variation of De Moivre's formula: To find the cube roots of write the quaternion in the form Then the cube roots are given by: Consider the following matrix formula_31. Then formula_32. This fact (although it can be proven in the very same way as for complex numbers) is a direct consequence of the fact that the space of matrices of type formula_33 is isomorphic to the space of complex numbers. De"
"real roots. The formula involves starting with an angle whose cosine is known, trisecting the angle by multiplying it by 1/3, and taking the cosine of the resulting angle and adjusting for scale. The distinction between the reducible and irreducible cubic cases with three real roots is related to the issue of whether or not an angle with rational cosine or rational sine is trisectible by the classical means of compass and unmarked straightedge. If the cosine of an angle is known to have a particular rational value, then one third of this angle has a cosine that is one"
"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:"
"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"
"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"
"The cubic function has point symmetry about its inflection point. All of the above assumes that the coefficients are real as well as the variable . This section is about how to solve the cubic equation using various methods. For details and proofs see . The general cubic equation has the form: with . The algebraic solution of the cubic equation can be derived in a number of different ways. (See for example Cardano's method and Vieta's method below.) The numbers of real and complex roots are determined by the discriminant of the cubic equation, It turns out that: The"
"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"