FP3: Complex Numbers - Schoolworkout.co.uk
... * to understand de Moivre’s theorem for positive and negative powers * to find the square roots of a complex number using this theorem. Suppose z is a complex number with modulus r and argument θ, i.e. z r (cos i sin ) OR, using shorthand notation, z = [r, θ]. ...
... * to understand de Moivre’s theorem for positive and negative powers * to find the square roots of a complex number using this theorem. Suppose z is a complex number with modulus r and argument θ, i.e. z r (cos i sin ) OR, using shorthand notation, z = [r, θ]. ...
0915.pdf file.
... remainder r = a − bq (i.e., fractional part = − q = − b c). Then one continues b b b b b with the reciprocal of the fractional part, (which is now greater than 1) and repeat the r process! So one can implement the Euclidean algorithm as repeated use of the ‘integer part’ function bxc, subtraction, a ...
... remainder r = a − bq (i.e., fractional part = − q = − b c). Then one continues b b b b b with the reciprocal of the fractional part, (which is now greater than 1) and repeat the r process! So one can implement the Euclidean algorithm as repeated use of the ‘integer part’ function bxc, subtraction, a ...
Full text
... is Pascal’s square defined by (1.10). Thus it can be argued that Pascal’s square is a natural extension of Pascal’s triangle. The remainder of this article is devoted to justifying the matrix formulas presented in this section, all of which is known except for (1.16) relating to hypergeometric Berno ...
... is Pascal’s square defined by (1.10). Thus it can be argued that Pascal’s square is a natural extension of Pascal’s triangle. The remainder of this article is devoted to justifying the matrix formulas presented in this section, all of which is known except for (1.16) relating to hypergeometric Berno ...
(°1)+ - Art of Problem Solving
... smallest possible value of n such that we can assign 5 integers of set S to vertices A ,B ,C ,D ,and E and 5 integers of set S to vertices F,G, H , I and J (that have not already been assigned to vertices A ,B ,C ,D ,and E ), such that the product of the integers at A ,B ,C ,D ,and E is equal to the ...
... smallest possible value of n such that we can assign 5 integers of set S to vertices A ,B ,C ,D ,and E and 5 integers of set S to vertices F,G, H , I and J (that have not already been assigned to vertices A ,B ,C ,D ,and E ), such that the product of the integers at A ,B ,C ,D ,and E is equal to the ...
Discussion
... using the both the commutative and associative laws of addition, could be to change the order and groupings of the numbers. In our example, the first grouping could be the largest number with the smallest number (i.e. 1 + 16), next grouping the second largest number with the second smallest number ( ...
... using the both the commutative and associative laws of addition, could be to change the order and groupings of the numbers. In our example, the first grouping could be the largest number with the smallest number (i.e. 1 + 16), next grouping the second largest number with the second smallest number ( ...
Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.