Data and Graphs coordinate grid: a grid used to show ordered pairs
... calculations. Work inside parentheses is done first. Next, terms with exponents are evaluated. Then multiplication and division are done in order from left to right, and finally addition and subtraction are done in order from left to right. 8. parentheses: Used in mathematics as grouping symbols for ...
... calculations. Work inside parentheses is done first. Next, terms with exponents are evaluated. Then multiplication and division are done in order from left to right, and finally addition and subtraction are done in order from left to right. 8. parentheses: Used in mathematics as grouping symbols for ...
Lecture 12
... So far we have concentrated on analysis over the field of real numbers. In our initial discussion, the construction of the real numbers was given in order to ensure certain polynomial equations, such as x2 − 2, have solutions. Although we have developed a rich theory, the real numbers are lacking in ...
... So far we have concentrated on analysis over the field of real numbers. In our initial discussion, the construction of the real numbers was given in order to ensure certain polynomial equations, such as x2 − 2, have solutions. Although we have developed a rich theory, the real numbers are lacking in ...
File - Ms. Fujie`s Math Class
... 7. Add, subtract, multiply, divide, and evaluate functions. 8. Compose functions and apply to real-life problems. 9. Differentiate the purposes of vertical and horizontal line tests 10. Identify a function as one-to-one and its implications on inverse functions. 11. Find the inverse of a function an ...
... 7. Add, subtract, multiply, divide, and evaluate functions. 8. Compose functions and apply to real-life problems. 9. Differentiate the purposes of vertical and horizontal line tests 10. Identify a function as one-to-one and its implications on inverse functions. 11. Find the inverse of a function an ...
Solutions to problem sheet 4.
... (b) 3 − i = 2e−πi/6 . Hence one root is α = (2)1/7 e−iπ/42 . Hence every root is of the form αω for one of the 7th roots of unity. Hence it is one of the seven complex numbers ei(12k−1)π/42 for k = 0, 1, . . . , 6. (c) By standard trigonometry it is enough to show that the three sides of the triangl ...
... (b) 3 − i = 2e−πi/6 . Hence one root is α = (2)1/7 e−iπ/42 . Hence every root is of the form αω for one of the 7th roots of unity. Hence it is one of the seven complex numbers ei(12k−1)π/42 for k = 0, 1, . . . , 6. (c) By standard trigonometry it is enough to show that the three sides of the triangl ...
Solving linear, const.-coeff. ODEs
... Solving linear, const.-coeff. ODEs This is a summary of the method for finding the general solution of linear, first and second order, constant-coefficient ODEs. Details of the theory can be found in most introductory books on ordinary differential equations such as Boyce & DiPrima. The ODE to be so ...
... Solving linear, const.-coeff. ODEs This is a summary of the method for finding the general solution of linear, first and second order, constant-coefficient ODEs. Details of the theory can be found in most introductory books on ordinary differential equations such as Boyce & DiPrima. The ODE to be so ...
Exponents and Radicals
... VOCABULARY CHECK: Fill in the blanks. 1. In the exponential form an, n is the ________ and a is the ________. 2. A convenient way of writing very large or very small numbers is called ________ ________. 3. One of the two equal factors of a number is called a __________ __________ of the number. 4. T ...
... VOCABULARY CHECK: Fill in the blanks. 1. In the exponential form an, n is the ________ and a is the ________. 2. A convenient way of writing very large or very small numbers is called ________ ________. 3. One of the two equal factors of a number is called a __________ __________ of the number. 4. T ...
Example sheet 4
... 12. Show that a complex number α is an algebraic integer if and only if the additive group of the ring Z[α] is finitely generated (i.e. Z[α] is a finitely generated Z-module). Furthermore if α and β are algebraic integers show that the subring Z[α, β] of C generated by α and β also has a finitely ge ...
... 12. Show that a complex number α is an algebraic integer if and only if the additive group of the ring Z[α] is finitely generated (i.e. Z[α] is a finitely generated Z-module). Furthermore if α and β are algebraic integers show that the subring Z[α, β] of C generated by α and β also has a finitely ge ...
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.