Rabu, 04 November 2009
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I. Properties of Logarithm
When x and b are restricted to positive real numbers, logb(x) is a unique real number. The magnitude of the base b must be neither 0 nor 1; the base used is typically 10, e, or 2. Logarithms are defined for real numbers and for complex numbers.
The major property of logarithms is that they map multiplication to addition. This ability stems from the following
A related property is reduction of exponentiation to multiplication. Using the identity:
In words, to raise a number to a power p, find the logarithm of the number and multiply it by p. The exponentiated value is then the inverse logarithm of this product; that is, number to power = bproduct
Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction, and roots to division.
Logarithms make lengthy numerical operations easier to perform by converting multiplications to additions. The manual computation process is made easy by using tables of logarithms, or a slide rule. The property of common logarithms pertinent to the use of log tables is that any decimal sequence of the same digits, but different decimal-point positions, will have identical mantissas and differ only in their characteristics.
II. Common Factor and Grouping
In mathematics, factorization (also factorisation in 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 factorization is expansion. This is the process of multiplying together factors to recreate the original, "expanded" polynomial. 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.
III.Trigonometry Function
• The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.
• The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.
• The tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.
IV. Function Terminology
a function is a relation between a given set of elements called the domain and a set of elements called the codomain. The function associates each element in the domain with exactly one element in the codomain. The elements so related can be any kind of thing (words, objects, qualities) but are typically mathematical quantities, such as real numbers. An example of a function with domain {A,B,C} and codomain {1,2,3} associates A with 1, B with 2, and C with 3. An example of a function with the real numbers as both its domain and codomain is the function f(x) = 2x, which associates every real number with the real number twice as big. In this case, we can write f(5) = 10.
There are many ways to represent or visualize functions: a function may be described by a formula, by a plot or graph, by an algorithm that computes it, by arrows between objects, or by a description of its properties. Sometimes, a function is described through its relationship to other functions (for example, inverse functions). In applied disciplines, functions are frequently specified by tables of values or by formulae. In a setting where outputs of functions are numbers, functions may be added and multiplied, yielding new functions. Collections of functions with certain properties, such as continuous functions and differentiable functions, usually closed under certain operations, are called function spaces and are studied as objects in their own right, in such disciplines as real analysis and complex analysis. An important operation on functions, which distinguishes them from numbers, is composition of functions. The composite function is obtained by using the output of one function as the input of another. This operation provides the theory of functions with its most powerful structure. In pure mathematics, functions are defined using set theory, and there are theorems that show the existence of uncountably many different functions, most of which cannot be expressed with a formula or algorithm
V. Parallelogram
• Opposite sides of a parallelogram are equal in length.
• Opposite angles of a parallelogram are equal in measure.
• The area, A, of a parallelogram is A = bh, where b is the base of the parallelogram and h is its height.
• Opposite sides of a parallelogram will never intersect.
• The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
• The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides.
• The diagonals of a parallelogram bisect each other.
• Any non-degenerate affine transformation takes a parallelogram to another parallelogram.
There is an infinite number of affine transformations which take any given parallelogram to a square
To prove that the diagonals of a parallelogram bisect each other, we will use congruent triangles:
angle ABE congruent angle CDE (alternate interior angles are equal in measure)
angle BAE congruent angle DCE (alternate interior angles are equal in measure).
(since these are angles that a transversal makes with parallel lines AB and DC ).
Also, side AB is equal in length to side DC, since opposite sides of a parallelogram are equal in length.
Therefore triangles ABE and CDE are congruent (ASA postulate, two corresponding angles and the included side).
Therefore,
AE = CE
BE = DE.
Since the diagonals AC and BD divide each other into segments of equal length, the diagonals bisect each other.
In addition, the diagonals AC and BD are each bisected by point E. Therefore, point E is the midpoint of each diagonal.
When x and b are restricted to positive real numbers, logb(x) is a unique real number. The magnitude of the base b must be neither 0 nor 1; the base used is typically 10, e, or 2. Logarithms are defined for real numbers and for complex numbers.
The major property of logarithms is that they map multiplication to addition. This ability stems from the following
A related property is reduction of exponentiation to multiplication. Using the identity:
In words, to raise a number to a power p, find the logarithm of the number and multiply it by p. The exponentiated value is then the inverse logarithm of this product; that is, number to power = bproduct
Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction, and roots to division.
Logarithms make lengthy numerical operations easier to perform by converting multiplications to additions. The manual computation process is made easy by using tables of logarithms, or a slide rule. The property of common logarithms pertinent to the use of log tables is that any decimal sequence of the same digits, but different decimal-point positions, will have identical mantissas and differ only in their characteristics.
II. Common Factor and Grouping
In mathematics, factorization (also factorisation in 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 factorization is expansion. This is the process of multiplying together factors to recreate the original, "expanded" polynomial. 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.
III.Trigonometry Function
• The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.
• The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.
• The tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.
IV. Function Terminology
a function is a relation between a given set of elements called the domain and a set of elements called the codomain. The function associates each element in the domain with exactly one element in the codomain. The elements so related can be any kind of thing (words, objects, qualities) but are typically mathematical quantities, such as real numbers. An example of a function with domain {A,B,C} and codomain {1,2,3} associates A with 1, B with 2, and C with 3. An example of a function with the real numbers as both its domain and codomain is the function f(x) = 2x, which associates every real number with the real number twice as big. In this case, we can write f(5) = 10.
There are many ways to represent or visualize functions: a function may be described by a formula, by a plot or graph, by an algorithm that computes it, by arrows between objects, or by a description of its properties. Sometimes, a function is described through its relationship to other functions (for example, inverse functions). In applied disciplines, functions are frequently specified by tables of values or by formulae. In a setting where outputs of functions are numbers, functions may be added and multiplied, yielding new functions. Collections of functions with certain properties, such as continuous functions and differentiable functions, usually closed under certain operations, are called function spaces and are studied as objects in their own right, in such disciplines as real analysis and complex analysis. An important operation on functions, which distinguishes them from numbers, is composition of functions. The composite function is obtained by using the output of one function as the input of another. This operation provides the theory of functions with its most powerful structure. In pure mathematics, functions are defined using set theory, and there are theorems that show the existence of uncountably many different functions, most of which cannot be expressed with a formula or algorithm
V. Parallelogram
• Opposite sides of a parallelogram are equal in length.
• Opposite angles of a parallelogram are equal in measure.
• The area, A, of a parallelogram is A = bh, where b is the base of the parallelogram and h is its height.
• Opposite sides of a parallelogram will never intersect.
• The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
• The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides.
• The diagonals of a parallelogram bisect each other.
• Any non-degenerate affine transformation takes a parallelogram to another parallelogram.
There is an infinite number of affine transformations which take any given parallelogram to a square
To prove that the diagonals of a parallelogram bisect each other, we will use congruent triangles:
angle ABE congruent angle CDE (alternate interior angles are equal in measure)
angle BAE congruent angle DCE (alternate interior angles are equal in measure).
(since these are angles that a transversal makes with parallel lines AB and DC ).
Also, side AB is equal in length to side DC, since opposite sides of a parallelogram are equal in length.
Therefore triangles ABE and CDE are congruent (ASA postulate, two corresponding angles and the included side).
Therefore,
AE = CE
BE = DE.
Since the diagonals AC and BD divide each other into segments of equal length, the diagonals bisect each other.
In addition, the diagonals AC and BD are each bisected by point E. Therefore, point E is the midpoint of each diagonal.
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