Euler's number e Video

 

 

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The number e mathematical constant is the base of the natural logarithm.

Euler's number e

The number e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828,^[and is the limit of (1 + 1/n)ⁿ as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series^[2]

The constant can be defined in many ways; for example, e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = e^x at the point x = 0 is equal to 1.^[3] The function e^x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case, e is the number whose natural logarithm is 1. There are also more alternative characterizations.

Alternative characterizations

The area between the x-axis and the graph y = 1/x, between x = 1 and x = e is 1.

See also: Representations of e

Other characterizations of e are also possible: one is as the limit of a sequence, another is as the sum of an infinite series, and still others rely on integral calculus. So far, the following two (equivalent) properties have been introduced:
1. The number e is the unique positive real number such that

2. The number e is the unique positive real number such that

The following three characterizations can be proven equivalent:
3. The number e is the limit

Similarly:

4. The number e is the sum of the infinite series

where n! is the factorial of n.

5. The number e is the unique positive real number such that

research in graph theory prove that all maps could be colored using only four colors

Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be colored using only four colors so that no areas of the same color touched. Kenneth Appel and Wolfgang Haken proved this in 1976.^[5]

College Intermediate Algebra

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut3_sets.htm

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Matrix Rotation Analysis

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix

rotates points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation using a rotation matrix R, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv. Since matrix multiplication has no effect on the zero vector (i.e., on the coordinates of the origin), rotation matrices can only be used to describe rotations about the origin of the coordinate system.
Rotation matrices provide a simple algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In two-dimensional space, a rotation can be simply described by an angle θ of rotation, but it can also be represented by the four entries of a rotation matrix with two rows and two columns. In three-dimensional space, every rotation can be interpreted as a rotation by a given angle about a single fixed axis of rotation (see Euler's rotation theorem), and hence it can be simply described by an angle and a vector with three entries. However, it can also be represented by the nine entries of a rotation matrix with three rows and three columns. The notion of rotation is not commonly used in dimensions higher than 3; there is a notion of a rotational displacement, which can be represented by a matrix, but not associated single axis or angle.
Rotation matrices are square matrices, with real entries. More specifically they can be characterized as orthogonal matrices with determinant 1:

.

The set of all such matrices of size n forms a (generally not commutative) group, known as the special orthogonal group SO(n).