Higher Mathematics

The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828 which can be characterized in many ways. It is the base of the natural logarithms. It is the limit

an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series

.

Euler's Formula: Geometry and Graphs

Euler's Formula for Complex Numbers

e (Euler's Number)

Jai Ganesh · 2022-07-31 00:37:30

Scalar and Vector

Scalar (mathematics)

A scalar is an element of a field which is used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector.

In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers). Then scalars of that vector space will be elements of the associated field (such as complex numbers).

A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space.

The real component of a quaternion is also called its scalar part.

The term scalar is also sometimes used informally to mean a vector, matrix, tensor, or other, usually, "compound" value that is actually reduced to a single component. Thus, for example, the product of a 1 × n matrix and an n × 1 matrix, which is formally a 1 × 1 matrix, is often said to be a scalar.

The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix.

Vector (mathematics and physics)

In mathematics and physics, vector is a term that refers colloquially to some quantities that cannot be expressed by a single number, or to elements of some vector spaces.

Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers.

The term vector is also used, in some contexts, for tuples, which are finite sequences of numbers of a fixed length.

Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is a set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on above sorts of vectors. A vector space formed by geometric vectors is called a Euclidean vector space, and a vector space formed by tuples is called a coordinate vector space.

There are many vector spaces that are considered in mathematics, such as extension field, polynomial rings, algebras and function spaces. The term vector is generally not used for elements of these vectors spaces, and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces (for example, when discussing general properties of vector spaces).

Jai Ganesh · 2022-08-01 00:35:20

Number of Elements

Complement

Common Number Sets

Set Symbols

Jai Ganesh · 2022-08-02 00:19:08

Prime Numbers - Advanced Concepts

In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.

The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols,

where is the sum-of-divisors function.

For instance, 28 is perfect as

Prime and Composite Numbers

Prime Properties

Prime Numbers - Advanced.

Jai Ganesh · 2022-08-03 00:04:45

Area of a Triangle when the length of sides are known

Heron's Formula

Law of Cosines

Law of Sines

Jai Ganesh · 2022-08-03 23:20:24

Binomial Theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial

into a sum involving terms of the form where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example, for n = 4,

The coefficient a in the term of

is known as the binomial coefficient or (the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where gives the number of different combinations of b elements that can be chosen from an n-element set. Therefore
is often pronounced as "n choose b".

Binomial Theorem - Definition

Binomial Theorem : Detailed.

Jai Ganesh · 2022-08-05 01:05:00

Hyperbolic Functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively.

Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.

The basic hyperbolic functions are:

1) hyperbolic sine "sinh",
2) hyperbolic cosine "cosh",

from which are derived:

3) hyperbolic tangent "tanh",
4) hyperbolic cosecant "csch" or "cosech",
5) hyperbolic secant "sech",
6) hyperbolic cotangent "coth"

corresponding to the derived trigonometric functions.

Hyperbolic Functions - Definition

Hyperbolic Functions - Details.

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