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#26 2022-07-13 00:39:33

ganesh
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Re: Quantitative Aptitude and Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#27 2022-07-14 03:44:11

ganesh
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Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

Interactive Quadrilaterals

Quadrilaterals

Interactive Quadrilaterals.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#28 2022-07-15 00:58:18

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#29 2022-07-15 23:44:34

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#30 2022-07-17 01:48:26

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#31 2022-07-18 01:54:45

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#32 2022-07-19 00:10:28

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#33 2022-07-20 00:03:07

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#34 2022-07-21 00:22:02

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#35 2022-07-22 00:29:12

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

Boolean Algebra and Logic Gates

Boolean Algebra

Logic Gates

Truth Table


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#36 2022-07-23 00:14:40

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

Real Numbers, Imaginary Numbers, and Complex Numbers

Evolution of Numbers

Real Numbers

Imaginary Numbers

Complex Numbers.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#37 2022-07-24 00:58:22

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#38 2022-07-25 00:13:33

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#39 2022-07-25 13:04:02

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

Union, Intersection, Number of Elements

Union

Intersection

Complement

Sets and Venn Diagrams.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#40 2022-07-25 15:39:40

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

Pythagorean Theorem and Pythagorean Triples

What are Pythagorean Triples?

Three natural non-zero numbers such that

The simplest you can think of is


Another example:

There are infinite of them.

Pythagoras' Theorem

Pythagoreas Areas

Pythagorean Theorem Proof

Pythagoras' Theorem in 3D.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#41 2022-07-26 19:12:32

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

De Moivre's Formula

De Moivre's Formula.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#42 2022-07-28 00:11:47

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#43 2022-07-29 00:40:14

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

Complementary, Supplementary, Interior, and Exterior Angles

Complementary Angles

Supplementary Angles

Interior Angles

Exterior Angles.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#44 2022-07-30 00:04:39

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

Euler's Formula

e (mathematical constant)

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)


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#45 2022-07-31 00:37:30

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

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).


Scalar

Vectors

Dot Product

Cross Product

Scalars and Vectors.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#46 2022-08-01 00:35:20

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#47 2022-08-02 00:19:08

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

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.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#48 2022-08-03 00:04:45

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

Area of a Triangle when the length of sides are known

Heron's Formula

Law of Cosines

Law of Sines


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#49 2022-08-03 23:20:24

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

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.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#50 2022-08-05 01:05:00

ganesh
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Registered: 2005-06-28
Posts: 38,747

Re: Quantitative Aptitude and Higher Mathematics

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.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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