<![CDATA[Math Is Fun Forum / Quantitative Aptitude and Higher Mathematics]]> 2022-08-14T13:15:09Z FluxBB https://www.mathisfunforum.com/viewtopic.php?id=27202 <![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Prime Numbers - Advanced Concepts

Prime and Composite Numbers

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form

for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is
. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form
for some prime p.

The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ...  and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ....

Numbers of the form

without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is

Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes. Many of the largest known primes are Mersenne primes because Mersenne numbers are easier to check for primality.

As of October 2020, 51 Mersenne primes are known. The largest known prime number,

, is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project. In December 2020, a major milestone in the project was passed after all exponents below 100 million were checked at least once.

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Binary Number System

Number Bases

Octal

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Pascal's Triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy.

The rows of Pascal's triangle are conventionally enumerated starting with row n=0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k=0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number in the first (or any other) row is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.

Factorial

Combinations and Permutations

Pascal's Triangle

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Square and Square Roots

Cube and Cube Roots

nth Root

Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly n nth roots of unity, except when n is a multiple of the (positive) characteristic of the field.

General definition

An nth root of unity, where n is a positive integer, is a number z satisfying the equation

Unless otherwise specified, the roots of unity may be taken to be complex numbers (including the number 1, and the number -1 if n is even, which are complex with a zero imaginary part), and in this case, the nth roots of unity are

However, the defining equation of roots of unity is meaningful over any field (and even over any ring) F, and this allows considering roots of unity in F. Whichever is the field F, the roots of unity in F are either complex numbers, if the characteristic of F is 0, or, otherwise, belong to a finite field. Conversely, every nonzero element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details.

An nth root of unity is said to be primitive if it is not an mth root of unity for some smaller m, that is if

If n is a prime number, then all nth roots of unity, except 1, are primitive.

In the above formula in terms of exponential and trigonometric functions, the primitive nth roots of unity are those for which k and n are coprime integers.

Subsequent sections of this article will comply with complex roots of unity.

Trigonometric expression

De Moivre's formula, which is valid for all real x and integers n, is

Setting

gives a primitive nth root of unity – one gets

but

for k = 1, 2, …, n - 1. In other words,

is a primitive nth root of unity.

This formula shows that in the complex plane the nth roots of unity are at the vertices of a regular n-sided polygon inscribed in the unit circle, with one vertex at 1 (see the plots for n = 3 and n = 5 on the right.) This geometric fact accounts for the term "cyclotomic" in such phrases as cyclotomic field and cyclotomic polynomial; it is from the Greek roots "cyclo" (circle) plus "tomos" (cut, divide).

Euler's formula

which is valid for all real x, can be used to put the formula for the nth roots of unity into the form

It follows from the discussion in the previous section that this is a primitive nth-root if and only if the fraction

is in lowest terms; that is, that k and n are coprime. An irrational number that can be expressed as the real part of the root of unity; that is, as

is called a trigonometric number.

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Solution of a Quadratic Equation

Here, the sum of roots is -b/a and product of roots c/a.

In algebra, a quadratic equation (from Latin quadratus 'square') is any equation that can be rearranged in standard form as

where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0. If a = 0, then the equation is linear, not quadratic, as there is no

term. The numbers a, b, and c are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.
The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the expression on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a single real double root, or two complex solutions. A quadratic equation always has two roots, if complex roots are included; and a double root is counted for two. A quadratic equation can be factored into an equivalent equation

where r and s are the solutions for x.

expresses the solutions in terms of a, b, and c. Completing the square is one of several ways for getting it.

Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC.

Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two.

Completing the Square

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Phi - Golden Ratio

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with

where the Greek letter phi

or
represents the golden ratio. It is an irrational number that is a solution to the quadratic equation
with a value of

The golden ratio is also called the golden mean or golden section (Latin: sectio aurea). Other names include extreme and mean ratio, medial section, divine proportion (Latin: proportio divina), divine section (Latin: sectio divina), golden proportion, golden cut, and golden number.

Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.

Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing this to be aesthetically pleasing. These often appear in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio.

Definition of Golden Ratio

Golden Ratio

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Euler Number

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 of

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

It is also the unique positive number a such that the graph of the function

has a slope of 1 at x = 0.

The (natural) exponential function

is the unique function f that equals its own derivative and satisfies the equation f(0) = 1; hence one can also define e as f(1). The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one. There are various other characterizations.

e is sometimes called Euler's number (not to be confused with Euler's constant

), after the Swiss mathematician Leonhard Euler, or Napier's constant, after John Napier. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.

The number e is of great importance in mathematics, alongside

and i. All five appear in one formulation of Euler's identity, and play important and recurring roles across mathematics. Like the constant π, e is irrational (that is, it cannot be represented as a ratio of integers) and transcendental (that is, it is not a root of any non-zero polynomial with rational coefficients). To 50 decimal places the value of e is:

2.71828182845904523536028747135266249775724709

e - Euler's number

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Pi

The number

(spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulas across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as 22/7 are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only sums, products, powers, and integers. The transcendence of
implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of
π appear to be randomly distributed, but no proof of this conjecture has been found.

For thousands of years, mathematicians have attempted to extend their understanding of

, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of
for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate
with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated
to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706.

The invention of calculus soon led to the calculation of hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of

to many trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test supercomputers.

Because its definition relates to the circle,

is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as cosmology, fractals, thermodynamics, mechanics, and electromagnetism. In modern mathematical analysis, it is often instead defined without any reference to geometry; therefore, it also appears in areas having little to do with geometry, such as number theory and statistics. The ubiquity of
makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to
have been published, and record-setting calculations of the digits of

Irrational Number

Approximation of Pi

Digits of Pi

Is Pi Normal?

Transcendental Number.

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Sine, Cosine, and Tangent

Sine, Cosine, and Tangent

Inverse Sine, Cosine, and Tangent

Small Angle Approximations

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Angle

In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles are also formed by the intersection of two planes. These are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.

Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

Angles Around a Point

Angles On One Side of A Straight Line

Interior Angle

Exterior Angle

Alternate Interior Angles

Alternate Exterior Angles

Vertical Angles

Vertically Opposite Angles

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<![CDATA[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

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<![CDATA[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

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<![CDATA[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

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<![CDATA[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

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Number of Elements

Complement

Common Number Sets

Set Symbols

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