Higher Mathematics

Jai Ganesh · 2022-09-22 00:40:44

In this section, we show that factoring over Q (the rational numbers) and over Z (the integers) is essentially the same problem.

The content of a polynomial p ∈ Z[X], denoted "cont(p)", is, up to its sign, the greatest common divisor of its coefficients. The primitive part of p is primpart(p)=p/cont(p), which is a primitive polynomial with integer coefficients. This defines a factorization of p into the product of an integer and a primitive polynomial. This factorization is unique up to the sign of the content. It is a usual convention to choose the sign of the content such that the leading coefficient of the primitive part is positive.

For example,

is a factorization into content and primitive part.

Every polynomial q with rational coefficients may be written

,
where p ∈ Z[X] and c ∈ Z: it suffices to take for c a multiple of all denominators of the coefficients of q (for example their product) and p = cq. The content of q is defined as:

,
and the primitive part of q is that of p. As for the polynomials with integer coefficients, this defines a factorization into a rational number and a primitive polynomial with integer coefficients. This factorization is also unique up to the choice of a sign.

For example,

is a factorization into content and primitive part.

Gauss proved that the product of two primitive polynomials is also primitive (Gauss's lemma). This implies that a primitive polynomial is irreducible over the rationals if and only if it is irreducible over the integers. This implies also that the factorization over the rationals of a polynomial with rational coefficients is the same as the factorization over the integers of its primitive part. Similarly, the factorization over the integers of a polynomial with integer coefficients is the product of the factorization of its primitive part by the factorization of its content.

In other words, an integer GCD computation reduces the factorization of a polynomial over the rationals to the factorization of a primitive polynomial with integer coefficients, and the factorization over the integers to the factorization of an integer and a primitive polynomial.

Everything that precedes remains true if Z is replaced by a polynomial ring over a field F and Q is replaced by a field of rational functions over F in the same variables, with the only difference that "up to a sign" must be replaced by "up to the multiplication by an invertible constant in F". This reduces the factorization over a purely transcendental field extension of F to the factorization of multivariate polynomials over F.

Jai Ganesh · 2022-09-23 00:05:22

Exponentiation - I

Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent or power n, and pronounced as "b (raised) to the (power of) n". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:

The exponent is usually shown as a superscript to the right of the base. In that case, bn is called "b raised to the nth power", "b (raised) to the power of n", "the nth power of b", "b to the nth power", or most briefly as "b to the nth".

Starting from the basic fact stated above that, for any positive integer n,

is n occurrences of b all multiplied by each other, several other properties of exponentiation directly follow. In particular:

Fractional Exponents.

Jai Ganesh · 2022-09-24 00:56:44

Exponentiation - II

In other words, when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that

must be equal to 1, as follows. For any n, Dividing both sides by gives .

The fact that

can similarly be derived from the same rule. For example, . Taking the cube root of both sides gives

The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. Consider the question of what

should mean. In order to respect the "exponents add" rule, it must be the case that Dividing both sides by gives , which can be more simply written as , using the result from above that . By a similar argument, .

The properties of fractional exponents also follow from the same rule. For example, suppose we consider

and ask if there is some suitable exponent, which we may call r, such that . From the definition of the square root, we have that . Therefore, the exponent r must be such that . Using the fact that multiplying makes exponents add gives . The b on the right-hand side can also be written as , giving . Equating the exponents on both sides, we have . Therefore, , so .

The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.

Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.

Jai Ganesh · 2022-09-25 00:29:20

Complex Number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation

; every complex number can be expressed in the form a + bi, where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation

has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions -1 + 3i and -1 - 3i.

Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule

combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers also form a real vector space of dimension two, with {1, i} as a standard basis.

This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm.

In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.

Imaginary Numbers

Complex Numbers

Complex Number Multiplication

Complex Plane

Common Number Sets.

Jai Ganesh · 2022-09-25 01:24:15

Complex Number : Definition

An illustration of the complex number z = x + iy on the complex plane. The real part is x, and its imaginary part is y.

A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying

For example, 2 + 3i is a complex number.

This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation

is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation induces the equalities , and which hold for all integers k; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in i, again of the form a + bi with real coefficients a, b.

The real number a is called the real part of the complex number a + bi; the real number b is called its imaginary part. To emphasize, the imaginary part does not include a factor i; that is, the imaginary part is b, not bi.

Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate i, by the ideal generated by the polynomial

Notation

A real number a can be regarded as a complex number a + 0i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. As with polynomials, it is common to write a for a + 0i and bi for 0 + bi. Moreover, when the imaginary part is negative, that is, b = −|b| < 0, it is common to write a − |b|i instead of a + (−|b|)i; for example, for b = -4, 3 - 4i can be written instead of 3 + (-4)i.

Since the multiplication of the indeterminate i and a real is commutative in polynomials with real coefficients, the polynomial a + bi may be written as a + ib. This is often expedient for imaginary parts denoted by expressions, for example, when b is a radical.

The real part of a complex number z is denoted by Re(z),

, or ; the imaginary part of a complex number z is denoted by Im(z), , or For example,

The set of all complex numbers is denoted by

(blackboard bold) or C (upright bold).

In some disciplines, particularly in electromagnetism and electrical engineering, j is used instead of i as i is frequently used to represent electric current. In these cases, complex numbers are written as a + bj, or a + jb.

Jai Ganesh · 2022-09-26 00:53:10

Refresher - Calculus

Introduction to Derivatives

Slope of a Function at a Point

Derivatives as dy/dx

The derivative of a function is the rate of change of the function's output relative to its input value.

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.

Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

Jai Ganesh · 2022-09-27 00:48:37

Finding Maxima and Minima using Derivatives

Concave Upward and Downward

Inflection Points

Arc Length

Fourier Series

Limits (An Introduction)

Jai Ganesh · 2022-09-28 01:06:35

Variable with Exponents

Index Notation and Powers of 10

Exponents of Negative Numbers

When multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that

must be equal to 1, as follows. For any n, . Dividing both sides by gives .

The fact that

can similarly be derived from the same rule. For example, {\displaystyle (b^{1})^{3}=b^{1}\cdot b^{1}\cdot b^{1}=b^{1+1+1}=b^{3}}[/math]. Taking the cube root of both sides gives

The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. Consider the question of what

should mean. In order to respect the "exponents add" rule, it must be the case that . Dividing both sides by gives , which can be more simply written as , using the result from above that . By a similar argument, .

The properties of fractional exponents also follow from the same rule. For example, suppose we consider

and ask if there is some suitable exponent, which we may call r, such that . From the definition of the square root, we have that . Therefore, the exponent r must be such that . Using the fact that multiplying makes exponents add gives . The b on the right-hand side can also be written as , giving . Equating the exponents on both sides, we have . Therefore, , so

The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.

Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.

Jai Ganesh · 2022-09-29 01:04:05

Second Derivative Animation

Implicit Differentiation

Proof of the Derivatives of sin, cos, and tan.

Jai Ganesh · 2022-09-29 13:59:04

Refresher

Introduction to Integration

Integration Rules

Integration by Parts

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.

The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function. In this case, they are called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known.

Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral; it is more robust than Riemann's in the sense that a wider class of functions are Lebesgue-integrable.

Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.

Jai Ganesh · 2022-09-30 00:38:14

Special Binomial Products

Difference of Two Cubes.

Jai Ganesh · 2022-09-30 23:46:53

Trigonometric Identities

Power series expansion

Applying the differential equations to power series with indeterminate coefficients, one may deduce recurrence relations for the coefficients of the Taylor series of the sine and cosine functions. These recurrence relations are easy to solve, and give the series expansions

The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane.

Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions, that is functions that are holomorphic in the whole complex plane, except some isolated points called poles. Here, the poles are the numbers of the form

for the tangent and the secant, or for the cotangent and the cosecant, where k is an arbitrary integer.

Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence. Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets.

Jai Ganesh · 2022-10-01 02:24:39

Trigonometry Identities

In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.

These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

The basic relationship between the sine and cosine is given by the Pythagorean identity:

where means and means

This can be viewed as a version of the Pythagorean theorem, and follows from the equation

for the unit circle. This equation can be solved for either the sine or the cosine:

where the sign depends on the quadrant of ..

Dividing this identity by

, or both yields the following identities:

..

Jai Ganesh · 2022-10-02 02:02:16

Remainder Theorem and Factor Theorem

Polynomial remainder theorem

In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial

by a linear polynomial is equal to In particular, is a divisor of if and only if a property known as the factor theorem.

Jai Ganesh · 2022-10-03 00:33:21

Factor Theorem

In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.

The factor theorem states that a polynomial

has a factor if and only if (i.e. is a root).

Factorization of polynomials

Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent.

The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros intact, thus producing a lower degree polynomial whose zeros may be easier to find. Abstractly, the method is as follows:

1. "Guess" a zero

of the polynomial . (In general, this can be very hard, but math textbook problems that involve solving a polynomial equation are often designed so that some roots are easy to discover.)

2. Use the factor theorem to conclude that

is a factor of .

3. Compute the polynomial Compute the polynomial

, for example using polynomial long division or synthetic division, for example using polynomial long division or synthetic division.

4. Conclude that any root

of is a root of . Since the polynomial degree of is one less than that of , it is "simpler" to find the remaining zeros by studying .

Jai Ganesh · 2022-10-04 00:06:16

Calculus

Continuous Functions

Differentiable

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.

Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their definition is the basis of topology.

A stronger form of continuity is uniform continuity. In order theory, especially in domain theory, a related concept of continuity is Scott continuity.

As an example, the function H(t) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M(t) denoting the amount of money in a bank account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.

Jai Ganesh · 2022-10-06 00:52:45

Greatest Common Divisor and Least Common Multiple

Greatest Common Divisor.

Least Common Multiple.

Greatest Common Divisor

In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted

. For example, the GCD of 8 and 12 is 4, that is,

In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor (hcf), etc. Historically, other names for the same concept have included greatest common measure.

Jai Ganesh · 2022-10-08 00:24:21

Least Common Miltiple

In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) as 0 for all a, since 0 is the only common multiple of a and 0.

The lcm is the "lowest common denominator" (lcd) that can be used before fractions can be added, subtracted or compared.

The least common multiple of more than two integers a, b, c, . . . , usually denoted by lcm(a, b, c, . . .), is also well defined: It is the smallest positive integer that is divisible by each of a, b, c, . .

Overview

A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of -5 and -2 as well.

Notation

The least common multiple of two integers a and b is denoted as lcm(a, b). Some older textbooks use [a, b].

Example

Multiples of 4 are:

Multiples of 6 are:

Common multiples of 4 and 6 are the numbers that are in both lists:

In this list, the smallest number is 12. Hence, the least common multiple is 12.

Jai Ganesh · 2022-10-08 13:31:53

Polynomials - Addition and Subtraction.

Jai Ganesh · 2022-10-10 01:20:15

Polynomials - Multiplication

Multiplying Polynomials

Multiplication

Polynomials can also be multiplied. To expand the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other.

For example,

and

Carrying out the multiplication in each term produces

Combining similar terms yields

which can be simplified to

As in the example, the product of polynomials is always a polynomial.

Jai Ganesh · 2022-10-11 02:15:18

Polynomials - Division

Polynomials

Polynomials - Long Division

How Polynomials Behave

Jai Ganesh · 2022-10-13 01:58:39

Prime and Composite Numbers

Fundamental Theorem of Arithmetic

Whole Numbers and Integers

Commutative, Associative and Distributive Laws.

Jai Ganesh · 2022-10-15 01:37:00

Special Binomial Products

Binomial (polynomial)

In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of sparse polynomial after the monomials.

Definition

A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate (also known as a univariate binomial) can be written in the form

where a and b are numbers, and m and n are distinct nonnegative integers and x is a symbol which is called an indeterminate or, for historical reasons, a variable. In the context of Laurent polynomials, a Laurent binomial, often simply called a binomial, is similarly defined, but the exponents m and n may be negative.

More generally, a binomial may be written as:

.

Examples:

.

Jai Ganesh · 2022-10-17 00:57:40

Interest

Compound Interest

Compound Interest Formula Derivations

Compound Interest: Periodic Compounding

Compound Interest Calculator.

Jai Ganesh · 2022-10-19 00:52:09

Complex Plane

Complex Number Multiplication

Math Is Fun Forum

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Re: Higher Mathematics

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