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#101 2022-09-22 00:40:44

ganesh
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Registered: 2005-06-28
Posts: 38,663

Re: Quantitative Aptitude and Higher Mathematics

Polynomials

Solving Polynomials

How Polynomials Behave

Polynomials: The Rule of Signs.

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.


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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#102 2022-09-23 00:05:22

ganesh
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Registered: 2005-06-28
Posts: 38,663

Re: Quantitative Aptitude and Higher Mathematics

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:

Exponents

Negative Exponents

Cubes and Cube Roots

nth root

Surds

Fractional Exponents.


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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#103 2022-09-24 00:56:44

ganesh
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Registered: 2005-06-28
Posts: 38,663

Re: Quantitative Aptitude and Higher Mathematics

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.


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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#104 Yesterday 00:29:20

ganesh
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Registered: 2005-06-28
Posts: 38,663

Re: Quantitative Aptitude and Higher Mathematics

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.


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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#105 Yesterday 01:24:15

ganesh
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Registered: 2005-06-28
Posts: 38,663

Re: Quantitative Aptitude and Higher Mathematics

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.


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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#106 Today 00:53:10

ganesh
Administrator
Registered: 2005-06-28
Posts: 38,663

Re: Quantitative Aptitude and Higher Mathematics

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.


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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