<![CDATA[Math Is Fun Forum / Quantitative Aptitude and Higher Mathematics]]> 2022-09-26T12:53:10Z FluxBB https://www.mathisfunforum.com/viewtopic.php?id=27202 <![CDATA[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.

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<![CDATA[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.

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<![CDATA[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

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<![CDATA[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.

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<![CDATA[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

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

Solving Polynomials

How Polynomials Behave

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.

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

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate x is

An example in three variables is

Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry.

Polynomials

Adding and Subtracting Polynomials

Multiplying Polynomials

Dividing Polynomials

Polynomials - Long Multiplication

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Moser's number

It has been proven that in Conway chained arrow notation,

and, in Knuth's up-arrow notation,

Therefore, Moser's number, although incomprehensibly large, is vanishingly small compared to Graham's number:

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Steinhaus–Moser notation

In mathematics, Steinhaus–Moser notation is a notation for expressing certain large numbers. It is an extension (devised by Leo Moser) of Hugo Steinhaus's polygon notation.

Definitions

* n in a triangle a number n in a triangle means

.
* n in a square a number n in a square is equivalent to "the number n inside n triangles, which are all nested."
* n in a pentagon a number n in a pentagon is equivalent with "the number n inside n squares, which are all nested."
etc.: n written in an (m + 1)-sided polygon is equivalent with "the number n inside n nested m-sided polygons". In a series of nested polygons, they are associated inward. The number n inside two triangles is equivalent to
inside one triangle, which is equivalent to
raised to the power of
.

Steinhaus defined only the triangle, the square, and the circle n in a circle, which is equivalent to the pentagon defined above.

Special values

Steinhaus defined:

mega is the number equivalent to 2 in a circle: ②
megiston is the number equivalent to 10 in a circle: ⑩
Moser's number is the number represented by "2 in a megagon". Megagon is here the name of a polygon with "mega" sides (not to be confused with the polygon with one million sides).

Alternative notations:

use the functions square(x) and triangle(x)
let M(n, m, p) be the number represented by the number n in m nested p-sided polygons; then the rules are:

and
mega =

megiston =

moser =

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Graham's Number

Graham's number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is much larger than many other large numbers such as Skewes's number and Moser's number, both of which are in turn much larger than a googolplex. As with these, it is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe. Thus Graham's number cannot be expressed even by physical universe-scale power towers of the form

However, Graham's number can be explicitly given by computable recursive formulas using Knuth's up-arrow notation or equivalent, as was done by Ronald Graham, the number's namesake. As there is a recursive formula to define it, it is much smaller than typical busy beaver numbers. Though too large to be computed in full, the sequence of digits of Graham's number can be computed explicitly via simple algorithms; the last thirteen digits are ...7262464195387. With Knuth's up-arrow notation, Graham's number is

, where

Graham's number was used by Graham in conversations with popular science writer Martin Gardner as a simplified explanation of the upper bounds of the problem he was working on. In 1977, Gardner described the number in Scientific American, introducing it to the general public. At the time of its introduction, it was the largest specific positive integer ever to have been used in a published mathematical proof. The number was described in the 1980 Guinness Book of World Records, adding to its popular interest. Other specific integers (such as TREE(3)) known to be far larger than Graham's number have since appeared in many serious mathematical proofs, for example in connection with Harvey Friedman's various finite forms of Kruskal's theorem. Additionally, smaller upper bounds on the Ramsey theory problem from which Graham's number derived have since been proven to be valid.

Definition

Using Knuth's up-arrow notation, Graham's number G (as defined in Gardner's Scientific American article) is

where the number of arrows in each subsequent layer is specified by the value of the next layer below it; that is,

where

and where a superscript on an up-arrow indicates how many arrows there are. In other words, G is calculated in 64 steps: the first step is to calculate
with four up-arrows between 3s; the second step is to calculate
with
with
up-arrows between 3s; and so on, until finally calculating
with
up-arrows between 3s.

Equivalently,

and the superscript on f indicates an iteration of the function, e.g.,
. Expressed in terms of the family of hyperoperations
, the function f is the particular sequence
, which is a version of the rapidly growing Ackermann function A(n, n). (In fact,
for all n.) The function f can also be expressed in Conway chained arrow notation as
, and this notation also provides the following bounds on G:

Magnitude

To convey the difficulty of appreciating the enormous size of Graham's number, it may be helpful to express—in terms of exponentiation alone—just the first term (

) of the rapidly growing 64-term sequence. First, in terms of tetration
) alone:

where the number of 3s in the expression on the right is

Now each tetration
) operation reduces to a power tower
according to the definition

where there are X 3s.

Thus,

becomes, solely in terms of repeated "exponentiation towers",

Thus,

becomes, solely in terms of repeated "exponentiation towers",

and where the number of 3s in each tower, starting from the leftmost tower, is specified by the value of the next tower to the right.

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Conway chained arrow notation

Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers. It is simply a finite sequence of positive integers separated by rightward arrows, e.g.

As with most combinatorial notations, the definition is recursive. In this case the notation eventually resolves to being the leftmost number raised to some (usually enormous) integer power.

Definition and overview

A "Conway chain" is defined as follows:

* Any positive integer is a chain of length 1.
* A chain of length n, followed by a right-arrow → and a positive integer, together form a chain of length n+1.

Any chain represents an integer, according to the six rules below. Two chains are said to be equivalent if they represent the same integer.

Let a,b,c denote positive integers and let

denote the unchanged remainder of the chain. Then:

1. An empty chain (or a chain of length 0) is equal to 1.
2. The chain p represents the number p.
3. The chain

represents the number

4. The chain
represents the number
(see Knuth's up-arrow notation)
5. The chain
represents the same number as the chain

6. Else, the chain
represents the same number as the chain
.

Properties

1. A chain evaluates to a perfect power of its first number
Therefore,

is equal to 1
is equivalent to X
is equal to 4
is equivalent to
(not to be confused with
)

Interpretation

One must be careful to treat an arrow chain as a whole. Arrow chains do not describe the iterated application of a binary operator. Whereas chains of other infixed symbols (e.g. 3 + 4 + 5 + 6 + 7) can often be considered in fragments (e.g. (3 + 4) + 5 + (6 + 7)) without a change of meaning (see associativity), or at least can be evaluated step by step in a prescribed order, e.g.

from right to left, that is not so with Conway's arrow chains.

For example:

The fourth rule is the core: A chain of 4 or more elements ending with 2 or higher becomes a chain of the same length with a (usually vastly) increased penultimate element. But its ultimate element is decremented, eventually permitting the second rule to shorten the chain. After, to paraphrase Knuth, "much detail", the chain is reduced to three elements and the third rule terminates the recursion.

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Knuth's up-arrow notation

In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.

In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc.

Various notations have been used to represent hyperoperations. One such notation is

. Another notation is
, an infix notation which is convenient for ASCII. The notation
is known as 'square bracket notation'.

Knuth's up-arrow notation

is an alternative notation. It is obtained by replacing [n] in the square bracket notation by n-2 arrows.

For example:

the single arrow

represents exponentiation (iterated multiplication)

the double arrow
represents tetration (iterated exponentiation)

the triple arrow
represents pentation (iterated tetration)

The general definition of the up-arrow notation is as follows (for

Here,
stands for n arrows, so for example

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

Large numbers are significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical mechanics. They are typically large positive integers, or more generally, large positive real numbers, but may also be other numbers in other contexts. The study of nomenclature and properties of large numbers is googology.

Examples

Googol =

Centillion =
or
, depending on number naming system
Millinillion =
or
, depending on number naming system

The largest known Mersenne prime =

(as of December 21, 2018)
Googolplex =

Skewes's numbers: the first is approximately
, the second

Graham's number, larger than what can be represented even using power towers (tetration). However, it can be represented using Knuth's up-arrow notation
Kruskal's tree theorem is a sequence relating to graphs. TREE(3) is larger than Graham's number.
Rayo's number is a large number named after Agustín Rayo which has been claimed to be the largest named number. It was originally defined in a "big number duel" at MIT on 26 January 2007.

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

Metric Numbers

Activity: Count to a Billion

Billion - Definition

Giga- Definition

Trillion - Definiton

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<![CDATA[Re: Quantitative Aptitude and Higher Mathematics]]> Decimals (Recap)

Decimals

Terminating Decimals - Definition

Recurring Decimals

Irrational Numbers

Algebraic Number

Transcendental number

In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are

and e.

Though only a few classes of transcendental numbers are known — partly because it can be extremely difficult to show that a given number is transcendental — transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers comprise a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping rational, algebraic non-rational and transcendental real numbers. For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation

. The golden ratio (denoted
or
) is another irrational number that is not transcendental, as it is a root of the polynomial equation
. The quality of a number being transcendental is called transcendence.

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