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#76 2022-08-28 02:58:07

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Statistics - I

Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.

When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation.

Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, and inferential statistics, which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation). Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena.

A standard statistical procedure involves the collection of data leading to a test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors (null hypothesis is falsely rejected giving a "false positive") and Type II errors (null hypothesis fails to be rejected and an actual relationship between populations is missed giving a "false negative"). Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis.

Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic (bias), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also occur. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems.

Definitions

Range

Mean

How to find the Mean

How to Find the Median Value

How to Find the Mode or Modal Value

Mean, Median and Mode from Grouped Frequencies

Frequency Distribution

Grouped Frequency Distribution

Finding a Central Value.


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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#77 2022-08-29 00:57:14

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Statistics - II

Least Squares Regression

Variance

Standard Deviation

Standard Deviation and Variance

In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

Standard Deviation

Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case Greek letter σ (sigma), for the population standard deviation, or the Latin letter s, for the sample standard deviation.

The standard deviation of a random variable, sample, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same unit as the data.

The standard deviation of a population or sample and the standard error of a statistic (e.g., of the sample mean) are quite different, but related. The sample mean's standard error is the standard deviation of the set of means that would be found by drawing an infinite number of repeated samples from the population and computing a mean for each sample. The mean's standard error turns out to equal the population standard deviation divided by the square root of the sample size, and is estimated by using the sample standard deviation divided by the square root of the sample size. For example, a poll's standard error (what is reported as the margin of error of the poll), is the expected standard deviation of the estimated mean if the same poll were to be conducted multiple times. Thus, the standard error estimates the standard deviation of an estimate, which itself measures how much the estimate depends on the particular sample that was taken from the population.

In science, it is common to report both the standard deviation of the data (as a summary statistic) and the standard error of the estimate (as a measure of potential error in the findings). By convention, only effects more than two standard errors away from a null expectation are considered "statistically significant", a safeguard against spurious conclusion that is really due to random sampling error.

When only a sample of data from a population is available, the term standard deviation of the sample or sample standard deviation can refer to either the above-mentioned quantity as applied to those data, or to a modified quantity that is an unbiased estimate of the population standard deviation (the standard deviation of the entire population).


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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#78 2022-08-30 00:13:30

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Statistics - III

Standard Deviation Formulas

Binomial Distribution - Definition

Binomial Distribution

Normal Distribution.

Normal distribution

In statistics, a normal distribution (also known as Gaussian, Gauss, or Laplace–Gauss distribution) is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

The parameter

  is the mean or expectation of the distribution (and also its median and mode), while the parameter
  is its standard deviation. The variance of the distribution is
. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.

Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors, often have distributions that are nearly normal.

Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of a fixed collection of normal deviates is a normal deviate. Many results and methods, such as propagation of uncertainty and least squares parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed.

A normal distribution is sometimes informally called a bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions).

The univariate probability distribution is generalized for vectors in the multivariate normal distribution and for matrices in the matrix normal distribution.


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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#79 2022-08-31 00:38:37

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

A quincunx is a geometric pattern consisting of five points arranged in a cross, with four of them forming a square or rectangle and a fifth at its center. The same pattern has other names, including "in saltire" or "in cross" in heraldry (depending on the orientation of the outer square), the five-point stencil in numerical analysis, and the five dots tattoo. It forms the arrangement of five units in the pattern corresponding to the five-spot on six-sided dice, playing cards, and dominoes.

Quincunx

Quincunx Explained

Standard Normal Distribution 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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#80 2022-09-01 00:38:57

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Inequalities

Solving Inequalities.

In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities:

* The notation a < b means that a is less than b.
* The notation a > b means that a is greater than b.

In either case, a is not equal to b. These relations are known as strict inequalities, meaning that a is strictly less than or strictly greater than b. Equivalence is excluded.

In contrast to strict inequalities, there are two types of inequality relations that are not strict:

* The notation a ≤ b or a ⩽ b means that a is less than or equal to b (or, equivalently, at most b, or not greater than b).
* The notation a ≥ b or a ⩾ b means that a is greater than or equal to b (or, equivalently, at least b, or not less than b).

The relation not greater than can also be represented by a ≯ b, the symbol for "greater than" bisected by a slash, "not". The same is true for not less than and a ≮ b.

The notation a ≠ b means that a is not equal to b; this inequation sometimes is considered a form of strict inequality. It does not say that one is greater than the other; it does not even require a and b to be member of an ordered set.

In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude.

* The notation a ≪ b means that a is much less than b.
* The notation a ≫ b means that a is much greater than b.

This implies that the lesser value can be neglected with little effect on the accuracy of an approximation (such as the case of ultrarelativistic limit in physics).

In all of the cases above, any two symbols mirroring each other are symmetrical; a < b and b > a are equivalent, etc.


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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#81 2022-09-02 01:13:57

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Inquality

Equal, Less, Greater

Inequalities between means

There are many inequalities between means. For example, for any positive numbers a1, a2, …, an we have H ≤ G ≤ A ≤ Q, where

    (harmonic mean),

    (geometric mean),

    (arithmetic mean),

    (quadratic mean).

Inequality of arithmetic and geometric means

In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which case they are both that number).

The simplest non-trivial case – i.e., with more than one variable – for two non-negative numbers x and y, is the statement that

with equality if and only if x = y. This case can be seen from the fact that the square of a real number is always non-negative (greater than or equal to zero) and from the elementary case

of the binomial formula:

Hence

, with equality precisely when
, i.e. x = y. The AM–GM inequality then follows from taking the positive square root of both sides and then dividing both sides by 2.

For a geometrical interpretation, consider a rectangle with sides of length x and y, hence it has perimeter 2x + 2y and area xy. Similarly, a square with all sides of length

has the perimeter
and the same area as the rectangle. The simplest non-trivial case of the AM–GM inequality implies for the perimeters that
and that only the square has the smallest perimeter amongst all rectangles of equal area.

Extensions of the AM–GM inequality are available to include weights or generalized means.


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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#82 2022-09-02 21:33:49

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Commutative, Associative, and Distributive Laws

Long Division with Remainders

Long Division to Decimal Places

Long Division Worksheets.

Long Division

In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu-Arabic numerals (Positional notation) that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps.

As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps. The abbreviated form of long division is called short division, which is almost always used instead of long division when the divisor has only one digit. Chunking (also known as the partial quotients method or the hangman method) is a less mechanical form of long division prominent in the U.K which contributes to a more holistic understanding of the division process.

While related algorithms have existed since the 12th century, the specific algorithm in modern use was introduced by Henry Briggs c. 1600.


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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#83 2022-09-03 00:56:36

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Trigonometry Refresher

Introduction

Trigonometric Identities

Proof of the Derivatives of sin, cos, and tan

Trigonometry Index.

Trigonometry

Trigonometry (from Ancient Greek 'triangle', and 'measure') is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.

Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.

Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation.


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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#84 2022-09-04 00:25:59

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Modulo (mathematics)

In mathematics, the term modulo ("with respect to a modulus of", the Latin ablative of modulus which itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor. It was initially introduced into mathematics in the context of modular arithmetic by Carl Friedrich Gauss in 1801. Since then, the term has gained many meanings—some exact and some imprecise (such as equating "modulo" with "except for"). For the most part, the term often occurs in statements of the form:

A is the same as B modulo C

which means

A and B are the same—except for differences accounted for or explained by C.

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in 7 + 8 = 15, but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic modulo 12. In terms of the definition below, 15 is congruent to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock.

Modulo Operation (Definition)

Modulo Operation.


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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#85 2022-09-05 00:57:17

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Coprime integers

In mathematics, two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides a does not divide b, and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also a is prime to b or a is coprime with b.

The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition.

Coprime

Coprime Calculator

Greatest Common Factor

Prime Factorization Calculator

Prime Factorization.


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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#86 2022-09-06 00:30:13

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Roman Numeral - Definition

Roman Numerals

Conversion

Decimal Numbers

Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet. Modern style uses seven symbols, each with a fixed integer value:

Symbol  :  I  :  V  :  X  :  L  :  C  :  D  :  M
Value  :  1  :  5  :  10  :  50  :  100  :  500  :  1000

The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced by Arabic numerals; however, this process was gradual, and the use of Roman numerals persists in some applications to this day.

One place they are often seen is on clock faces. For instance, on the clock of Big Ben (designed in 1852), the hours from 1 to 12 are written as:

I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII

The notations IV and IX can be read as "one less than five" (4) and "one less than ten" (9), although there is a tradition favouring representation of "4" as "IIII" on Roman numeral clocks.[2]

Other common uses include year numbers on monuments and buildings and copyright dates on the title screens of movies and television programs. MCM, signifying "a thousand, and a hundred less than another thousand", means 1900, so 1912 is written MCMXII. For the years of this century, MM indicates 2000. The current year is MMXXII (2022).


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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#87 2022-09-07 00:34:10

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.

For example,

is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the following ordered triple.

since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually.

In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system, a helpful technique when making a mathematical model or computer simulation of a relatively complex system.

Very often, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Integer linear programming is a collection of methods for finding the "best" integer solution (when there are many). Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure.

Linear Equations

Simultaneous Equations - Definition

Systems of Linear Equations.


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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#88 2022-09-09 00:46:56

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Lengths From Very Small To Very Large

Definition - Googol

Scientific Notation - Definition

Scintific Notation - Elaborate

Infinity

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.

Numbers expressible in scientific notation:

Approximate number of atoms in the observable universe =

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.


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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#89 2022-09-10 00:28:01

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Cardinal, Ordinal, and Nominal Numbers

Ordinal number

In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets.

A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number

that is greater than every natural number, along with ordinal numbers
etc., which are even greater than
.

A linear order such that every subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to an initial segment of the other. So, ordinal numbers exist, and are essentially unique.

Ordinal numbers are distinct from cardinal numbers, which measure the size of sets. Although the distinction between ordinals and cardinals is not always apparent on finite sets (one can go from one to the other just by counting labels), they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although none of these operations are commutative.

Ordinals were introduced by Georg Cantor in 1883 in order to accommodate infinite sequences and classify derived sets, which he had previously introduced in 1872 - while studying the uniqueness of trigonometric series.

Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The transfinite cardinal numbers, often denoted using the Hebrew symbol

  (aleph) followed by a subscript, describe the sizes of infinite sets.

Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set—something that cannot happen with proper subsets of finite sets.

There is a transfinite sequence of cardinal numbers:

There is a transfinite sequence of cardinal numbers:

This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.

Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra and mathematical analysis. In category theory, the cardinal numbers form a skeleton of the category of sets.

Nominal number

Nominal numbers are numerals used as labels to identify items uniquely. Importantly, the actual values of the numbers which these numerals represent are less relevant, as they do not indicate quantity, rank, or any other measurement.

Labelling referees Smith and Jones as referees "1" and "2" is a use of nominal numbers. Any set of numbers (a subset of the natural numbers) will be consistent labels as long as a distinct number is uniquely used for each distinct term which needs to be labelled. Nonetheless, sequences of integers may naturally be used as the simplest way to begin labelling; for example, 1, 2, 3, and so on.


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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#90 2022-09-11 00:36:34

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Iteration

Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration.

In mathematics and computer science, iteration (along with the related technique of recursion) is a standard element of algorithms.

In mathematics

In mathematics, iteration may refer to the process of iterating a function, i.e. applying a function repeatedly, using the output from one iteration as the input to the next. Iteration of apparently simple functions can produce complex behaviors and difficult problems – for examples, see the Collatz conjecture and juggler sequences.

Another use of iteration in mathematics is in iterative methods which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method is an example of an iterative method. Manual calculation of a number's square root is a common use and a well-known example.

Relationship with recursion

In algorithmic situations, recursion and iteration can be employed to the same effect. The primary difference is that recursion can be employed as a solution without prior knowledge as to how many times the action will have to repeat, while a successful iteration requires that foreknowledge.

Some types of programming languages, known as functional programming languages, are designed such that they do not set up block of statements for explicit repetition as with the for loop. Instead, those programming languages exclusively use recursion. Rather than call out a block of code to be repeated a pre-defined number of times, the executing code block instead "divides" the work to be done into a number separate pieces, after which the code block executes itself on each individual piece. Each piece of work will be divided repeatedly until the "amount" of work is as small as it can possibly be, at which point algorithm will do that work very quickly. The algorithm then "reverses" and reassembles the pieces into a complete whole.

The classic example of recursion is in list-sorting algorithms such as merge sort. The merge sort recursive algorithm will first repeatedly divide the list into consecutive pairs; each pair is then ordered, then each consecutive pair of pairs, and so forth until the elements of the list are in the desired order.

The code below is an example of a recursive algorithm in the Scheme programming language that will output the same result as the pseudocode under the previous heading.

Iteration - Definition

Square Root

Scientific Calculator

Factorial

Pi

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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#91 2022-09-12 01:05:53

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Logarithms

Exponents, Roots and Logarithms.

Logarithm

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g. since

, the "logarithm base 10" of 1000 is 3, or
. The logarithm of x to base b is denoted as
, or without parentheses,
, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

The logarithm base 10 (that is b = 10) is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e (that is b ≈ 2.718) as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative. The binary logarithm uses base 2 (that is b = 2) and is frequently used in computer science.

Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors:

provided that b, x and y are all positive and b ≠ 1. The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century, and who also introduced the letter e as the base of natural logarithms.

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting.

The concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete logarithm is the multi-valued inverse of the exponential function in finite groups; it has uses in 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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#92 2022-09-13 00:42:37

ganesh
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Re: Quantitative Aptitude and Higher Mathematics

Decimals (Recap)

Decimals

Terminating Decimals - Definition

Recurring Decimals

Irrational Numbers

Algebraic Number

Transcendental Numbers.

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.


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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#93 2022-09-14 00:17:19

ganesh
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Posts: 38,662

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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#94 2022-09-15 00:41:17

ganesh
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Registered: 2005-06-28
Posts: 38,662

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.


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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#95 2022-09-16 00:05:52

ganesh
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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


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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#96 2022-09-17 00:48:41

ganesh
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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.


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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#97 2022-09-18 02:05:31

ganesh
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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.


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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#98 2022-09-19 00:35:33

ganesh
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Posts: 38,662

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 =


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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#99 2022-09-20 00:34:15

ganesh
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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:


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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#100 2022-09-21 00:24:22

ganesh
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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

Polynomials - Long Division.


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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