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Functions

A function is a relationship between two sets of numbers. We may think of this as a mapping; a function maps a number in one set to a number in another set. Notice that a function maps values to one and only one value. Two values in one set could map to one value, but one value must never map to two values: that would be a relation, not a function.

For example, if we write (define) a function as:

f(x) = x2

then we say:

'f of x equals x squared'

and we have

f( - 1) = 1

f(1) = 1

f(7) = 49

f(1 / 2) = 1 / 4

f(4) = 16

and so on.

Exponentiation

Exponentiation is a mathematical operation, written as aⁿ, involving two numbers, the base 'a' and the exponent 'n'.

When n is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of n factors of a:

Just as multiplication by a positive integer corresponds to repeated addition:

The exponent is usually shown as a superscript to the right of the base.

The power aⁿ can be defined also when n is a negative integer, for nonzero a. No natural extension to all real a and n exists, but when the base a is a positive real number, aⁿ can be defined for all real and even complex exponents n via the exponential function e^z.

Trigonometric functions can be expressed in terms of complex exponentiation.

Exponentiation where the exponent is a matrix is used for solving systems of linear differential equations.

Graphs of y=ax for various bases a: base 10 (green), base e (red), base 2 (blue), and base ½ (cyan). Each curve passes through the point (0,1) because any nonzero number raised to the power 0 is 1. At x=1, the y-value equals the base because any number raised to the power 1 is itself.

Algebra

Algebra is the branch of mathematics that concerns with the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, and number theory, algebra is one of the main branches of pure mathematics.

Elementary algebra introduces the concept of variables representing numbers. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition. This can be done for a variety of reasons, including equation solving. Algebra is much broader than elementary algebra. Addition and multiplication can be generalized and their precise definitions lead to structures such as groups, rings and fields, studied in the area of mathematics called abstract algebra.

In 1545, the Italian mathematician Girolamo Cardano published Ars magna -The great art, a 40-chapter masterpiece in which he gave for the first time a method for solving the general quadratic equation.

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Decimal Representation

The decimal numeral system (also called base ten or occasionally denary) has ten as its base. It is the numerical base most widely used by modern civilizations.

Decimal notation often refers to a base-10 positional notation. 

Decimals also refer to decimal fractions, either separately or in contrast to vulgar fractions. In this context, a decimal is a tenth part, and decimals become a series of nested tenths. There was a notation in use like 'tenth-metre', meaning the tenth decimal of the metre, currently an Angstrom. The contrast here is between decimals and vulgar fractions, and decimal divisions and other divisions of measures, like the inch. It is possible to follow a decimal expansion with a vulgar fraction; this is done with the recent divisions of the troy ounce, which has three places of decimals, followed by a trinary place.

A decimal representation of a non-negative real number r is an expression of the form

where a0 is a nonnegative integer, and a1, a2, … are integers satisfying 0 ≤ ai ≤ 9, called the digits of the decimal representation. The sequence of digits specified may be finite, in which case any further digits ai are assumed to be 0. Some authors forbid decimal representations with an infinite sequence of digits 9.[1] This restriction still allows a decimal representation for each non-negative real number, but additionally makes such a representation unique. The number defined by a decimal representation is often written more briefly as

 

That is to say, a0 is the integer part of r, not necessarily between 0 and 9, and a1, a2, a3, … are the digits forming the fractional part of r.

Both notations above are, by definition, the following limit of a sequence:

.

Number Line

In basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point.[1] Often the integers are shown as specially-marked points evenly spaced on the line. Although this image only shows the integers from −9 to 9, the line includes all real numbers, continuing "forever" in each direction, and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers.

It is divided into two symmetric halves by the origin, i.e. the number zero.

In advanced mathematics, the expressions real number line, or real line are typically used to indicate the above-mentioned concept that every point on a straight line corresponds to a single real number, and vice versa.

The number line is usually represented as being horizontal. Customarily, positive numbers lie on the right side of zero, and negative numbers lie on the left side of zero. An arrowhead on either end of the drawing is meant to suggest that the line continues indefinitely in the positive and negative real numbers, denoted by  . The real numbers consist of irrational numbers and rational numbers, as well as the integers, whole numbers, and the natural numbers (the counting numbers).

A line drawn through the origin at right angles to the real number line can be used to represent the imaginary numbers. This line, called imaginary line, extends the number line to a complex number plane, with points representing complex numbers.

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ABSOLUTE VALUE

In mathematics, the absolute value (or modulus) |a| of a real number a is a's numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.

Generalizations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

For any real number a the absolute value or modulus of a is denoted by | a | (a vertical bar on each side of the quantity) and is defined as

Since, the complex numbers are not ordered, the definition given above for the real absolute value cannot be directly generalized for a complex number. However the identity given in equation    :

can be seen as motivating the following definition.

The graph of the absolute value function for real numbers.

The absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its complex conjugate z have the same absolute value.

Estimation problems

To save time on the GRE, you should get comfortable with estimating.  Even if estimating doesn’t give you the 100% accurate answer, it generally narrows it down to one obvious choice (if you’re good at estimating and round up and down appropriately).  Some questions even tell you to approximate, so there really is no point calculating the precise answer there.

Numerical Estimations

Practice estimating with percentages.  This will save you a lot of time, particular on the questions with charts and graphs. I tend to like figuring out 1%, 5% or 10% represents and working from there depending on the question. 

Visual Estimations

Visual estimations usually work for things like graphs or simple diagrams. The important lesson in visual estimations is not to do it for triangles.  You should always assume that triangles are never drawn to scale and when looking at diagrams of triangles, you should only apply rules of triangles e.g. sum of interior angles is 180, isosceles triangles have two equal angles and two equal sides etc.

PROPORTION

In mathematics, two quantities are proportional if they vary in such a way that one of them is a constant multiple of the other.

The mathematical symbol '∝' is used to indicate that two values are proportional. For example, A ∝ B.

If the two or more ratio quantities encompass all of the quantities in a particular situation, for example two apples and three oranges in a fruit basket containing no other types of fruit, it could be said that "the whole" contains five parts, made up of two parts apples and three parts oranges. In this case, 2/5 , or 40% of the whole are apples and 3/5, or 60% of the whole are oranges. This comparison of a specific quantity to "the whole" is sometimes called a proportion. Proportions are sometimes expressed as percentages as demonstrated above.

Direct proportionality

Given two variables x and y, y is (directly) proportional to x (x and y vary directly, or x and y are in direct variation) if there is a non-zero constant k such that

y = kx

The relation is often denoted

y ∝ x

and the constant ratio

k = y/x

is called the proportionality constant or constant of proportionality.

Inverse proportionality

As noted in the definition above, two proportional variables are sometimes said to be directly proportional. This is done so as to contrast direct proportionality with inverse proportionality.

Two variables are inversely proportional (or varying inversely, or in inverse variation, or in inverse proportion or reciprocal proportion) if one of the variables is directly proportional with the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant. It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that

y = k/x

PERCENTAGE

In mathematics, a percentage is a way of expressing a number as a fraction of. It is often denoted using the percent sign, "%", or the abbreviation "pct". For example, 45% is equal to 45/100, or 0.45.

Percentages are used to express how large/small one quantity is, relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity, which should be greater than zero.

The fundamental concept to remember when performing calculations with percentages is that the percent symbol can be treated as being equivalent to the pure number constant 1 / 100 = 0.01 , for example 35% of 300 can be written as (35/100) × 300 = 105.

To find the percentage that a single unit represents out of a whole of N units, divide 100% by N.

For instance, if you have 1250 apples, and you want to find out what percentage of these 1250 apples a single apple represents, 100%/1250 = (100/1250)% provides the answer of 0.08%. So, if you give away one apple, you have given away 0.08% of the apples you had. Then, if instead you give away 100 apples, you have given away 100 × 0.08% = 8% of your 1250 apples.

RATIO

In mathematics, a ratio is a relationship between two numbers of the same kind[1] (i.e., objects, persons, students, spoonfuls, units of whatever identical dimension), usually expressed as "a to b" or a:b.

The ratio of numbers A and B can be expressed as:[4]

• the ratio of A to B
• A is to B
• A:B

The numbers A and B are sometimes called terms with A being the antecedent and B being the consequent.

The quantities being compared in a ratio might be physical quantities such as speed, or may simply refer to amounts of particular objects.

A common example of the latter case is the weight ratio of water to cement used in concrete, which is commonly stated as 1:4. This means that the weight of cement used is four times the weight of water used. It does not say anything about the total amounts of cement and water used, nor the amount of concrete being made.

Older televisions have a 4:3 ratio, which means that the height is 3/4 of the width. Widescreen TVs have a 16:9 ratio, which means that the width is nearly double the height.

Exponents and radicals (Understanding Radicals)

Once you have the idea of exponents down and can solve problems using circles, marbles or multiplication, it is time to work with radicals. Where exponents usually make numbers bigger, radicals are their opposite and make numbers smaller. When solving equations, radicals are used to get rid of exponents, just like division is used to get rid of multiplication. This example shows the third root of 8. This means, which number appears three times as a factor of 8 or what put into 3 multiplication blanks would get an answer of 8. If you think about it in terms of exponents, it is saying what number multiplied by itself 3 times makes 8.

Exponents and radicals (Understanding Exponents)

Exponents and radicals are a core skill when it comes to transitioning to algebra, and many students struggle with the way that the concepts are presented in texts. Some have a hard time differentiating between exponents and multiplication as well as radicals and division.

Understanding Exponents

Using visual aids such as marbles is a great way to wrap your head around the idea of exponents. If you have two marbles in a group and there are three groups, then there are six marbles all together. This is like saying 3 groups of 2 makes 6, a multiplication problem. If you think, ___ x___ , the numbers represent what goes in the blanks. Exponents are a special type of multiplication where the exponent represents how many blanks there are instead of what goes in them. The base is the big number and the exponent, or power, is the small number. If you have 2 to the fourth power, that means you have four blanks: ___ x ____ x ____ x ____. The base goes in every blank, so 2 to the fourth power is 2 x 2 x 2 x 2. To represent this with marbles, think two groups with two groups in each with two groups in each of those and with two marbles in each of the last groups. Draw circles for the groups: two big ones with two smaller ones in each, two tiny ones in each of those, and two marbles in each tiny circle. Notice that numbers get bigger much faster with exponents than with multiplication.

GRE - Math (Arithmetic)

Arithmetic operations

The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, square roots, exponentiation, and logarithmic functions.

Arithmetic is performed according to an order of operations. The standard order of operations, or precedence, is expressed in the following chart:

1. terms inside brackets
2. exponents and roots
3. multiplication and division
4. addition and subtraction

This means that if a number or other symbol, or an expression grouped by one or more symbols of grouping, is preceded by one operator and followed by another, the operator higher on the list should be applied first.

Any set of objects upon which all four arithmetic operations (except division by zero) can be performed, and where these four operations obey the usual laws, is called a field. In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fields, p-adic fields, and so forth.

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Numeric Entry, Quantitative Comparison)

Explanations for both correct and incorrect answer options help you understand the concept being tested by the question

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Cracking the GRE, uCertify way

uCertify's Prepkits contain enough material for you to practice on the topics that are being questioned in the real exam. They cover all the topics and subjects on which the questions come in the real exam. Their kits contain various kinds of simulation questions, study notes, and practice tests that will help you gain good marks in the test.

GRE Questions ( Analytical Reasoning)

Directions:All GRE sample analytical reasoning questions are based on a passage or set of conditions. While answering a few of the GRE sample analytical reasoning questions, you would find it useful to draw a rough diagram. To answer any question choose the answer you think is most appropriate among the given options.

Questions 

Three men (Tom, Peter and Jack) and three women (Eliza, Anne and Karen) are spending a few months at a hillside. They are to stay in a row of nine cottages, each one living in his or her own cottage. There are no others staying in the same row of houses.

1.        Anne, Tom and Jack do not want to stay in any cottage, which is at the end of the row.

2.        Eliza and Anne are unwilling to stay besides any occupied cottage..

3.        Karen is next to Peter and Jack.

4.        Between Anne and Jack’s cottage there is just one vacant house.

5.        None of the girls occupy adjacent cottages.

6.        The house occupied by Tom is next to an end cottage.

1.        Which of the above statements can be said to have been derived from two other statements ?

A.        Statement 1

B.        Statement 2

C.        Statement 3

D.        Statement 5

E.        Statement 6

Ans : D

2.        How many of them occupy cottages next to a vacant cottage ?

A.        2

B.        3

C.        4

D.        5

E.        6

Ans : C

3.        Which among these statement(s) are true ?

                                 I.            Anne is between Eliza and Jack.

                                II.            At the most four persons can have occupied cottages on either side of them. .

                              III.            Tom stays besides Peter.

A.        I only

B.        II only

C.        I and III only

D.        II and III only

E.        I, II and III

Ans : C

GRE ( Sentence Completion)

The __ tones of the flute succeeded in __ his tense nerves.

1. rhapsodic - minimising

2. blatant - enhancing

3. hovendous - calming

4. vibrant - portraying

5. mellifluous - soothing

Ans :E

Without the psychiatrist's promise of confidentiality, trust is __ and the patient's communication limited; even though confidentiality can thus be seen to be precious in thercopy, moral responsibility sometimes requires a willingness to __ it.

1. lost - forget

2. implicit - extend

3. impaired - sacrifise

4. ambiguous - apply

5. assumed - examine

Ans :C

Parts of seventeenth-century Chinese pleasure gardens were not necessarily intended to look __ they were designed expressly to evoke the agreeable melancholy resulting from a sense of the __ of natural beauty and human glory.

1. great - immutability

2. joyful - mortality

3. conventional - wildness

4. cheerful - transitoriness

5. colorful - abstractness

Ans :D

Despite the __ of many of their colleagues, some scholars have begun to emphasize ''pop culture'' as a key for __ the myths, hopes, and fears of contemporary society.

1. pedantry - reinstating

2. enthusiasm - symbolizing

3. skepticism - deciphering

4. antipathy - involving

5. discernment - evaluating

Ans :C

Clearly refuting sceptics, researches have __ not only that gravitational radiation exists but that it also does exactly what the theory __ it should do.

1. supposed - asserted

2. voubted -warranted

3. assumed - deduced

4. demonstrated - predicted

5. estimated - accepted

Ans :D

The Neolatonists' conception of a deity, in which perfection was measured by abundant fecundity, was contradicted by that of the Aristotelians, in which perfection was displayed in the __ of creation.

1. variety

2. economy

3. profusion

4. clarity

5. precision

Ans :B

It is a great __ to be able to transfer useful genes with as little extra gene material as possible, because the donor's genome may contain, in addition to desirable genes, many genes with __ effects.

1. Disappointment - superfluous

2. Convenience - exquisite

3. Advantage - deleterious

4. Accomplishment - profound

5. Misfortune - unpredictable

Ans :C

While admitting that the risks incurred by use of the insecticide were not - the manufacturer's spokesperson argued that effective __ were simply not available.

1. indeterminable - safeguards

2. unusual - alternatives

3. inconsequential - substitutes

4. proven - antidotes

5. increasing - procedures

Ans :C

Human reaction to the realm of though is often as strong as that to sensible presences; our higher moral life is based on the fact that __ sensations actually present may have a weaker influence on our action than do ideas of __ facts.

1. emotional - impersonal

2. familiar : symbolic

3. disturbing - ordinary

4. material - remote

5. defenitive - controvoisial

Ans :D

GRE ( Sentence Completion)

Sentence Completions

That the Third Battalion's fifty percent casually rate transformed its assault on Hill 306 from a brilliant stratagem into a debacle does not ___ eyewitness reports of its commander's extra-ordinary ___ in deploying his forces.

1. invalidate - brutality

2. gainsay - cleverness

3. underscore - ineptitude

4. justify - rapidity

5. corroborate -determination

Ans : B

No longer __ by the belief that the world around us was expressly designed for humanity, many people try to find intellectual __ for that lost certainty in astrology and in mysticism.

1. satisfied - reasons

2. reassured - justifications

3. restricted - parallels

4. sustained - substitutes

5. hampered - equivalents

Ans : D

In eighth-century Japan, people who __ wasteland were rewarded with official ranks as part of an effort to overcome the shortage of __ fields.

1. cultivated - domestic

2. located - desirable

3. conserved - forested

4. reclaimed - arable

5. irrigated - accessible

Ans :D

Clearly refuting sceptics, researchers have __ not only that gravitational radiation exists but that it also does exactly what the theory __ it should do.

1. assumed - deducted

2. estimated - accepted

3. supposed - asserted

4. doubted - warranted

5. demonstrated - predicted

Ans :E

Melodramas, which presented stark oppositions between innocence and criminality, virtue and corruption, good and evil, were popular precisely because they offered the audience a world __ of __.

1. deprived - polarity

2. full - circumstantiality

3. bereft - theatricality

4. devoid - neutrality

5. composed - adversity

Ans :D

Sponsors of the bill were __ because there was no opposition to it within the legislative, until after the measure had been signed into law.

1. well-intentioned

2. persistent

3. detained

4. unreliable

5. relieved

Ans :B

Ecology, like economics, concerns itself with the movement of valuable __ through a complex network of producers and consumers.

1. nutrients

2. dividends

3. communications

4. artifacts

5. commodities

Ans :C

Having fully embraced the belief that government by persuasion is preferable to government by __ the leaders of the movement have recently __ most of their previous statements supporting totalitarianism.

1. proclamation - codified

2. coercion - repudiated

3. participation - moderated

4. intimidation - issued

5. demonstration - deliberated.

Ans :B

It would be difficult for one so __ to be led to believe that all men are equal and that we must disregard race, color and creed.

1. tolerant

2. democratic

3. broadminded

4. emotional

5. intolerant

Ans :E

Many philosophers agree that the verbal aggression of profanity in certain redical newspapers is not - or childish, but an assault on - essential to the revolutionary's purpose.

insolent - sociability

trivial - decorum

belligerent - fallibility

serious - propriety

deliberate - affectation.

Ans :B

GRE Verbal Test

Directions: Each GRE sample sentence completion question below has one or two blanks. Each blank shows that something has been omitted. Under each GRE sample sentence completion question five words are given as choice. Choose the one correct word for each blank that best fits the meaning of the sentences as a whole.

The Sociologist responded to the charge that her new theory was - by pointing out that it did not in fact contradict accepted sociological principles.

1. unproven

2. banal

3. superficial

4. complex

5. heretical

Ans :E

Despite assorted effusion to the contrary, there is no necessary link between scientific skill and humanism, and quite possibly, there may be something of a - between them.

1. dichotomy

2. congruity

3. reciprocity

4. fusion

5. generosity

Ans :E

The most technologically advanced societies have been responsible for the greatest - indeed savagery seems to be indirect proposition to -

1. inventions - know-how

2. wars - viciousness

3. triumphs - civilizations

4. atrocities - development

5. catastrophes - ill-will

Ans :D

Ironically, the party leaders encountered no greater - their efforts to build as Progressive Party than the - of the progressive already elected to the legislature.

1. obstacle to - resistance

2. support for - advocacy

3. praise for - reputation

4. threat to - promise

5. benefit - success

Ans :A

The simplicity of the theory - its main attraction - is also its - for only by - the assumptions of the theory is it possible to explain the most recent observations made by researchers.

1. glory - rejecting

2. liability - accepting

3. undoing - supplementing

4. downfall - considering

5. virtue - qualifying

Ans : C

GRE Verbal Test

Directions: Each GRE sample sentence completion question below has one or two blanks. Each blank shows that something has been omitted. Under each GRE sample sentence completion question five words are given as choice. Choose the one correct word for each blank that best fits the meaning of the sentences as a whole.

The fact that the- of confrontation is no longer as popular as it once was - progress in race relations.

1. insidiousness - reiterates

2. practice - inculcates

3. glimmer - foreshadows

4. technique - presages

5. reticence - indicates

Ans :D

A child should not be - as being either very shy or over - aggressive.

1. categorized

2. instructed

3. intoned

4. distracted

5. refrained

Ans :A

President Anwar el - Sadat of Egypt, disregarding - criticism in the Alab world and in his own Government, - accepted prime minister Menahem Begin's invitation to visit Israel in order to address the Israeli parliament.

1. acrimonious - formally

2. blemished - stiffly

3. categorical - previously

4. malignant - plaintively

5. charismatic - meticulously

Ans :A

In his usual - manner, he had insured himself against this type of loss.

1. pensive

2. providential

3. indifferent

4. circumspect

5. caustic

Ans :D

We never believed that he would resort to - in order to achieve his goal; we always regarded him as a - man.

1. charm - insincere

2. necromancy - pietistic

3. logic - honorable

4. prestidigitation - articulate

5. subterfuge - honest

Ans :E