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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.