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## DIVISION

In mathematics division is an operation which is the opposite of multiplication. If c times b equals a , written: where b is not zero, then a divided by b equals c , written: For instance, since . In the above expression, a is called the dividend, b the divisor and c the quotient. Division by zero ...is not defined. Notation Division is most often shown by placing the dividend over the divisor with a horizontal line, also called a vinculum, between them. For example, a divided by b is written T

## Roman numerals

Roman numerals Roman numerals stem from the numeral system of ancient Rome . They are based on certain letters of the alphabet which are combined to signify the sum (or, in some cases, the difference) of their values. The first ten Roman numerals are: I, II, III, IV, V, VI, VII, VIII, IX, and X. The Roman numeral system is decimal but not directly positional and does not include a zero . It is a cousin of the Etruscan numerals , and the letters derive from earlier non-alphabetical symbols; over

## Coordinates

Coordinates are numbers which describe the location of points in a plane or in space. For example, the height above sea level is a coordinate which is useful for describing points near the surface of the earth. A coordinate system , in a plane or in space, is a systematic method of assigning a pair or a triple of numbers to each point in the plane or in space (respectively) which describe its position uniquely. For example, the triple consisting of latitude , longitude and altitude (height abov

## Calculus-Partial Fractions

Partial Fractions Consider the integral ó õ 3x 3 -2x 2 -19x-7 x 2 -x-6 dx. The integrand is an improper rational function. By "long division" of polynomials, we can rewrite the integrand as the sum of a polynomial and a proper rational function "remainder": 3x + 1 x 2 -x-6 ø 3x 3 - 2x 2 - 19x - 7 3x 3 - 3x 2 - 18x x 2 - x - 7 x 2 - x - 6 - 1 So ó õ 3x 3 -2x 2 -19x-7 x 2 -x-6 dx = ó õ æ ç è 3x+1+ -1 x 2 -x-6 ö ÷ ø dx. This looks much easier to work with! We can integrate 3x+1 immediately, but wh

## Divisibility rule

A divisibility rule is a shorthand way of discovering whether a given number is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, and they are all different, we present rules only for decimal numbers.

## Calculus-Lines, Planes, and Vectors

Lines, Planes, and Vectors In this tutorial, we will use vector methods to represent lines and planes in 3-space. Displacement Vector The displacement vector v with initial point (x 1 ,y 1 ,z 1 ) and terminal point (x 2 ,y 2 ,z 2 ) is v = (x 2 -x 1 ,y 2 -y 1 ,z 2 -z 1 ) Why? That is, if vector v were positioned with its initial point at the origin, then its terminal point would be at (x 2 -x 1 ,y 2 -y 1 ,z 2 -z 1 ). Example The vector v with initial point (-1,4,5) and final point (4,-3,2) is v

## Circumference

The circumference is the distance around a closed curve. Circumference is a special perimeter. Circumference of a circle The circumference of a circle is the length around it. The circumference of a circle can be calculated from its diameter using the formula: Or, substituting the radius for the diameter: where r is the radius and d is the diameter of the circle, and the Greek letter p is defined as the ratio of the circumference of the circle to its diameter. The numerical value of p is 3.141

## PYTHAGOREAS THEOREM - A REVIEW

You may be familiar with some of the concepts presented in this and the next page, allowing you to go through them very quickly. However, I hope that you will find some interesting concepts. Angles and Triangles -- Let's first define an angle. When two lines intersect in a point, called a "vertex", the circular span between the lines is called an angle. The following figure shows the angle n between the lines A and B: The size of the angle n describes how open or closed the lines are in a c

## MODEL THEORY

In mathematics, model theory is the study of (classes of) mathematical structures such as groups, fields, graphs, or even models of set theory, using tools from mathematical logic. Model theory has close ties to algebra and universal algebra. This article focuses on finitary first order model theory of infinite structures. Finite model theory, which concentrates on finite structures, diverges significantly from the study of infinite structures in both the problems studied and the techniques use

## Derivative

In calculus, a branch of mathematics, the derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point. For example, the derivative of the position or distance of a car at some point in time is the instantaneous velocity, or instantaneous speed (respectively), at which that car is traveling (conversely the integral of the velocity is the car's position). A c

## Law of cosines

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule ) is a statement about a general triangle which relates the lengths of its sides to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states that where c is the side opposite of angle ? , and that a and b are the sides that form the angle ? . The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle ? is a right angle (of measure 90

## ASSOCIATIVITY

In mathematics, associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order that the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value. Consider for instance the equation Even though the parentheses were rearranged (the left side requires adding 5 and

## calculus-Change of Basis

Change of Basis Let V be a vector space and let S = { v 1 , v 2 , ¼ , v n } be a set of vectors in V. Recall that S forms a basis for V if the following two conditions hold: S is linearly independent . S spans V. If S = { v 1 , v 2 , ¼ , v n } is a basis for V, then every vector v Î V can be expressed uniquely as a linear combination of v 1 , v 2 , ¼ , v n : v = c 1 v 1 + c 2 v 2 + ¼ + c n v n . Think of é ê ê ê ê ê ë c 1 c 2 : c n ù ú ú ú ú ú û as the coordinates of v relative to the basis S.

## ADDITION

Addition is the mathematical process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples — meaning three apples and two other apples — which is the same as five apples, since 3 + 2 = 5. Besides counts of fruit, addition can also represent combining other physical and abstract quantities using different kinds of numbers: negative numbers, fractions, irrational numbers, vectors, and more. As a mathem

## Solenoidal vector field

In vector calculus a solenoidal vector field (also known as an incompressible vector field ) is a vector field v with divergence zero: The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as: automatically results in the identity (as can

## Primary mathematics:Dividing numbers

When first teaching young children to divide, you may like to try getting a small group together and sharing out a number of objects evenly between them. This will ensure that the students get a clear understanding of the concept of division before moving on to the written process of short and long division. The process of division should be taught as the reverse of multiplication, again the times tables are needed. The links between addition and subtraction and multiplication and division beco

## ABSTRACT ALGEBRA

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Most contemporary authors simply write algebra instead of abstract algebra. The term abstract algebra now refers to the study of all algebraic structures, as distinct from the elementary algebra ordinarily taught to children, which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex number

## Calculus-Solving Systems of Linear Equations; Row Reduction

Solving Systems of Linear Equations; Row Reduction Systems of linear equations arise in all sorts of applications in many different fields of study. The method reviewed here can be implemented to solve a linear system a 11 x 1 + a 12 x 2 + ¼ + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + ¼ + a 2n x n = b 2 : : · · · : : a m1 z 1 + a m2 x 2 + ¼ + a mn x n = b m of any size. We write this system in matrix form as é ê ê ê ê ê ê ë a 11 a 12 ¼ a 1n a 21 a 22 ¼ a 2n : : · · · · · · a m1 a m2 ¼ a mn ù ú ú ú ú

## Calculus-Volume

Volume Many three-dimensional solids can be generated by revolving a curve about the x-axis or y-axis. For example, if we revolve the semi-circle given by f(x) = ______ Ö r 2 -x 2 about the x-axis, we obtain a sphere of radius r. We can derive the familiar formula for the volume of this sphere. Finding the Volume of a Sphere Consider a cross-section of the sphere as shown. This cross-section is a circle with radius f(x) and area p [f(x)] 2 . Informally speaking, if we "slice" the sphere vertica

## DISTANCE

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria (e.g. "two counties over"). In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific set of rules, and provides a concrete way of describing what it means for elements of some space to be "close to" or "far awa

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