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Home >> Educational Materials >> Mathematics## Negative and non-negative numbers

Being negative or non-negative is a property of a number which is real, or a member of a subset of real numbers such as rational and integer numbers. A negative number is one that is less than zero, such as - , -1.414, -1. A positive number (e.g., positive real number, positive rational number, positive integer) is one that is greater than zero, such as , 1.414, 1. Zero itself is neither positive nor negative . The non-negative numbers are the numbers that are not negative (they are positive or## Multiplicative inverse

From Wikipedia, the free encyclopedia Jump to: navigation , search The reciprocal function: y = 1 / x . For every x except 0, y represents its multiplicative inverse. In mathematics , a multiplicative inverse or reciprocal for a number x , denoted by 1 / x or x -1 , is a number which when multiplied by x yields the multiplicative identity , 1. The multiplicative inverse of x is also called the reciprocal of x . The multiplicative inverse of a fraction p / q is q / p . For the multiplicative inv## Hamming graphs

Hamming graphs are a special class of graphs used in several branches of mathematics and computer science. Let S be a set of q elements and d a positive integer. The Hamming graph H ( d , q ) has vertex set S d , the set of ordered d -tuples of elements of S , or sequences of length d from S . Two vertices are adjacent if they differ in precisely one coordinate. The special case in which q = 2 is also known as the hypercube graph, denoted Q d . The special cases in which d = 1 and d = 2 are the## Elementary algebra

Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic . While in arithmetic only numbers and their arithmetical operations (such as +, -, ×, ÷) occur, in algebra one also uses symbols (such as x and y , or a and b ) to denote numbers. These are called variables . This is useful because: It allows the generalization of arithmetical equations (and inequalities ) to be stat## Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. [ 1 ] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exh## Calculus - Differentiating Special Functions

Differentiating Special Functions In this tutorial, we review the differentiation of trigonometric, logarithmic, and exponential functions. Trigonometric Functions The derivatives of the basic trigonometric functions are given here for reference. f(x) f ¢ (x) sin(x) cos(x) cos(x) -sin(x) tan(x) sec 2 (x) sec(x) sec(x)tan(x) csc(x) -csc(x)cot(x) cot(x) -csc 2 (x) The derivatives of sin(x) and cos(x) can be derived using the limit definition of the derivative . For sin(x), d dx [sin(x)] = lim D x## Multiplication

Multiplication (symbol " × ") is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic (the others being addition, subtraction and division). Because the result of scaling by whole numbers can be thought of as consisting of some number of copies of the original, whole-number products greater than 1 can be computed by repeated addition; for example, 3 multiplied by 4 (often said as "3 times 4") can be calculated by adding 4 c## Quotient

In mathematics, a quotient is the end result of a division problem. For example, in the problem 6 ÷ 3, the quotient would be 2. Then 6 would be called the dividend, and 3 the divisor. The quotient can also be expressed as the number of times the divisor divides into the dividend. A quotient can also mean just the integral part of the result of dividing two integers. For example, the quotient of 13 ÷ 5 would be 2 while the remainder would be 3. Quotients also come up in certain tests, like the I## DIVISION

In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. Specifically, 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 . Conceptually, division describes two distinct but related settings. Partitioning involves taking a set of size a and forming b groups that are equal in s## About Mathematics

Euclid, Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens . [1] Mathematics (colloquially, maths or math ), is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions". [2] Other practitioners of mathematics [3] [4] maintain that mathematics is the science of pattern, that mathematicians## Calculus-Partial Differentiation

Partial Differentiation Suppose you want to forecast the weather this weekend in Los Angeles. You construct a formula for the temperature as a function of several environmental variables, each of which is not entirely predictable. Now you would like to see how your weather forecast would change as one particular environmental factor changes, holding all the other factors constant. To do this investigation, you would use the concept of a partial derivative... Let the temperature T depend on vari## Axiom

In traditional logic , an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident , or subject to necessary decision . Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths. In mathematics , the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms" . In both senses, an axiom is any mathematical statement that ser## Hasse diagram

In order theory, a branch of mathematics, a Hasse diagram (pronounced /ˈhæsə/ , German: /ˈhasə/ ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of itstransitive reduction. Concretely, for a partially ordered set (S, =) one represents each element of S as a vertex in the plane and draws a line segment or curve that goes upward from x to y whenever y covers x (that is, whenever x < y and there is no z such that x <## Calculus-Multiple Integration

Multiple Integration Recall our definition of the definite integral of a function of a single variable: Let f(x) be defined on [a,b] and let x 0 ,x 1 , ¼ ,x n be a partition of [a,b]. For each [x i-1 ,x i ], let x i * Î [x i-1 ,x i ]. Then ó õ b a f(x) dx = lim max D x i ® 0 n å i = 1 f(x i * ) D x i Take a quick look at the Riemann Sum Tutorial We can extend this definition to define the integral of a function of two or more variables. Double Integral of a Function of Two Variables Let f(x,y)## NATURAL NUMBER

In mathematics, a natural number (also called counting number) can mean either an element of the set {1, 2, 3, ...} (the positive integers) or an element of the set {0, 1, 2, 3, ...} (the non-negative integers). The former is generally used in number theory, while the latter is preferred in mathematical logic, set theory, and computer science. A more formal definition will follow. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), and they## A Gentle Introduction to Continuity and Limits

Determining the definition for continuity was a major accomplishment of nineteenth century mathematics. The compactness of the definition belies the effort expended in arriving at it. The French mathematician Augustin Cauchy is credited for the delta and epsilon definition found in calculus courses. I can remember being bewildered the first time I saw the definition of continuity. In what follows I am going to go through a series of steps to show how the definition of continuity could be arrive## Decimal separator

The decimal separator or decimal point or decimal comma is a symbol used to mark the boundary between the integral and the fractional parts of a decimal number in a positional numeral system. Different symbols have been and are used for the decimal separator. The choice of symbol for the decimal separator affects the choice of symbol for the thousands separator used in digit grouping. Consequently the latter is treated in this article as well. The decimal separator is mathematically a radix poi## ABSTRACT ALGEBRA

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra , the study of the rules for manipulating formulas and algebraic expressions involving real or complex numbers, and unknowns. The distinction is rarely made in more recent writings. Contemporary mathem## 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 that relates the lengths of its sides to the cosine of one of itsangles. Using notation as in Fig. 1, the law of cosines states that where ? denotes the angle contained between sides of lengths a and b and opposite the side of length c . The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle ? is a right angle (of meas## Field of sets

In mathematics a field of sets is a pair where X is a set and is an algebra over X i.e., a non-empty subset of the power set of X closed under the intersection and union of pairs of sets and under complements of individual sets. In other words forms a subalgebra of the power set Boolean algebra of X . (Many authors refer to itself as a field of sets.) Elements of X are called points and those of are called complexes . Fields of sets play an essential role in the representation theory of Boolean$.ajax({
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