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Real Numbers
In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limit (mathematics), limits, continuous function, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally mathematical notation, denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction (mathematics), fraction . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blackboard Bold
Blackboard bold is a style of writing Emphasis (typography), bold symbols on a blackboard by doubling certain strokes, commonly used in mathematical lectures, and the derived style of typeface used in printed mathematical texts. The style is most commonly used to represent the Set (mathematics)#Special sets of numbers in mathematics, number sets \N (natural numbers), \Z (integers), \Q (rational numbers), \R (real numbers), and \C (complex numbers). To imitate a bold typeface on a typewriter, a character can be typed over itself (called ''double-striking''); symbols thus produced are called double-struck, and this name is sometimes adopted for blackboard bold symbols, for instance in Unicode glyph names. In typography, a typeface with characters that are not solid is called ''inline'', ''handtooled'', or ''open face''. History Traditionally, various symbols were indicated by boldface in print but on blackboards and in manuscript (publishing), manuscripts "by wavy underscoring, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number Line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely. The association between numbers and points on the line links arithmetical operations on numbers to geometric relations between points, and provides a conceptual framework for learning mathematics. In elementary mathematics, the number line is initially used to teach addition and subtraction of integers, especially involving negative numbers. As students progress, more kinds of numbers can be placed on the line, including fractions, decimal fractions, square roots, and transcendental numbers such as the circle constant : Every point of the number line corresponds to a unique real number, and every real number to a unique point. Using a number line, numerical concepts can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Numbers
In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . The quality of a number being transcendental is called transcendence. 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 form 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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number
In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is an algebraic number, because it is a root of the polynomial X^2 - X - 1, i.e., a solution of the equation x^2 - x - 1 = 0, and the complex number 1 + i is algebraic as a root of X^4 + 4. Algebraic numbers include all integers, rational numbers, and nth root, ''n''-th roots of integers. Algebraic complex numbers are closed under addition, subtraction, multiplication and division, and hence form a field (mathematics), field, denoted \overline. The set of algebraic real numbers \overline \cap \R is also a field. Numbers which are not algebraic are called transcendental number, transcendental and include pi, and . There are countable set, countably many algebraic numbers, hence almost all real (or complex) numbers (in the sense of Lebesgue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of 2
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Technically, it should be called the ''principal'' square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a Unit square, square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational number, irrational. The fraction (≈ 1.4142857) is sometimes used as a good Diophantine approximation, rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 60 decimal places: : History The Babylonian clay tablet YBC 7289 (–1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates 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 problem (mathematics education), 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; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Of A Function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6=(x-2)(x-3) has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fraction (mathematics)
A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A ''common'', ''vulgar'', or ''simple'' fraction (examples: and ) consists of an integer numerator, displayed above a line (or before a slash like ), and a division by zero, non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction , the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates of a cake. Fractions can be used to represent ratios and division (mathematics), division. Thus the fraction can be used to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4^2 = (-4)^2 = 16. Every nonnegative real number has a unique nonnegative square root, called the ''principal square root'' or simply ''the square root'' (with a definite article, see below), which is denoted by \sqrt, where the symbol "\sqrt" is called the '' radical sign'' or ''radix''. For example, to express the fact that the principal square root of 9 is 3, we write \sqrt = 3. The term (or number) whose square root is being considered is known as the ''radicand''. The radicand is the number or expression underneath the radical sign, in this case, 9. For non-negative , the principal square root can also be written in exponent notation, as x^. Every positive number has two square roots: \sqrt (which is positive) and -\sqrt (which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
