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Precalculus
In mathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level that is designed to prepare students for the study of calculus, thus the name precalculus. Schools often distinguish between algebra and trigonometry as two separate parts of the coursework. Concept For students to succeed at finding the derivatives and antiderivatives with calculus, they will need facility with algebraic expressions, particularly in modification and transformation of such expressions. Leonhard Euler wrote the first precalculus book in 1748 called ''Introductio in analysin infinitorum'' (Latin: Introduction to the Analysis of the Infinite), which "was meant as a survey of concepts and methods in analysis and analytic geometry preliminary to the study of differential and integral calculus." He began with the fundamental concepts of variables and functions. His innovation is noted for its use of exponentiation to introduce the transcendent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometry
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History Sumerian astronomers studied angle me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called '' systems of linear equations''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Expression
In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers), variables, and the basic algebraic operations: addition (+), subtraction (-), multiplication (×), division (÷), whole number powers, and roots (fractional powers).. For example, is an algebraic expression. Since taking the square root is the same as raising to the power , the following is also an algebraic expression: :\sqrt An ''algebraic equation'' is an equation involving polynomials, for which algebraic expressions may be solutions. If you restrict your set of constants to be numbers, any algebraic expression can be called an arithmetic expression. However, algebraic expressions can be used on more abstract objects such as in Abstract algebra. If you restrict your constants to integers, the set of numbers that can be described with an algebraic expression are called Algebraic numbers. By contrast, transcendental numbers like and are not algebraic, since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conic Section
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes considered a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a '' focus'', and some particular line, called a ''directrix'', are in a fixed ratio, called the ''eccentricity''. The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pre-algebra
Pre-algebra is a common name for a course (education), course taught in middle school mathematics in the United States, usually taught in the 6th, 7th, 8th, or 9th grade. The main objective of it is to prepare students for the study of algebra. Usually, Algebra I is taught in the Eighth grade, 8th or Ninth grade, 9th grade. As an intermediate stage after arithmetic, pre-algebra helps students pass specific conceptual barriers. Students are introduced to the idea that an equals sign, rather than just being the answer to a question as in basic arithmetic, means that two sides are Equality (mathematics), equivalent and can be manipulated together. They may also learn how numbers, Variable (mathematics), variables, and words can be used in the same ways. Subjects Subjects taught in a pre-algebra course may include: * Review of natural number arithmetic * Types of numbers such as integers, Fraction (mathematics), fractions, decimals and negative numbers * Ratios and Percentage, perce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factorization, polynomial factoring, number theory, and algebraic geometry. The discriminant of the quadratic polynomial ax^2+bx+c is :b^2-4ac, the quantity which appears under the square root in the quadratic formula. If a\ne 0, this discriminant is zero if and only if the polynomial has a double root. In the case of real number, real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots. Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grégoire De Saint-Vincent
Grégoire de Saint-Vincent () - in Latin : Gregorius a Sancto Vincentio, in Dutch : Gregorius van St-Vincent - (8 September 1584 Bruges – 5 June 1667 Ghent) was a Flemish Jesuit and mathematician. He is remembered for his work on quadrature of the hyperbola. He is also known as Gregorio a San Vincente. Grégoire gave the "clearest early account of the summation of geometric series." Margaret E. Baron (1969) ''The Origins of the Infinitesimal Calculus'', Pergamon Press, republished 2014 by ElsevierGoogle Books preview/ref> He also resolved Zeno's paradox by showing that the time intervals involved formed a geometric progression and thus had a finite sum. Life Grégoire was born in Bruges 8 September 1584. After reading philosophy in Douai, he entered the Society of Jesus 21 October 1605. His talent was recognized by Christopher Clavius in Rome. Grégoire was sent to Louvain in 1612, and was ordained a priest 23 March 1613. Grégoire began teaching in association with Fran� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Interest
Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower. Compound interest is contrasted with Interest#Calculation, simple interest, where previously accumulated interest is not added to the principal amount of the current period. Compounded interest depends on the simple interest rate applied and the frequency at which the interest is compounded. Compounding frequency The ''compounding frequency'' is the number of times per given unit of time the accumulated interest is capitalized, on a regular basis. The frequency could be yearly, half-yearly, quarterly, monthly, weekly, daily, continuous compounding, continuously, or not at all until maturity. For example, monthly capitalization with interest expressed as an annual rate means that the compounding frequency is 12, with time periods measured in m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Equation
In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and then the equation is linear equation, linear, not quadratic.) The numbers , , and are the ''coefficients'' of the equation and may be distinguished by respectively calling them, the ''quadratic coefficient'', the ''linear coefficient'' and the ''constant coefficient'' or ''free term''. The values of that satisfy the equation are called ''solution (mathematics), solutions'' of the equation, and ''zero of a function, roots'' or ''zero of a function, zeros'' of the quadratic function on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a single real double root, or two comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagram Showing How To Derive The Power Reducing Formula For Sine
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word '' graph'' is sometimes used as a synonym for diagram. Overview The term "diagram" in its commonly used sense can have a general or specific meaning: * ''visual information device'' : Like the term " illustration", "diagram" is used as a collective term standing for the whole class of technical genres, including graphs, technical drawings and tables. * ''specific kind of visual display'' : This is the genre that shows qualitative data with shapes that are connected by lines, arrows, or other visual links. In science the term is used in both ways. For example, Anderson (1997) stated more generally: "diagrams are pictorial, yet abstract, repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number , one has e^ = \cos x + i \sin x, where is the base of the natural logarithm, is the imaginary unit, and and are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted ("cosine plus ''i'' sine"). The formula is still valid if is a complex number, and is also called ''Euler's formula'' in this more general case. Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". When , Euler's formula may be rewritten as or , which is known as Euler's identity. History In 1714, the English mathematician Roger Cotes prese ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
