COMMON TERMS IN MATHEMATICS

Dilara DORAK & M.Tevfik DORAK

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Absolute value: The magnitude of a number. It is the number with the sign (+ or -) removed and is symbolised using two vertical straight lines ( |5| ). Also called modulus.

Abstract number: A number with no associated units.

Acute angle: An angle with degree measure less than 90. See MathWorld: Geometry: Trigonometry: Angles.

Addition: The process of finding the sum of two numbers, which are called addend and the augend (sometimes both are called the addend).

Algorithm: Any mathematical procedure or instructions involving a set of steps to solve a problem.

Arctan: The inverse of the trigonometric function tangent shown as arctan(x) or tan-1(x). It is useful in vector conversions and calculations. See Wikipedia: Mathematics: Trigonometric Functions.

Arithmetic mean: M = (x1 + x2 + .... xn) / n (n = sample size).

Arithmetic sequence: A sequence of numbers in which each term (subsequent to the first) is generated by adding a fixed constant to its predecessor.

Associative property: A binary operation (*) is defined associative if, for a*(b*c) = (a*b)*c. For example, the operations addition and multiplication of natural numbers are associative, but subtraction and division are not.

Asymptote: A straight line that a curve approaches but never meets or crosses. The curve is said to meet the asymptote at infinity. In the equation y = 1/x, y becomes infinitely small as x increases but never reaches zero.

Axiom: Any assumption on which a mathematical theory is based.

Average: The sum of several quantities divided by the number of quantities (also called mean).

Avogadro's number: The number of molecules in one mole is called Avogadro’s number (approximately 6.022 × 1023 particles/mole).

Binary operation: An operation that is performed on just two elements of a set at a time.

Brownian motion: See an article (by Lee & Hoon) and an animation and a second one..

Butterfly effect: In a system when a small change results in an unpredictable and disproportionate disturbance, the effect causing this is called a butterfly effect.

Calculus: Branch of mathematics concerned with rates of change, gradients of curves, maximum and minimum values of functions, and the calculation of lengths, areas and volumes. It involves determining areas (integration) and tangents (differentiation), which are mutually inverse. Also called real analysis. See also Dr. Vogel's Gallery of Calculus Pathologies; MathWorld: Calculus; Wikipedia:: CalculusPlanetMathCalculus Animations with Mathcad.

Cartesian coordinates: Cartesian coordinates (x,y) specify the position of a point in a plane relative to the horizontal x and the vertical y axes. The x and y axes form the basis of two-dimensional Cartesian coordinate system.

Chaos: Apparent randomness whose origins are entirely deterministic. A state of disorder and irregularity whose evolution in time, though governed by simple exact laws, is highly sensitive to starting conditions: a small variation in these conditions will produce wildly different results, so that long-term behaviour of chaotic systems cannot be predicted. This sensitivity to initial conditions is also known as the butterfly effect (when a butterfly flaps its wings in Mexico, the result may be a hurricane in Florida a month later).

Chord: A straight line joining two points on a curve or a circle. See also secant line.

Circle: A circle is defined as the set of points at a given distance (or radius) from its centre. If the coordinates of the centre of a circle on a plane is (a,b) and the radius is r, then (x-a)2 + (y-b)2 = r2. The equation that characterises a circle has the same coefficients for x2 and y2. The area of a circle is A = pr2 and circumference is C = 2pr. A circle with centre (a,b) and radius r has parametric equations: x = a + r.cos q and y = b + r.sin q (0 ≤ q ≤ 2p). A ‘tangent’ is a line, which touches a circle at one point (called the point of tangency) only. A ‘normal’ is a line, which goes through the centre of a circle and through the point of tangency (the normal is always perpendicular to the tangent). A straight line can be considered a circle; a circle with infinite radius and centre at infinity. See a Lecture Note, BBC Bitesize: Circle; Wikipedia: Mathematics: Circle; MathWorld: Geometry: Circles.

Circumference: A line or boundary that forms the perimeter of a circle.

Closure property: If the result of doing an operation on any two elements of a set is always an element of the set, then the set is closed under the operation. For example, the operations addition and multiplication of natural numbers (the set) are closed, but subtraction and division are not.

Coefficient: A number or letter before a variable in an algebraic expression that is used as a multiplier.

Common denominator: A denominator that is common to all the fractions within an equation. The smallest number that is a common multiple of the denominators of two or more fractions is the lowest (or least) common denominator (LCM).

Common factor: A whole number that divides exactly into two or more given numbers. The largest common factor for two or more numbers is their highest common factor (HCF).

Common logarithm: Logarithm with a base of 10 shown as log10 [log1010x = x].

Common ratio: In a geometric sequence, any term divided by the previous one gives the same common ratio.

Commutative property: A binary operation (*) defined on a set has the commutative property if for every two elements, a and b, a*b = b*a. For example, the operations addition and multiplication of natural numbers are commutative, but subtraction and division are not.

Complementary angles: Two angles whose sum is 90o. See also supplementary angles.

Complex numbers: A combination of real and imaginary numbers of the form a + bi where a and b are real numbers and i is the square root of -1 (see imaginary number). While real numbers can be represented as points on a line, complex numbers can only be located on a plane. See Types of Numbers.

Composite number: Any integer which is not a prime number, i.e., evenly divisible by numbers other than 1 and itself.

Congruent: Alike in all relevant respects.

Constant: A quality of a measurement that never changes in magnitude.

Coordinate: A set of numbers that locates the position of a point usually represented by (x,y) values.

Cosine law: For any triangle, the side lengths a, b, c and corresponding opposite angles A, B, C are related as follows: a2 = b2 + c2 - 2bc cosA etc. The law of cosines is useful to determine the unknown data of a triangle if two sides and an angle are known. See Wikipedia: Cosine Law.

Counting number: An element of the set C = {1,2,3,...}.

Cube root: The factor of a number that, when it is cubed (i.e., x3) gives that number.

Curve: A line that is continuously bent.

Decimal: A fraction having a power of ten as denominator, such as 0.34 = 34/100 (102) or 0.344 = 344/1000 (103). In the continent, a comma is used as the decimal point (between the unit figure and the numerator).

Degree of an angle: A unit of angle equal to one ninetieth of a right angle. Each degree ( 0 ) may be further subdivided into 60 parts, called minutes (60’), and in turn each minute may be subdivided into another 60 parts, called seconds (60’’). Different types of angles are called acute (<900)< right (900) < obtuse (900-1800) < reflex (1800-3600). See also radian (the SI unit of angle).

Denominator: The bottom number in a fraction.

Derivative: The derivative at a point on a curve is the gradient of the tangent to the curve at the given point. More technically, a function (f'(x0)) of a function y = f(x), representing the rate of change of y and the gradient of the graph at the point where x = x0, usually shown as dy/dx. The notation dy/dx suggests the ratio of two numbers dy and dx (denoting infinitesimal changes in y and x), but it is a single number, the limit of a ratio (k/h) as they both approach zero. Differentiation is the process of calculating derivatives. The derivatives of all commonly occurring functions are known. See Derivative Calculator

Differential Equations: Equations containing one or more derivatives (rate of change). As such these equations represent the relationships between the rates of change of continuously varying quantities. The solution contains constant terms (constant of integration) that are not present in the original differential equation. Two general types of differential equations are ordinary differential equations (ODE) and partial differential equations (PDE). When the function involved in the equation depends upon only a single variable, the differential equation is an ODE. If the function depends on several independent variables (so that its derivatives are partial derivatives) then the differential equation is a PDE. See Internet Resources for Differential Equations; S.O.S Mathematics Review: Differential Equations.

Diameter: A straight line that passes from side to side thorough the centre of a circle.

Differential calculus: Differentiation is concerned with rates of change and calculating the gradient at any point from the equation of the curve, y = f(x).

Differential equation: Equations involving total or partial differentiation coefficients and the rate of change; the difference between some quantity now and its value an instant into the future. See also Wikipedia

Digit: In the decimal system, the numbers 0 through 9.

Dimension: Either the length and/or width of a flat surface (two-dimensional); or the length, width, and/or height of a solid (three-dimensional).

Distributive property: A binary operation (*) is distributive over another binary operation (^) if, a*(b^c) = (a*b)^(a*c). For example, the operation of multiplication is distributive over the operations of addition and subtraction in the set of natural numbers.

Division: The operation of ascertaining how many times one number, the divisor, is contained in another, the dividend. The result is the quotient, and any number left over is called the remainder. The dividend and divisor are also called the numerator and denominator, respectively.

Dynamics: The branch of mathematics, which studies the way in which force produces motion.

e: Symbol for the base of natural logarithms (2.7182818285...), defined as the limiting value of (1 + 1/m)m.

Equilibrium: The state of balance between opposing forces or effects.

Even number: A natural number that is divisible by two.

Exponent (power, index): A number denoted by a small numeral placed above and to the right of a numerical quantity, which indicates the number of times that quantity is multiplied by itself. In the case of Xn, it is said that X is raised to the power of n. When a and b are non-zero real numbers and p and q are integers, the following rules of power apply:

ap x aq = ap+q;  (ap)q = apq;  (a1/n)m = am/n;  a1/2 x b1/2 = (ab)1/2.

Exponential function: A function in the form of f(x) = ax where x is a real number, and a is positive and not 1. One exponential function is f(x) = ex

Extrapolation: Estimating the value of a function or a quantity outside a known range of values. See also interpolation.

Factorial: The product of a series of consecutive positive integers from 1 to a given number (n). It is expressed with the symbol ( ! ). For example, 5! = 5x4x3x2x1 = 120. As a rule (n!+n) is evenly divisible by n.

Factor: When two or more natural numbers are multiplied, each of the numbers is a factor of the product. A factor is then a number by which another number is exactly divided (a divisor).

Factorisation: Writing a number as the product of its factors which are prime numbers.

Fermat's little theorem: If p is a prime number and b is any whole number, then bp-b is a multiple of p (23 - 2 = 6 and is divisible by 3).

Fermat prime: Any prime number in the form of 22n + 1 (see also Mersenne prime).

Fibonacci sequence: Sequence of integers, where each is the sum of the two preceding it. 1,1,2,3,5,8,13,21,... The number of petals of flowers forms a Fibonacci series.

Fractals: Geometrical entities characterised by basic patterns that are repeated at ever decreasing sizes. They are relevant to any system involving self-similarity repeated on diminished scales (such as a fern's structure) as in the study of chaos.

Fraction (quotient): A portion of a whole amount. The term usually applies only to ratios of integers (like 2/3, 5/7). Fractions less than one are called common, proper or vulgar fractions; and those greater than 1 are called improper fraction.

Function (f): The mathematical operation that transforms a piece of data into a different one. For example, f(x) = x2 is a function transforming any number to its square.

Geometric mean: G = (x1.x2...xn)1/n where n is the sample size. This can also be expressed as antilog ((1/n) S log x). See Applications of the Geometric Mean; Spizman, 2008: Geometric Mean in Forensic Economy.

Geometric sequence: A sequence of numbers in which each term subsequent to the first is generated by multiplying its predecessor by a fixed constant (the common ratio).

Goldbach conjecture: Every even number greater than 4 is the sum of two odd primes (32 = 13 + 19). Every odd number greater than 7 can be expressed as the sum of three odd prime numbers (11 = 3 + 3 + 5).

Gradient: The slope of a line. The gradient of two points on a line is calculated as rise (vertical increase) divided by run (horizontal increase), therefore, the gradient of a line is equal to the tangent of the angle it makes with the positive x-axis (y/x). See Mathlets: Lines and Slopes; Curve Bank: Slope.

Harmonic mean: Of a set of numbers (y1 to yi), the harmonic mean is the reciprocal of the arithmetic mean of the reciprocal of the numbers [H = N / S (1/y)]. See also Wikipedia: Mathematics: Harmonic Mean. Not to be confused with Harmonic Ratio.

Hierarchy of operations: In an equation with multiple operators, operations proceed in the following order: (brackets), exponentiation, division/multiplication, subtraction/summation and from left to right.

Highest common factor (HCF): The greatest natural number, which is a factor of two or more given numbers.

Hypotenuse: The longest side of a right triangle, which lies opposite the vertex of the right angle.

i: The square root of -1 (an imaginary number).

Identity element: The element of a set which when combined with any element of the same set leaves the other element unchanged (like zero in addition and subtraction, and 1 in multiplication or division).

Imaginary number: The product of a real number x and i, where i2 + 1 = 0. A complex number in which the real part is zero. In general, imaginary numbers are the square roots of negative numbers. See Types of Numbers.

Improper fraction: A fraction whose numerator is the same as or larger than the denominator; i.e., a fraction equal to or greater than 1.

Infinite: Having no end or limits. Larger than any quantified concept. For many purposes it may be considered as the reciprocal of zero and shown as an 8 lying on its side ().

Infinitesimal: A vanishingly small part of a quantity. It equals almost zero.

Integer: Any whole number: positive and negative whole numbers and zero.

Integral calculus: This is the inverse process to differentiation; i.e., a function which has a given derived function. For example, x2 has derivative 2x, so 2x has x2 as an integral. A classic application of integral is to calculate areas. Wikipedia

Integration: The process of finding a function given its derived function.

Intersection: The intersection of two sets is the set of elements that are in both sets.

Intercept: A part of a line/plane cut off by another line/plane.

Interpolation: Estimating the value of a function or a quantity from known values on either side of it.

Inverse function: A function which 'does the reverse' of a given function. For example, functions with the prefix arc are inverse trigonometric functions; e.g. arcsin x for the inverse of sin(x). See also Wikipedia: Mathematics: Inverse Functions

Irrational number: A real number that cannot be expressed as the ratio of two integers, and therefore that cannot be written as a decimal that either terminates or repeats. The square root of 2 is an example because if it is expressed as a ratio, it never gives 2 when multiplied by itself. The numbers p = 3.141592645..., and e = 2.7182818... are also irrational numbers. See also transcendental numbers, real numbers, and Types of Numbers.

Iteration: Repeatedly performing the same sequence of steps. Simply, solving an algebraic equation with an arbitrary value for the unknown and using the result to solve it again, and again.

Least squares method: A method of fitting a straight line or curve based one minimisation of the sum of squared differences (residuals) between the predicted and the observed points. Given the data points (xi, yi), it is possible to fit a straight line using a formula, which gives the y=a+bx. The gradient of the straight line b is given by [S(xi - mx)(yi-my)] / [(S(x-mx))2], where mx and my are the means for xi and yi. The intercept a is obtained by my - bmx. See Wikipedia: Least Squares.

Linear: A model or function where the input and output are proportional.

Linear expression: A polynomial expression with the degree of polynomial being 1, i.e., that does not include any terms as the power of a variable. It will be something like, f(x)=2x1+3, but not x2+2x+4 (the latter is a quadratic expression). Linear equations are closely related to a straight line.

Literal numbers: Letters representing numbers (as in algebraic equations).

Logarithm: The logarithm of a number N to a given base b is the power to which the base must be raised to produce the number N. Written as logb N. Naturally, logb bx = x. In any base, the following rules apply: log (ab) = log a + log b; log (a/b) = log a - log b; log (1/a) = -log a; log ab = b log a; log 1 = 0 and log 0 is undefined. See

Logistic model (map, sequence)

Lowest common multiple (LCM): The smallest non-zero natural number that is a common multiple of two or more natural numbers (compare with the highest common factor).

Matrix: A matrix (plural: matrices) is a rectangular table of data. See Basics of Matrix Algebra; Linear Algebra Review; ‘Introduction to Matrix Algebra’ Book; Matrix Algebra Tool and Interactive Exercises.

Mechanics: Study of the forces acting on bodies, whether moving (dynamics) or stationary (statics).

Mersenne prime: A Mersenne number, Mp, has the form 2p-1, where p is a prime. If Mp itself a prime, then it is called a Mersenne prime. There are 32 such primes known (i.e., not all primes yield a Mersenne prime). (See also Fermat prime.)

Mixed number: A number that contains both a whole number and a fraction.

Modulus: The absolute value of a number regardless of its sign, shown as | x | or mod x. For a vector u, the modulus  | u | is used to indicate its magnitude calculated using Pythagoras’ theorem: | u | = (a2 + b2)1/2.

Multiplication: The process of finding the product of two quantities that are called the multiplicand and the multiplier.

Natural logarithm: Logarithm with a base of e, usually abbreviated ln (ln ex = x).

Natural number: Any element of the set N = {0,1,2,3,...} (positive integers). The inclusion of zero is a matter of definition. See Types of Numbers.

Numerator: The top number in a fraction.

Obtuse angle: An angle with a degree measure between 90 and 180. See MathWorld: Geometry: Trigonometry: Angles: Obtuse Angle.

Odd number: A natural number that is not divisible by 2.

Odds: The odds of a success is defined to be the ratio of the probability of a success to the probability of a failure (p/(1-p)).

Ordinate: The vertical coordinate on a plane.

Origin: The point on a graph that represents the point where the x and y axes meet: (x,y) = (0,0).

Parallel: Lines or planes that are equidistant from each other and do not intersect.

Perfect number: A number which is equal to the sum of its proper divisors. 6, 28, and 496 are the three of seven known perfect numbers. [6 is a perfect number because its proper divisors (1,2, and 3) total 6.] See Types of Numbers.

Permutation: A permutation of a sequence of objects is just a rearrangement of them.

Perpendicular: At right angles to a line or plane.

Pi (p): The ratio of the circumference of a circle to its diameter. The value of p is 3.1415926, correct to seven decimal places. The sum of the three angles of a triangle is p radians.

Poisson distribution: The probability distribution of the number of occurrences of random (usually rare and independent) events in an interval or time or space. See a Lecture Note.

Polar equation: A system which describes a point in the plane not by its Cartesian coordinates (x,y) but by its polar coordinates: angular direction (q) and distance r from the origin (r, q).

Polygon: A geometric figure that is bound by many straight lines such as triangle, square, pentagon, hexagon, heptagon, octagon etc.

Polynomial: An algebraic expression of the form a0xn + a1xn-1 + ... + an, where a0, a1, ..., an are members of a field (or ring), and n is the degree of the polynomial. See Wikipedia: Polynomial.

Precalculus: A foundational mathematical discipline. Pre-calculus intends to prepare students for the study of calculus. Pre-calculus typically includes a review of algebra, as well as an introduction to exponential, logarithmic and trigonometric functions as preparation for the study of calculus. See Wikipedia: Mathematics: Precalculus.

Prime factors: Prime factors of a number are a list of prime numbers the product of which is the number concerned. When n=1, for example, f(x)=2x1+3, this is a linear expression. If n=2, it is quadratic (for example, x2 + 2x + 4); if n=3, it is cubic, if n=4, it is quartic and if n=5, it is quintic.

Prime number: A natural number other than 1, evenly divisible only by 1 and itself. The numbers 2,3,5,7,11,13,17,19,... Apart from 2, all primes are odd numbers and odd primes fall into two groups: those that are one less than a multiple of four (3,7,11,19) and those one more than a multiple of four (5,13,17). Every natural number greater than 1 may be resolved into a product of prime numbers; eg 8316 = 22 x 33 x 7 x 11. See Types of Numbers, Prime Numbers.

Probability distributions: See Gallery of Probability Distributions in Engineering Statistics Handbook.

Product: The result of a multiplication problem.

Proper divisor: Any number divides another without leaving a remainder.

Proper fraction: A fraction in which the numerator is smaller than the denominator; i.e., a fraction smaller than 1.

Proportion: A type of ratio in which the numerator is included in the denominator. It is the ratio of a part to the whole (0.0 ≤ p ≤ 1.0) that may be expressed as a decimal fraction (0.2), vulgar fraction (1/5) or percentage (20%).

Pythagoras’ Theorem: For any right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides. See . Wikipedia: Mathematics: Pythagoras’ theorem.

Quadratic equation: An algebraic equation of the second degree (having one or more variables raised to the second power). The general quadratic equation is ax2 + bx + c = 0, in which a, b, and c are constants (or parameters) and ‘a’ is not equal to 0.

Quotient (fraction): An algebraic expression in which the numerator is divided by the denominator. Turning a fraction upside down gives the fraction’s reciprocal.

Radian (rad): The SI unit for measuring an angle formally defined as ‘the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle’ (the angle of an entire circle is 2p radians; p radians equal 1800 (sum of the three angles of a triangle); this is the basis of circumference of a circle formula 2pr). Sum of angles of a triangle equals p radians. See an Animation of Radian.

Radius: The distance between the centre of a circle and any point on the circle's circumference.

Rate: The relationship between two measurements of different units such as change in distance with respect to time (miles per hour).

Ratio: The relationship between two numbers or measurements, usually with the same units like the ratio of the width of an object to its length. The ratio a:b is equivalent to the quotient a/b.

Rational number: A number that can be expressed as the ratio of two integers, e.g. 6/7. The set of rational numbers is denotes as ‘Q’ for quotient. See Types of Numbers.

Real number: Rational (fractions) and irrational (numbers with non-recurring decimal representation) numbers. The set of real numbers is denoted as ‘R’ for real. In computing, any number with a fractional (or decimal) part. Basically, real numbers are all numbers except imaginary numbers (such as the square root of -1). See Types of Numbers.

Reciprocal: The multiplicative inverse of a number (i.e., 1/x). It can be shown with a negative index (x-1).

Reflex angle: An angle with a degree measure between 180 and 360. See MathWorld: Geometry: Trigonometry: Angles: Reflex Angle.

Repeating decimal: A decimal that can be written using a horizontal bar to show the repeating digits.

Right angle: An angle with a degree measure 90. An angle which is not an right angle is called oblique angle. See MathWorld: Geometry: Trigonometry: Angles: Right Angle.

Root: If, when a number is raised to the power of n gives the answer a, then this number is the nth root of a (a1/n).

Rounding: To give a close approximation of a number by dropping the least significant numbers. For example 15.88 can be rounded up to 15.9 (or 16) and 15.12 can be rounded down to 15.1 (or 15).

Scalar: A real number and also a quantity that has magnitude but no direction, such as mass and density. See Wikipedia: Scalar.

Scientific notation (exponential notation, standard form): One way of writing very small or very large numbers. In this notation, numbers are shown as (0<N<10) x 10q. An equivalent form is N.Eq. For example; 365,000 is 3.65x105 or 3.65E5. See Wikipedia: Scientific Notation.

Secant line: A line that intersects a curve. The intercept is a chord of the curve. Wikipedia: Mathematics: Secant Line; Mathlets: Secant Line; CTK Glossary: Secant.

Sequence: An ordered set of numbers derived according to a rule, each member being determined either directly or from the preceding terms. See Real Analysis Glossary: Sequences & Context.

Sigma (S, s ): Represents summation (S, s). See Greek Letters.

Significant figure (s.f.): The specific degree of accuracy denoted by the number of digits used. For example 434.64 has five s.f. but at 3 s.f. accuracy it would be shown as ‘435 (to 3 s.f.)’. From the left, the first nonzero digit in a number is the first significant figure, after the first significant number, all digits, including zeros, count as significant numbers (Both 0.3 and 0.0003 have 1 s.f.; both 0.0303 and 0.303000 have 3 s.f.). If a number has to be reduced to a lower s.f., the usual rounding rules apply (2045.678 becomes 2046 to 4 s.f. and 2045.7 to 5 s.f.). The final zero even in a whole number is not a s.f. as it only shows the order of magnitude of the number (2343.2 is shown as 2340 to 3 s.f.).

Sine law: For any triangle, the side lengths a, b, c and corresponding opposite angles A, B, C are related as follows: sin A / a = sin B / b = sin C / c. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. See Wikipedia: Law of Sines.

Skew lines: Two lines in three-dimensional space, which do not lie in the same plane (and do not intersect).

Stationary point: Point at which the derivative of a function is zero. Includes maximum and minimum turning points, but not all stationary points are turning points.

Straight line: A straight line is characterised by an equation (y = a + bx), where a is the intercept and b is the gradient/slope. One of the methods for fitting a straight line is the least squares method.

Subtend: To lie opposite and mark out the limits of an angle.

Subtraction: The inverse operation of addition. In the notation a - b = c, the terms a, b, and c are called the minuend, subtrahend and difference, respectively.

Supplementary angles: Two angles whose sum is 180o. See also complementary angles.

Tangent: The tangent of an angle in a right-angled triangle is the ratio of the lengths of the side opposite to the side adjacent [tan(x) = sin(x) / cos(x)]. A tangent line is a line, which touches a given curve at a single point. The slope of a tangent line can be approximated by a secant line.

Tangent law:  For any triangle, the side lengths a, b, c and corresponding opposite angles A, B, C are related as follows: (a+b) / (a-b) = {tan[1/2(A+B)]} / {tan[1/2(A-B)]}. See Wikipedia: Law of Tangents.

Taylor Expansions: A special kind of power series used as a basis of approximation. A Taylor expansion (series) is the sum of functions based on continually increasing derivatives (till one gets a zero value) if they exist. The accuracy of the approximation improves as the order of the approximation is increased (like fourth derivative or fifth). As the nonlinearity of the function increases the higher order terms become more important (i.e., the error increases as fewer terms of the Taylor series are included in the approximation). The main use of Taylor polynomial approximations is not to estimate the value of a function at a given point (this can be done by a calculator) but to approximate functions over an interval (representing complicated nonlinear functions as series ‘infinite polynomials’ makes life easier; see for example: Applications of Taylor Series). Maclaurin’s expansion is a special case of Taylor’s expansion. Wikipedia: Taylor Series; MathWorld: Taylor Series; a Lecture Note on Taylor Expansions by Luca Sbano; Taylor Polynomials Quizzes; Taylor Series Applet

Transcendental number: A real number that does not satisfy any algebraic equation with integral coefficients, such as x3 - 5x + 11 = 0. All transcendental numbers are irrational and most irrational numbers (non-repeating, non-terminating decimals) are transcendental. Transcendental functions (such as exponential, sine and cosine functions) can burst into chaos under certain circumstances. See Types of Numbers.

Triangle: A three-sided figure that can take several shapes. The three inside angles add up to 180o. Triangles are divided into three basic types: obtuse, right and acute; they are also named by the characteristics of their sides: equilateral, isosceles, and scalene. The area of a triangle is 1/2 x perpendicular height x base.

Trigonometry: The branch of mathematics that is concerned with the trigonometric functions. Trigonometric identities are the results that hold true for all angles. Sin, Cos and Tan are trigonometric ratios; Cosec, Sec and Cot are reciprocal of trigonometric ratios; Arcsin (sin-1), Arccos (cos-1) and Arctan (tan-1) are inverse of trigonometric functions. See Syvum Math: Trigonometric Functions; CTK: Trigonometric Functions; Trigonometry Realms; S.O.S. Math: Trigonometric Identities Table; Wikipedia: Mathematics: Trigonometric Functions / Uses of Trigonometry; ; Dave's Short Trig Course (JAVA).

Union: The union of two sets is the set of elements that are in either of the two sets (compare with intersection).

Unit: A standard measurement.

Variable: An amount whose value can change.

Vector: A quantity characterised by a magnitude and a direction represented by (1) column form: two numbers (components) in a 2x1 matrix; (2) geometric form: by arrows in the (x,y)-plane; or (3) component form: the Cartesian unit vectors i (x-axis unit vector) and j (y-axis unit vector). The magnitude of a vector | u | is the length of the corresponding arrow and the direction is the angle (θ) the vector makes with the positive x-axis. When a1 and a­2­ are the components of the vector a (magnitude | a | = (a12 + a22)1/2), it equals to a = a1i + a2j in component form, which equals to a = | a | cos(θ)i + | a | sin(θ)j. The angle (θ) can be found as arctan (a2 / a1). Cosine rule and sine rule are used for conversion of vectors from one form to another. See Wikipedia: Algebra: Vector, Vector Calculus; Eigenvector.

Vertex: The point where lines intersect.

Whole number: Zero or any positive number with no fractional parts.

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Compiled by  Dilara DORAK & M.Tevfik DORAK

Last updated on 1 July 2013