In the ocean of mathematics, there are uncountable drops of different terms, words, definitions, and glossary. When you start searching for a specific topic and its meaning, you seem to get lost in the amazing world of numbers. Mathematics is the queen of all sciences, and this is reflected in the usage of numbers in our every day life. There is hardly any field, be it biology, physics, chemistry, astronomy, or economics where numbers don't come into play. Our lives would almost come to standstill without this subject. To help you look for the right expressions, this article has a glossary of math terms and definitions, which are alphabetically listed below. Basics Mathematical definitions are deduced from the extensive research and theories. Unless an explanation is not proved correct for an expression, it is always an area of research and debate. The terminology enlisted here have been picked up from a plethora of different branches, like Algebra, Trigonometry, Mensuration, Geometry, Calculus, etc. Branches This field has applications in almost all aspects of life and work. The operations of addition, subtraction, multiplication, and division form the platform for the higher order. Kinematics, Dynamics, Linear algebra, Ring theory, Calculus, and Integration are the most popular research areas. The magical world of Permutations and Combinations, not to mention Probability, has its own wonderful applications in the real world. Read the article below in order to enter this wonderful world. A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |
A AA similarity According to the AA similarity, if two angles of a triangle are congruent to two angles of another triangle, then the triangles are said to be similar to each other. AAS Congruence AAS congruence is called the angle-angle-side congruence. If there are two pairs of corresponding angles and a pair of corresponding opposite sides that are equal in measure, then the triangle is said to be congruent. Abscissa The X-coordinate of a point on the coordinate system is called abscissa. For example, in the ordered pair P(2, 3, 5), 2 will be called the abscissa of the point P. In mathematical language, it will be called the length of the point (P) relative to the X-axis. Absolute Convergence A series that converges when all its expressions are replaced by their absolute values. To check if a series converges absolutely, it is only required to replace any subtraction in the series with addition. A series _n=1Σ^n=∞a_n is absolutely convergent if the series _n=1Σ^{n= ∞} |a_n| converges. Absolute Maximum The highest point of the function or relation over the entire domain is called absolute maximum. The first and second derivative tests are commonly used to find the absolute maximum of a function. Absolute Minimum The lowest point of the function or relation over the entire domain is called the absolute minimum. The first and second derivatives are the commonly used methods to find the absolute minimum. Global minimum is also called the absolute minimum. Absolute Value A general concept of absolute value is that it makes a negative number positive. Absolute value is also called a mod value. The absolute value of a number (say X) is denoted as |X|. Remember, the absolute value uses bars, so don't use parenthesis or any other symbol, else the meaning changes. To put it simply, |-7| = 7 and |7| = 7. Positive numbers and zero are left unchanged in the absolute value. A better and more accurate way of understanding is that the absolute value of a number denotes the distance between the number and origin. So, |x-a| = b, where b>0, says the quantity x-a is b units from 0, x-a is b units right of 0(origin) x-a is b units left of 0(origin). Absolute Value of Complex number The absolute value of a complex number |a + bi| = √a² + b². The absolute value of a complex number is the distance between the origin and the complex plane. For a complex number in the form of r(acosθ + bsinθ), the modulus is r, i.e., the value of radius of the circle carved by the trigonometrical equation. Acceleration The rate of change of velocity with time is called acceleration. Mathematically, the second derivative of the distance traveled by an object is called acceleration. Accuracy The measure of the closeness of a value to the actual value of a result is called accuracy. Acute Angle An angle whose measure is less than 90⁰ is called an acute angle. Acute-Angled Triangle A triangle in which all the interior angles are acute is known as an acute angled triangle. Addition Rule Of Probability Addition rule of probability is meant to find out the probability of occurrence of either or both the events. If P(A) and P(B) are mutually exclusive events, then the probability P(A or B) = P(A) + P(B) else P(A or B) = P(A) + P(B) - P(A and B). Additive Inverse of a Matrix If the sign of every matrix element is changed, then the matrix is said to be an inverse of the original matrix. If A is the matrix, then -A will be the inverse of the matrix. If add a matrix and its inverse, then the sum would be zero, since the each element in the original matrix is negative of the other. Additive Property of Equality Simply stated, additive property states that same number can be added on both side of the equation. For example, x - 3 = 5 is same as x - 3 + 3 = 5 + 3. Adjacent Angles If the two angles share a common vertex and common plane and even have a same side, but if they don't overlap or one of the angles is not contained in the other, then the angles are called adjacent angles. Adjoint Matrix When we take the transpose of the co-factor of the original matrix, then it is known as adjoint matrix. Affine Transformations Affine transformations refer to the combination of the process that can be performed on any co-ordinate system, like translation, rotation, horizontal and vertical stretches, and shrinks. It is to be kept in mind that concurrency and co-linearity are invariant under any type of transformation. Aleph Null The 1st letter of the Hebrew alphabet, Aleph (א)is used to denote the cardinal number of the infinite countable set. Basically, א₀ with a subscript is generally used to denote the cardinality of the infinitely countable set. Algebra It is a branch of pure mathematics that uses alphabets and letters as variables. The variables are the unknown quantities whose values can be determined with the help of other equations. For example, 3X - 7 = 78, is an algebraic equation in one unknown variable (here it is X). Now, by applying algebra techniques we can solve the equation. More on algebra tips. Algebraic Numbers All rational numbers are algebraic numbers. Numbers that are roots of the polynomials with integer coefficients and are under the surd are also included as algebraic numbers. Any number that is not a root of polynomial with integer coefficients is not an algebraic number. These numbers are called transcendental numbers. e and Π are called the transcendental numbers. Algorithm Algorithm is a simple step by step to arrive at the solution of any problem. Alpha Alpha is the 1st letter of Greek alphabet. It is denoted by A (in an upper case) and by α (in lower case). It is frequently used in science as a variable to denote angles, etc. Alternate Angles If two or more parallel lines are cut by a transversal, then the angles formed in the alternate direction to each other are called alternate angles. Alternate Exterior Angles When two or more parallel lines are cut by a transversal, the alternate angles that are exterior to one another is called alternate exterior angle. Alternate Interior Angles When two or more lines are cut by a transversal then the alternate angles that lie interior to each other are called alternate interior angles. Alternate Series Alternating series is a series that consists of alternate positive and negative sides. The alternating series is of the form: 1 - ½ + 1/3 - ¼ + 1/5.......up to infinity. Alternating Series Remainder The alternating series is as follows: _{n = 1} ∑^{n = ∞} = (-1)ⁿ⁺¹a_n = a₁ - a₂ + a₃ + ... If the series converges to S, by applying the alternating series test, then, the remainder, R_n = S - _k=1∑ⁿ(-1)^k+1a_k, for all n ≥ N, is called the alternating series remainder. Also, |R_n| ≤ a_{n + 1}. Altitude Altitude is the shortest distance between the base to the apex of a figure like cones, triangle etc. Altitude of a Cone The distance between the apex of the cone and its base is called the height or the altitude of the cone. Altitude of a Cylinder The distance between the circular bases of the cylinder or the length of the line segment between two of its bases is known as altitude of a cylinder. Altitude of a Parallelogram The distance between the opposite sides of a parallelogram is called altitude of a parallelogram. Altitude of a Prism The distance between the two bases of a prism is called the altitude of a prism. Altitude of a Pyramid The distance between the apex of the pyramid to the base is called altitude of the pyramid. Altitude of a Trapezoid The distance between the two bases of the trapezoid is called altitude of a trapezoid. Altitude of a Triangle The shortest distance between the vertex of the triangle and the opposite side is called altitude of the triangle. Amplitude It is the measure of half the distance between the maximum and minimum range. For example, if you consider a sine wave, then ½ of the distance between the positive and negative curves in called an amplitude. It is to be remembered that only periodic functions with bounded range have amplitude. Analytic Geometry Analytical geometry is a branch that deals with the study of geometric figures with the help of co-ordinate axes. The points are plotted and with the help of the points we can easily find out the required information. Analytic Methods If you are asked to analytically solve a problem, then it means that you are not supposed to use a calculator. Analytical methods are used to solve the problems by the help of algebraic and numeric methods. Angle Angle is defined as the figure formed by touching the ends of two rays. In other words, it means two rays sharing a common point. Angle Bisector The line that bisects an angle into two equal halves is called an angle bisector. Angle of Depression The angle below the horizontal line that a observer must see in order to site the object is called angle of depression. To understand it better, consider an observer at the top of the cliff, when he has to sight an object down at some distance from the base of the cliff, the angle subtended he will have to subtend in order to site the object is called angle of depression. Angle of Elevation Angle of elevation is geometrically congruent to the angle of depression. If a viewer is observing an object at some elevation, then he needs to raise his line of sight above a horizontal level, this is called the angle of elevation. Angle of Inclination of a Line The angle subtended by a line with the x-axis is called angle of inclination of the line. The angle of inclination is always measured in counter clockwise direction, that means positive direction of the x-axis. The angle of inclination is always between the range 0⁰ to 180⁰. Annulus The area between two concentric circles of a ring (say) is called annulus. Anticlockwise The direction opposite to that of the movements of the watch. In this case, it is an assumption that the anticlockwise direction is always measured positive. Antiderivative of a Function If F(x) = 2x² + 3, then, its derivative F'(x) = 4x. Here 4x is called the antiderivative of F(x). Antipodal Points In three dimensions, the points diametrically opposite on a sphere is called antipodal points. Apothem Apothem is the same as the inradius of an inscribed circle in a regular polygon. In other words, it would mean the distance from any of midpoint of the sides of the polygon to the center of the polygon. Approximation by Differentials By the rule of approximation of differentials, the value of a function is approximated and the principles of derivation are used in this method. The formula used in the approximation by differentials is, f(x + ∆x) = f(x) + ∆y = f(x) + f'(x)∆x, where f'(x) is the differential of the function. Arc Length of a Curve The length of the line of a curve is called the arc length. There are three formulas to determine the arc length of a curve. There is a rectangular form, polar form, and parametric form that can be used.

• Rectangular form- ds = [1 + (dy/dx)²dx]^1/2
• Parametric form- ds = (dx/dt)² + (dy/dt)²dt]^1/2
• In polar form- ds = [r² + (dr/dƟ)²]^1/2

Area of a Circle The area of a circle is given by the formula Πr². Arccos The inverse function of a cosine function is called the arccos function. For example, cos^-1(1/2) (read as cos inverse half) or"the angle whose cosine is equal to ½. As we all know it nothing but 60⁰. Arccosec The inverse of a cosec function is called arccosec function. For example, cosec^-1(2) means the angle whose cosecant is equal to 2. The answer is 30⁰. It is to be noted that there can be many more angles with the cosecant equal to 30⁰. What we want is the most basic angle that gives the cosecant equal to 30⁰. For other angles, we need to consider the range of the function. Arccot Arccot is the inverse of the cotangent function. For example, cot^-1(1) means the angle whose cotangent is equal to 1. Cot^-11 = 45⁰. Arcsec The inverse of a secant function is called the arcsec function. For example, sec^-12 means the angle whose secant is equal to 2. Sec^-12 = 60⁰. Arcsin The inverse of a sine function is called arcsin function. For example, sin^-1(1/2) = 30⁰. Arctan The inverse of a tangent function is called arctan function. For example, tan^-1(1) = 45⁰ Area below a Curve The area occupied by any curve is called area that the curve forms together with the x and y axes. The area of a function y = f(x) is given by the definite integral _aʃ^b, where a and b are the limits of the function. Area = _aʃ^b f(x) dx Area between Curves The area between two curves y = f(x) and y = g(x) is given by the formula, Area = _aʃ^b |f(x) - g(x)|dx, where f(x) and g(x) is the area bounded above and below the x and y axis whereas x= a and x=b, on the left and right. Area of a Convex Polygon If (x₁, y₁), (x₂, y₂), ... , (x_n, y_n) are the co-ordinates of a convex polygon then the area of the polygon is found out by the determinant method. The expanded form of the determinant is as follows: 1/2[(x₁y₂) + x₂y₃+ x₃y₁+ ... x_ny₁)] - [y₁x₂ + y₂x₃ + ... y_nx₁]. Area of an Ellipse The area of an ellipse is given by the formula ∏ab, where a and b are the lengths of the major and minor axis of the ellipse. If the ellipse has its center at (h, k) then, Area = [(x-h)²/a² + (y-k)²/b²] Area of an Equilateral Triangle The area of an equilateral triangle is given by: a²√3/4, where a = side of the equilateral triangle. Area of a Kite The area of a kite is given by: ½ (product of the diagonals) = ½ x d₁d₂. Area of a Parabolic Segment The area of a parabolic segment is given by 2/3 of the product of width and height. Area of a Parallelogram Area of parallelogram = height x base of the parallelogram. Area of a Rectangle Area of rectangle = length x breadth Area of a Regular Polygon Area of regular polygon = ½ x apothem x perimeter. Area of a Rhombus Diagonals of a rhombus are perpendicular to each other. Area = ½ x product of diagonals or Area= h x s, where h and s are the height and side of the rhombus. Area of a Segment of a Circle We all know the area of a circle, but what if the area of a segment is to be found out, well the formula for area of a segment of a circle is: Area = 1/2r²(θ - sinθ) (radians) Area of a Trapezoid Area of a trapezium = ½ x (sum of the non- parallel sides) x h = ½ x (b₁ + b₂) x h Area of a Triangle There are various formulas to calculate the area of a triangle that are as follows.

• Area = A = ½ x base x height
• A = ½ x ab SinC = ½ x bc SinA = I/2 x ca SinB, where A, B and C are the angles of the triangle respectively.
• Given s= a+b+c/2 (semi perimeter), by Heron's Formula, A= [s(s-a)(s-b)(s-c)]^1/2.
• If 'r' and 'R' are the inradius and circumradius of the incircle and outercirlce of a triangle, then the Area (A) = rs and A= abc/4R, a, b and c are the sides of the triangle.

Area Using Polar Coordinates When the polar co-ordinates are involved in computation of the area then the area is given by: The area between the graph r = r(θ) and the origin and also between the lines θ = α and θ = β is given by the formula: Area = ½ _αʃ^βr²dθ Argand Plane The complex plane is called the argand plane. Basically, argand plane is used to denote the complex numbers graphically. The x-axis is called the real axis and the y-axis is called the imaginary axis. Argument of a Complex Number In order to describe the angle or inclination of a complex number on the argand plane, we use the term argument. Argument of a complex number is measured in radians. The polar form of a complex number is given by r(cosθ + isinθ) and the argument in this is given by θ. Argument of a Function The expression in which the function operates is called argument of the function. The argument of the function y= √x is x. Argument of a Vector The measure of an angle describing a vector or a line in the complex number analysis is called the argument of the vector. Arithmetic Mean The most simple average technique that we use in day-to-day life. For example, if there are 4 quantities then there arithmetic mean is given by, Arithmetic mean = (a + b + c + c + d)/4 Arithmetic Progression A series that has same common difference among its terms. For example, 1, 3, 5, 7, 9 ... up to infinity. The nth expression of an arithmetic progression is given by, T_n = a + (n-1)d, where a = 1st term, n = number of terms, and d= common difference. It is also called arithmetic sequence. The sum of an arithmetic progression is given by: S = n/2[2a + (n-1)d] or S = n(a₁ + a_n)/2, here n= number of terms. Arm of an Angle One of the rays/line forming an angle with the other is called the arm of an angle. Arm of a Right Triangle Any of the sides of the right angled triangle is called the arm of a right angled triangle. Associative The operation a + (b+c) = (a + b) + c is called associative operation. Addition and multiplication are associative while division and subtraction are not. For example, (4+5)+ 7 = 4 + (5+7) Asymptote An asymptote is a curve or line that approaches the curve very closely. There are horizontal and oblique asymptotes but not vertical asymptotes. Augmented Matrix The matrix representation of a set of linear equations is called the augmented matrix. For example, 3x - 2y = 1 and 4x + 6y = 4, then in a matrix form 3, -2 and 1 (from 1st equation) and 4, 6 and 4 (from 2nd equation) form the elements of 3x3 matrix respectively. Average Average is same as the arithmetic mean. Average Rate of Change The change in the slope of a line is called the average rate of change of the line. Also, the change in value of a quantity divided by time is average rate of change. Average Value of a Function For a function y =f(x), in the domain [a,b] the average value is given by the formula (1/b-a)_aʃ^bf(x)dx Axes The x, y, and z axes are known as the axes of a co-ordinate system. Axiom A statement that has been assumed to be true without any proof. Axis of a Cylinder The line that passes exactly through the center of the cylinder and also passes through the bases of the cylinder. Simply stated, the line that divides the cylinder into two equal halves vertically. Axis of Reflection A line across which the reflection takes place. Axis of Rotation An axis along which the rotation of the axis takes place. Axis of Symmetry A line along which the geometrical figure or the shape is symmetrical. Axis of Symmetry of a Parabola The axis of symmetry of a parabola is the line that passes through the focus and vertex of parabola. B Back Substitution Back substitution is a technique that is used to solve a system of linear equations that has already been changed to row-echelon form and reduced row-echelon form. After changing the equation, the first equation is solved, then the second last, then the next before that and so forth. Base (Geometry) The bottom part of a geometrical figure, like a solid object or a triangle is called the base of the object. Base of an Exponential Expression Consider the expression a^x. Then 'a' can be called the base of the expression a^x. Base of an Isosceles Triangle The base of an isosceles triangle is the non-congruent side of the triangle. In other words, it is the side other than the legs of the triangle. Base of a Trapezoid The trapezoid has four sides with two sides parallel. Either of the two parallel sides can be considered as the base of the trapezoid. Base of a Triangle Base of a triangle is the side at which an altitude can be drawn. It is the side, which is perpendicular to the altitude. Bearing Bearing is a method used to represent the direction of a line. If there are two points A and B, then we can say that A has a bearing of θ degrees from point B, if the line connecting A and B makes an angle θ with a vertical line drawn through B. The angle is measured in clockwise direction. Bernoulli Trials In the field of statistics, Bernoulli trials are the experiments where the outcome can be either true or false. In Bernoulli trials, all events must be independent. The binomial probability formula is given by p (k successes in n trials) = ⁿC_rp^kq^{n - k}, where, n= number of trials, k = number of successes, n - k = number of failures, p = probability of successes in trials, q = 1 - p, probability of failure in one trial. Beta (Β β) A Greek letter used frequently as a symbol to denote variables. Biconditional It is the method of expressing a statement containing more than one condition, that means a condition and its converse. These statements are called biconditionals. They are represented by the symbol ⇔. For example, the following statements can be called biconditionals: "A given triangle is equilateral" is same as "All the angles of a triangle measure 60º." Binomial A binomial can be simply defined as a polynomial, which has two terms, but they are not like terms. For example, 3x - 5z³, 4x - 6y². Binomial Coefficients The coefficients of the various expressions in the binomial expansion of the binomial theorem are called binomial coefficients. Mathematically, a binomial coefficient equals the number of r items that can be selected from a set of n items. They are simply called binomial coefficients, because they are coefficients of the binomial expanded expressions. Generally, they are represented by ⁿC_r. Binomial Coefficients in Pascal's Triangle Pascal's triangle is an arithmetic triangle that is used to calculate the binomial coefficients of the various numbers. The binomial coefficients (ⁿC_r) in the pascal's triangle are called the binomial coefficients in pascal's triangle. Pascal's triangle finds major use in algebra and probability/binomial theorem. Binomial Probability Formula The probability of getting m successes in n trials is called binomial probability formula. The formula is given by: Formula: P(m successes in n trials) = _mCⁿp^kq^n-k, where, n = number of trials m = number of successes n - m = number of failures p = probability of success in one trial q = probability of failure in one trial. Binomial Theorem A theorem used to expand the powers of polynomial and equations. It is given by: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁a^n-1b + ... +ⁿC_n-1ab^n-1 + ⁿC_n. Boolean Algebra Boolean algebra deals with the logical calculus. Boolean algebra takes only two values in the logical analysis, either 1 or zero. Read more on Boolean Origination. Boundary Value Problem Any differential equation that has constrained on the values of the function (not that on the derivatives) is called the boundary value problem. Bounded Function A function that has a bounded range. For example, in the set [2, 9], 9 the upper bounded number and 2 is the lower bounded number. Bounded Sequence A sequence that is bounded with upper and lower bounds. Like the harmonic series, 1, ½, 1/3, ¼, ... up to infinity is a bounded function since the function lies between 0 and 1. Bounded Set of Geometric Points The bounded set of geometric points is called the figure or set of points that can be enclosed in a fixed space or co-ordinates. Bounded Set of Numbers A set of numbers with lower and upper bound. For example, [3, 7] is called the bounded set of numbers. Bounds of Integration For a definite integral, _aʃ^b f(x)dx, a and b are called the bounds or limits of integration. The bounds of integration also indicate limits of integration. Box A rectangular parallelepiped is often referred to as a box. The volume of such a rectangular box is given by the product of length, breadth, and height. Box and Whisker Plot The box and whisker plot is a beginning lesson for starters, in order to let them understand the basics of handling data value. Box and whisker plot shows certain data, not the complete statistics of the recorded data. Five number summary is another name for visual representation of the box and whisker plot. Boxplot A data that displays the five number summary in a diagrammatic form represented as:

Smallest

1st Quartile

Median

3rd Quartile

Largest

Braces The symbolic representation {or} that is used to indicate sets etc. Brackets The symbol [ ] which signifies grouping. They work in a similar way parentheses do. C Calculus The branch that deals with integration, differentiation, and various other forms of derivatives. Cardinal Numbers Cardinal numbers are used to indicate the number of elements in an infinite or finite sets. Cardinality It is same as cardinal numbers. It is to be noted that cardinality of every infinite set is same. Cartesian Coordinates The Cartesian coordinates are the axes that are used to represent the coordinates of a point. (x,y) and (x,y,z) are the Cartesian coordinates. Cartesian Plane The planes formed by horizontal and vertical axes like the x and y axis is called the Cartesian plane. Catenary The curve formed by a hanging a wire or a ring is called catenary. Generally, a catenary is confused with a parabola. However, though the looks are similar, it is not same as the parabola. The graph of a hyperbolic cosine function is called the catenary. Cavalieri's Principle A method to find the volume of solids by using the formula V = bh, where b = area of cross section of the base (cylinder/prism) and h = height of the solid. Central Angle An angle in a circle with vertex at the circle's center. Centroid The intersection point of the three medians of a triangle. Centroid Formula The centroid of the points (x₁, y₁, x₂, y₂, ... x_n, y_n) is given by: (x₁ + x₂ + x₃+ ... x_n)/n , (y₁ + y₂ + y₃+ ... y_n)/n Ceva's Theorem Ceva's theorem is a way that relates the ratio in which three concurrent cevians divide a triangle. If AB, BC, and CA are the three sides of a triangle, and AE, BF and CD are the three cevians of the triangle, then according to Ceva's theorem, (AD/DB)(BE/EC)(CF/FA) = 1. Cevian A line that extends from the vertex of a triangle to the opposite side like altitudes and medians. Chain Rule A method used in differential calculus to find the derivative of a composite function. (d/dx)f(g(x)) = f^'((g(x))g^'(x) or (dy/dx) = (dy/du)(du/dx) Change of Base Formula A very useful formula in logarithm that is used to express a certain logarithmic function in a different base. That's why it is called base change formula. Base Change Formula: log_ax = (log_bx/log_ba) Check a Solution Checking a solution means putting the value of corresponding variables in the equation and verify if the equations satisfy the given equation or systems of equation. Chord A chord is a line segment that joins the two points on a curve. In a circle, the largest chord is the diameter that joins the two ends of the circle. Circle The locus of all points that is always at a fixed distance from a fixed point. Circular Cone A cone with a circular base. The volume of circular cone is given by V = 1/3πr² Circular Cylinder A cylinder with circle as the base. Circumcenter The center of a circumcircle is called circumcenter. Circumcircle A circle that passes through all the vertices of a regular polygon and triangles is called circumcircle. Circumference The perimeter of a circular figure. Circumscribable A plan figure that has a circumcircle. Circumscribed A figure circumscribed by a circle. Circumscribed Circle The circle that touches the vertices of a triangle or a regular polygon. Clockwise The direction of the moving hands of a clock. Closed Interval A closed interval is the one in which, both the first and last terms are included while considering the entire set. For example, [3,4]. Coefficient The constant number that is multiplied with the variables and powers in an algebraic expression. For example, in 234x²yz, 243 is the coefficient. Coefficient Matrix The matrix formed by the coefficients of a linear system of equations is called the coefficient matrix Cofactor When a determinant is obtained by deleting the rows and columns of a matrix in order to solve the equation, it is called the cofactors. Cofactor Matrix A matrix with the elements of the cofactors, term by term, in a square matrix is called the cofactor matrix. Cofunction Identities Cofunction identities are the identities that show the relation between the trigonometrical functions like the sine, cosine, cotangent. Coincident If two figures are superimposed on each other, then they are said to be coincident. In other words, a figure is coincident when all points are coincident. Collinear Two points are said to be collinear if they lie on the same line. Column of a Matrix The vertical set of numbers in a matrix is called the column of the matrix. Combination A selection of objects from a group of objects. The order is irrelevant in the selection of the object. Combination Formula A formula that is used to determine the number of possible combinations of r objects from a set of n objects. The formula involves the binomial coefficients and is given as: ⁿC_r. It is read as 'n choose r' Combinatorics The branch that studies the permutations and combinations of objects and materials. Common Logarithm The logarithm to the base 10 is called common logarithm. Commutative An operation is said to be commutative if x ø y = y ø x, for all values of x and y. Addition and multiplication are commutative operations. For example, 4 + 5 = 5 + 4 or 6 X 5 = 5 X 6. Division and subtraction are not commutative. Compatible Matrices Two matrices are said to be compatible for multiplication if the number of columns of 1st matrix equals to the number of rows of the other. Complement of an Angle The complement of angle say 75º is 90º - 75º = 15º. Complement of an Event The set of all outcomes of an event that are not included in the event. The complement of set A is written as A^c. The formula is given as: P(A^c) = 1 - P(A) or P (not A) = 1- P(A). Complement of a Set The elements of a given set that are not contained in the given set. Complementary Angles If the sum of two angles is 90º, then they are said to be complementary angles. For example, 30º and 60º are complementary to each other as their sum equals 90º. Composite Number A positive integer whose factors are the numbers other than 1 and the number itself. For example, 4, 6, 9, 12 etc. 1 is not a composite number. Compound Fraction A compound fraction is a fraction that has at least one fraction term in the numerator and denominator. Compound Inequality When two or more than two inequalities are solved together it is known as compound inequality. Compound Interest While calculating compound interest, the amount that is earned as an interest for a certain sum/principal is added to the original principal, and from that, the interest is calculated on the new principal. Thus, the interest is not only calculated on the original balance, but the balance or principal obtained after adding the interest. Concave Concave is a shape of a figure or a solid that has a surface curving inwards or bulging inwards. It is also known as non - convex. Concave down or concave up are the other forms of concave shapes. Concentric The geometrical figures that are similar in shape and share a common center. Usually, the term concentric is used for concentric circles. Concurrent If two or more than two lines or curves intersect at the same point, then they are said to be concurrent at that point. Conditional Equation An equation that is true for some values of the variables and is false for other values of the variables. The equation has certain conditions imposed on it that are only satisfied by certain values of the variables. Cos^-1x The inverse of cos function is read as 'cos inverse x'. For example, cos^-1½ = 60º. Cot^-1x By cot^-1x, we mean the angle whose cotangent is equal to x. For example, when we are asked to find the smallest angle whose cotangent is equal to 1? The answer is 45º. Thus, cot^-11 = 45º. Cube A cube is a three dimensional figure bounded by six equal sides. The volume of cube is given by l³, where l is the side of a cube. Cube Root A cube root is a number denoted as x^⅓ , such that b³ = x For example, (64)^⅓ = 4. Cubic Polynomial A polynomial of degree 3 is known as the cubic polynomial. For example, x³ + 2x² + x. Cuboid Cuboid is a three dimensional box that has length, width, and height. It is also called a rectangular parallelepiped. D De Moivre's Theorem De Moiver's Theorem is a formula that is widely used in complex number system in order to calculate the powers and roots of complex numbers. It is given by: [r(cosθ + isinθ)]ⁿ = rⁿ(cosnθ + isinnθ). Decagon A 10 sided polygon is called decagon. Deciles In statistics, deciles are any of the nine values that divide the data into 10 equal parts. The first decile cuts off at the lowest 10% of the data that is called the 10th percentile. The 5th decile cuts off the lowest 50% of the data that is called 50th percentile or 2nd quartile or median. The 9th decile cuts off lowest 90% of the data that is the 90th percentile. Decreasing Function A function whose value decreases continuously as we move from left to right of its graph is called decreasing function. A line with negative slope is a perfect example of a decreasing function, where the value of the function decreases as we proceed on the x-axis. If the decreasing function is differentiable, then its derivative at all points (where the function is decreasing) will be negative. Definite Integral An integral that is evaluated over an interval. It is given by _aʃ^bf(x)dx. Here the interval is [a, b]. Degenerate Conic Sections If a double cone is cut with a plane passing through the apex of the plane, it is called the degenerate conic sections. It has the general equations of the form: Ax² + Bxy + Cy² + Dx + Ey + F = 0 Degree (angle measure) Degree is the measure of the slope or the angle that a line or a plane subtends. Degree is represented by the symbol '°'. Degree of a Polynomial The power of a highest term in an algebraic expression is called the degree of the polynomial. In the expression 2x⁵ + 3y⁴ + 5x³, the degree of the polynomial is 5. Degree of a Term In 5y⁷, degree of term is 7, in 5x²4y³, the degree of the term is the sum of the exponents of 5x and 4y, that means 5. Del Operator Del operator is denoted by symbol ∂(x, y, z)/∂x. The del operator ∇ = (∂/∂x, ∂/∂y) or ( ∂/∂x, ∂/∂y, ∂/∂z ) Deleted Neighborhood Deleted neighborhood of a set is defined as a set {x: 0 < ;|x - a| < δ}. For example, one deleted neighborhood of 2 is the set {x: 0 < |x - 2| < 0.1}, which is the same as (1.9, 2) ∪ (2, 2.1). Delta (Δ δ) A Greek letter representing the basic discriminant of a quadratic equation. Denominator The lower part of a fraction is called denominator. In fraction (4/5), 5 is the denominator. Dependent Variable Consider an expression y = 2x + 3, here, x is the independent variable and y is the dependent variable. It is a general notion to plot the graph by taking independent variable on x axis and dependent variable on Y-axis. Derivative The slope of a line tangent to a function is called the derivative of the function. This is the graphical interpretation of the derivative. As a differentiation operation, consider f(x) = x² then it's derivative is f^'(x) = 2x. Descartes' Rule of Signs A method for determining the maximum number of positive zeros of a polynomial. According to this rule, the number of changes in the sign of the algebraic expression gives the number of roots of the expression. Determinant Determinants are the mathematical objects that are very useful in determining the solution of a set of system of linear equations. Diagonal Matrix A square matrix that has zeros everywhere except the main diagonal. Diagonal of a Polygon A line segment joining non-adjacent vertices of a diagonal. If a polygon is of n-sides, then the number of diagonals is given by the formula: n(n-3)/2 diagonals. Diameter The longest chord of a circle is called diameter. It can be also defined as the line segment that passes through the center of the circle and touches both the ends of the circumference of the circle. Diametrically Opposed Two points directly opposite to each other on a circle. Difference The result of subtracting two numbers is called difference. Differentiable A curve that is continuous at all points of its domain is called a differentiable function. In other words, if a derivative exists for a curve at all points of the domains variable, it is said to be differentiable. Differential A tiny or infinitesimal change in the value of a variable. Differential Equation An equation involving the functions and derivatives. For example, (dy/dx)² = y Differentiation Performing the process of finding a derivative. Digit Any of the numbers among the nine digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Dihedral Angle The angle formed by the intersection of two planes. Dilation Dilation refers to the enlargement of a geometrical figure by transformation method. Dilation of a Geometric Figure A transformation in which all distances are increased by some common factor. The points are stretched from a common fixed point P. Dilation of a Graph In graphical dilation, the x-coordinates and y-coordinates are enlarged by some common factor. The factor by which the transformation of the graph is done, must be greater than 1. If the factor is less than 1, it is called compression. Dimensions The sides of a geometrical figure are often referred to as dimensions. Dimensions of a Matrix The number of rows and columns of matrix is called the dimensions of the matrix. For example if a matrix has 2 rows and 3 columns, its dimensions will be 2X3 (read as two cross three). Direct Proportion When one of the variables is a constant multiple of the other, it is called direct variation. For example, y = kx (here y and x are the variables and k is a constant factor). Directrices of an Ellipse Two parallel lines on the exterior of an ellipse that are perpendicular to the major axis. E e e is a transcendental number that has a value approximately equal to 2.718. It is frequently used while working with logarithms and exponential functions. Eccentricity A number that indicates the shape of a curve. It is represented by the small letter 'e' (This e is in no ways related to the exponential e = 2.718). In conic section, the eccentricity of the curves is a ratio between the distance from the center to focus, and either the horizontal or vertical distance from the center to the vertex. Echelon Form of a Matrix An echelon matrix is used to solve a system of linear equations. Edge of a Polyhedron One of the line segments that together make up the faces of the polyhedron. Element of a Matrix The numbers inside the matrix in the form of rows and columns is called the element of matrix. Element of a Set Any point, line, letter, number, etc. contained in a set is called the element of the set. Empty Set A set that doesn't contains any element. The empty set is represented by {} or Ø. Equality Properties of Equation The equality properties of algebra that are used to solve the algebraic equations. The definitions of these equality properties are as follows: x = y means, x is equal to y and y ≠ x means y is not equal to x. The operations of addition, subtraction, multiplication, and division all hold true for equality properties of equation. Reflexive Property- x = x; Symmetric Property- If x = y then y = x; Transitive Property- If x = y and y = z then x = z Equilateral Triangle An equilateral triangle has all its three sides equal and the measure of each angle is 60º. Equivalence Relation Any equation that is reflexive, symmetric, and transitive. Equivalent Systems of Equation Two sets of simultaneous equations that have same solution. Essential Discontinuity It is a type of discontinuity in the graph that cannot be removed by simply adding a point. At the point of essential discontinuity, the limit of the function doesn't exist. Euclidean Geometry The geometrical study of lines, points, angles, quadrilaterals, axioms, theorems, and other branches of geometry is called the Euclidean geometry. Euclidean geometry is named after Euclid, one of the greatest Greek mathematicians and known as the 'Father of Geometry'. Read more on Famous mathematicians. Euler's Formula Euler's formula is given by e^iπ + 1= 1. It is a widely used formula in complex number analysis. Euler's Formula in Polyhedron For any polyhedron, the following relation holds true: [Number of faces(n)] - [number of vertices(v)] - [number of edges(e)] = 2. This formula holds true for all convex and concave polyhedron. Even Function A function whose graph is symmetric about y-axis. Also, f(-x) = f(x). Even Number The set of all integers that are divisible by 2. E= {0, 2, 4, 6, 8......} Explicit Differentiation The derivative of an explicit function is called the explicit differentiation. For example, y = x³ + 2x² - 3x. Differentiating it gives, y'= 3x² + 4x - 3. Explicit Function In an explicit function, the dependent variable can be totally expressed in terms of independent variable. For example, y= 5x² - 6x. Exponent Rules The exponential rules are as follows.

Serial Number	Exponential Formula
1	anam = an+m
2	(a.b)n = an. bn
3	a0 = 1
4	(am)n = amn
5	am/n = n√am
6	a-m = 1/a-m
7	(am/an )= a(m-n)

Extreme Value Theorem According to this theorem, there is always at least one absolute maximum and one absolute minimum for any continuous function over a closed interval. Extreme Value of a Polynomial The graph of a polynomial of degree n has at most n-1 extreme values (either maxima or minima) F Face of Polyhedron Polygonal outer boundary of a solid object having no curved surfaces. Factor of an integer If the given integer is divided evenly by another integer then the resultant is called factor of an integer. For example: 2, 4, 8, 16, etc., are the factors of 32. Factor of polynomial If polynomial P(x) is completely divided into polynomial R(x) by Q(x), then Q(x) is called the factor of polynomial. For example: P(x)= x²+6x+8, Q(x)=x+4 then P(x)/Q(x)= x+2. Q(x)=x+4 is the factor. Factor theorem When x-a is factor of P(x), the value x in P(x) is replaced with a, then if the resultant value is 0, such a theorem is called Factor theorem. For example: P(x)= x²+6x+24. Q(x)= x-(-4). If x is replaced with a, that is -4, then P(x)= 0. Factorial The product of an integer with all the consecutive smaller integers is called a factorial. It is represented as "n!". For example: 5! = 5*4*3*2*1= 120. Factoring Rules These are the formulas that govern the factorization of a polynomial. For example,

• x²-(a+b)x +ab= (x-a)(x-b).
• x²+2(a)x+a²=(x+a)²
• x²-2(a)x +a²=(x-a)²

Read more on factor by grouping. Fibonacci series It is a series of numbers where the next number is found by adding the previous two numbers in the series. The first two numbers of the series are 0 and 1. The series is 0,1,2,3,5,8 ... Finite The term is used to describe a set in which all the elements can be counted using natural numbers. First Derivative A function F(a), which governs the slope of the curve at any given point, or the slope of the line drawn tangent to the curve from that point in the plane is called the first derivative. It is represented as F'. For F(x)= 5x². F'(x)=10x will be the slope of the curve. First Derivative test A technique which is used to determine the capacity of inflection point.(minimum, maximum, or neither) First Order of the differential equation A differential equation P(a) who's order is 1. For example: P(a)=3a, here the order of a is 1. Flip It is also known as axis of reflection. It is a line, which divides the plane or a geometric figure into two halves that are mirror images of each other. Floor Function (Greatest Integer Function) It is a function F(x), which is responsible for finding the greatest integer less than the actual value of P(x). For example: P(x)= 5.5, here the greatest integer less than 5.5 is 5. The function which gives F(x)=5 becomes floor function. Foci of the Ellipse They are the fixed two points inside the ellipse such that the vertical curve is governed according to the equation L1+L2= 2a and horizontal curve according to equation L1+L2=2b, where L is the distance between the focal point and the curve, a is the horizontal radius and b is the vertical radius. Foci of hyperbola They are fixed two points inside of the curve of hyperbola such that the determinant of the L1-L2 is always constant. L1 and L2 are the distances between point P (which is the curve) and respective focus of the curve. Focus The curves of the conic sections are governed according to distances from a special point called focus. Focus of Parabola In a parabola, the distance between a point P on the curve and an arbitrary point inside parabola, which is equal to the distance between the same point P and directrix of the curve. Such an arbitrary point is called focus of the parabola. FOIL method FOIL is an acronym for First Outer Inner Last. It is method by which binomials are multiplied. The Multiplication order is

• First terms of Binomials
• Outer terms of Binomial
• Inner terms of binomials
• Outer terms of Binomials.

For example: (a+b)(a-b)= a.a+a.(-b)+b.a +b.(-b) Formula The relationship between various variables (sometimes expressed in the form of an equation) depicted using symbols. For example: a+b=7 Fractal When every part of the figure is similar to every other part of other figure, then the figure is called fractal. Fraction It is a ratio between two numbers. For example: 9/11. Fraction Rules The rules of algebra used for uniting various the fractions. Fractional Equation The expression in the form of A/B on both the sides of equal sign is called fractional equation. For example: x/6= 4/3. Function Operation Various operations, such as additions, subtractions, multiplications, divisions and compositions which have a combining effect on various functions. For example: F(a/b) = F(a)/F(b). Fundamental theorem of Algebra Every polynomial characterized by single variable having complex coefficients, will have a minimum of at least one root which is also complex in nature. Fundamental Theorem of Arithmetic The statement that the factors of a prime number are always distinct and unequal is the fundamental theorem of arithmetic. Fundamental Theorem of Calculus Differentiation and integration are two most basic operations of the calculus. The theorem that establishes a relationship between them is called Fundamental theorem of Calculus. G Gauss-Jordan Elimination A method of solving a system of linear equations. In this process, the augmented form of the matrix system is reduced into row echelon form by means of row operations. Gaussian Elimination A method of solving a system of linear equations. In the Gauss elimination method, the augmented form of matrix is reduced to row echelon form and then the system is solved by back substitution. Gaussian Integer Gaussian integers are the integers in the complex numbers that are represented by a + bi. For example, 3 + 2i, 5i and 6i + 5 are called Gaussian integers. GCF The largest integer that divides a certain set of numbers. Its full form is called Greatest Common Factor. For example, the GCF of 20, 30, and 60 is 10. General Form for the Equation of a Line The general form of equation of a line is represented by the equation- Ax + By + C = 0, where, A, B and C are integers. Geometric Figure A geometric figure is a set of points on the plane or space that leads to the formation of figure. Geometric Mean Geometric mean is a method of finding the average of certain set of numbers. For example, if there are numbers a₁, a₂, a₃, ... , a_n, then multiply the numbers and take the nth root of the product. Geometric Mean = (a₁, a₂, a₃, ... , a_n)^½ Geometric Progression A geometric progression is a sequence whose terms are in a constant ratio with the previous terms. For example, 2, 4, 8, 16, 32, ... , 28 are the terms of a geometric progression. Here, the common ratio is 2. (as 4/2 = 8/4 = 16/8 ... ) Geometric Series Geometric series is a series whose successive terms are in a constant ratio. An example of geometric series is 2, 4, 8, 16, 32, ... Geometry The study of geometric figures in two and three dimensions is called geometry. Greatest lower bound The greatest of all lower bounds of a set of numbers is called the GLB or greatest lower bound. For example, in the set [2,7], the GLB is 2. Glide Reflection A transformation in which a figure has to go through a combination of steps of translation and reflection. Global Maximum The highest point on the graph of a function or a relation (in the domain of the function). The first and second derivative tests are used to find the maximum values of a function. It is also called global maximum, absolute maximum, and relative maximum. Global Minimum The lowest point on the graph of a function or a relation. The first and second derivative tests are used to find the minimum values of a function. It is also called the global minimum, absolute minimum, or global minimum. Golden Mean The ratio (1 + √5)/2 ≈ 1.61803 is called the golden mean. The unique property of golden mean is that the reciprocal of golden mean is about 0.61803. Hence, the golden mean is one plus its reciprocal. Golden Rectangle If the ratio of length and breadth of a rectangle is equal to the golden mean, then the rectangle is called the golden rectangle. It is believed that this rectangle is most pleasing to the eyes. Golden Spiral A spiral that can be drawn inside the golden rectangle. Googol The number 10¹⁰⁰ is called googol. Googolplex Googolplex can be written as 10¹⁰⁰¹⁰⁰. Graph of an Equation or Inequality The graph obtained by plotting all the points on the coordinate system. Graphic Methods The use of graphical methods to solve the mathematical problems. Great Circle A circle that is drawn on the surface of the sphere and shares a common center with the circle. Greatest Integer Function The greatest integer function of any number (say x) is an integer 'less than or equal to x'. The greatest integer function is represented as [x]. For example, [3.4] = 3 and [-2.5] = 3 H Half Angle identities The identities of trigonometry that are used to calculate the value of sine, cosine, tangent, etc., of half of a given angle. The trigonometric identities are as follows: sin²x = (1 - cos2x)/2 cos²x = (1 + cos2x)/2 Half Closed Interval/Half Open Interval It is a set of all numbers containing only one end point. Harmonic Mean The inversion of the summation of the reciprocals of a set of numbers. For example: (1, 2, 3) are in a set then their harmonic mean is 1/(1+ ½+ ⅓ ) Harmonic Progression It is a sequence in which every term is the reciprocal of the natural number. For example 1, ½, ⅓, ¼. Harmonic Series The summation of all the terms in harmonic progression. For example: 1+ ½+ ⅓+ ¼ Height The least measurable distance between the base and the top of a geometric figure is called the height. The top can be the opposite vertex, an apex, or even another base of the figure. Height of the Cone The distance between the center of the circular base and the vertex of the cone can be called the height of the cone. Height of Cylinder The distance between the centers of the circular bases of the cylinder is the height of the cylinder. Height of a Parallelogram The perpendicular distance between the parallel sides of a parallelogram (i.e. the base to the opposite side). Height of a Prism The length of the shortest line segment between the bases of the prism. Height of a Pyramid The shortest distance between the vertex and extended base of the pyramid. Height of a Triangle The length of the shortest line segment between a vertex and the opposite side of the triangle. Helix It is a spiral shape curve in three dimensional space. Heptagon A heptagon can be called a polygon, which has seven sides. Its other name is septagon, but heptagon is widely used. Hero's Formula Suppose all the three sides of the triangle are known. The formula used to calculate the area of the triangle in this scenario is called Hero's formula. For example: √[s(s-a)(s-b)(s-c)] Hexagon It is a special geometric figure, which has six sides and angles. Hexahedron A solid, which has no curved surfaces and the number of surfaces are equal to six. Higher Derivative The derivative of first derivative or the derivative of second derivative that have degree more than 1 are called higher derivatives. Homogeneous Equations When two or more linear equations have their constant term as zero, then such a set of equations are called homogeneous equations. Horizontal Line Equation It is an equation, which is used to describe a line parallel to Y-axis. Horizontal Line Test If a horizontal line intersects a graph twice [graph is made by the function F(x)], then the function is said be not one on one. This test to check one to one of function is called horizontal line test. Horizontal Reflection A geometric figure in the first or fourth quadrant, the reflection of which is present in second or third quadrant along X-axis and vice or versa is called horizontal reflection. Hyperbola A hyperbola is a geometric figure, which is a locus of two points called foci, where the difference between the distances to each point is constant. Hyperbolic Geometry Given two entities, a point and a line, there can be infinitely many lines passing through the point and are parallel to first point. This is called Hyperbolic geometry. Hyperbolic Trigonometry The trigonometric functions sine cosine tangent, etc., whose values are calculated using 'e'. The explanations of hyperbolic trigonometry are as follows: sinhx = (e^x - e^-x)/2, coshx = (e^x + e^-x)/2 tanhx = (sinhx/coshx) = (e^x - e^-x)/(e^x + e^-x)/2 Hypotenuse The hypotenuse is longest side of right-angled triangle. Hypotenuse-leg Congruence Two different right-angled triangles are said to be congruent when their hypotenuse and one of the corresponding legs are equal in length. Hypotenuse-leg Similarity In two right-angled triangles, when the ratio of the corresponding sides have equal ratios, then such triangles are having HL Similarity. I i In complex number analysis, the letter i denotes iota. Iota is given by negative square root of 1, that means √-1. = i Icosahedron Icosahedron is a polyhedron with 20 faces. In the case of a regular icosahedron, the faces are all equilateral triangles. Identity (Equation) An equation that is true for any values of the variable. For example, the identity, sin²θ + cos²θ = 1 is true for all values of θ. Identity Function The function f(x) = x is called the identity function. Identity Matrix A square matrix that has 1 as its element in the principal diagonal and rest all elements are zero. Image of a Transformation The image obtained after performing the operations of dilation or rotation or translation. Imaginary Numbers A complex number like 7i, that is free of the real part is called the complex number. Imaginary Part Consider a complex number -7 + 8i, the coefficient of i called the imaginary part of the complex number. Implicit Function or Relation A function in which the dependent variable can't be exactly expressed as a function of the independent variable. Implicit Differentiation Differentiating an implicit function. For example, consider 4x² + 5y⁵ - 6x = 1. Here, y can't be written explicitly as a function of x. Impossible Event An event that is impossible to happen or an event whose probability is zero. Improper Fraction A fraction that has denominator greater than its numerator. Improper Integral A integration in which the bounds of integration has discontinuities in the graph. They can also have limits between ∞ and -∞. The discontinuities between the bounds of integration makes the use of limits necessary in evaluating improper integrals. Improper Rational Expression If the degree of a numerator polynomial is more than or equal to the degree of a denominator polynomial than the rational expression is called the improper rational expression. Incenter The center of a circle inscribed in a triangle or a polygon. Geometrically, incenter is the point of intersection of the angle bisectors of a triangle. Incircle The largest possible circle that can be drawn inside a plane figure. All triangles and regular polygons have an incircle. Inconsistent System of Equations A system of equations that has no solutions. Increasing Function A function whose value increases continuously as we move from left to right of its graph is called an increasing function. A line with positive slope is a perfect example of increasing function where the value of the function increases as we proceed on the x-axis. If the increasing function is differentiable, then its derivative at all points (where the function is increasing) will be negative. Indefinite Integral I = ^a&#