## Glossary of math terms, a complete online math dictionary

A list of meanings, definitions, examples in this concise glossary of math terms. I am uploading each word everyday in alphabetical order. If you don't find the word you are looking for, chances are I am typing it up as we speak.

I consulted maths dictionaries Collins Gem BASIC FACTS mathematics, thefreedictionary.com, and wikipedia.com to guide me in providing you with the following user-friendly definitions. I simplified and clarified this glossary into my own words to make your learning more enjoyable.

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Abacus. A counting frame with wires holding beads on it. It is used for arithmetic and working out sums.

The beads are used to move up and down.

Abscissa The x-coordinate on a graph, or it's the distance from the vertical axis (y axis). The example shows abscissa of Q as -3 and P as 5.

Absolute The absolute value of a number is its numerical value without regard for its positive or negative sign.

The absolute value of the real number x (sometimes called modulus or mod x) is written as |x|.

For example: |2.3|=2.3, |-5.7|=5.7

In other words, the absolute value of a number would be the distance from 0 to the number. The distance of a number from zero will always be positive.

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Acceleration is the rate of change of velocity with time.

Acute An acute angle is an angle that measures between 0 and 90 degrees.

Addition Addition is one of the basic operations of arithmetic. When the sum of two numbers is determined by the operation - addition. For example 3+4=7.

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Affine An affine transformation of the plane is one that can be represented in the form

Examples include translations, rotations, shears, reflections, stretches, and enlargements.

Under any affine transformation the images of sets of parallel lines are themselves sets of parallel lines.

Algebra The generalization and representation of symbolic form, usually deals with letters instead of numbers. Algebra is now a general method applicable to all areas of mathematics.

Related pages: what is algebra?

Basic algebra

Importance of Algebra

Algebra for dummies

Rules of algebra

Algebra formulas

Collecting Algebra Terms

Algebra Substitution

History of algebra

Algorithm This is a standard process designed to solve a particular set of problems.

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Alternate Two angles are called alternate if they are on opposites sides of a transversal (cutting two lines), and each has one of the lines for one of its sides.

In the diagram, a and c are alternate pairs of angles, b and d are also alternate angles.
PQ and SR are parallel lines, and in this case the alternate pairs of angles are equal. p=r and q=s.

Altitude An altitude of a triangle is when a line that is perpendicular from a vertex (point) of the triangle to the opposite side of the triangle, called the base (the bottom part).

In this triangle AC is the base and the distance BD is the altitude.

Amplitude An amplitude is the measure of an oscillating motion, of the maximum measure from the mean or base position. For this example, periodic function f(x)=3sinx, the amplitude is 3, which is the highest point from the base 0.

Analysis Analysis is a branch of higher mathematics concerned with rigorous proofs of propositions mainly in the areas of calculus and algebra.

Angle The angle between two lines is the measure of the slant of the lines one to the other. Alternatively an angle can be viewed as a measure of the rotation (about the point of intersection of the lines) required to map one line onto the other.

The two most common measure of angle are degree and radian measure.

Types of angles

• Angles in a triangle add up to 180˚.

• Angles in a straight line add up to 180˚.

• Angles in a circle add up to 360˚.

Annulus An annulus is a region of a plane bounded by two concentric circles. In the example the blue part is called the annulus, which is bounded by two circles

If the circle of radii r and R,

with r < R, then the area of the annulus is: Area of large circle minus the area of small circle

Algebraically, this is represented as:

Antilogarithm The inverse of a logarithm function. Its particularly used when base 10 logarithms are done specifically for calculating. For example:

log10100=2 antilog10=100

Apex An apex is the name given to the highest point of a solid shape (3D) or plane (2D) figure relative to a base plane or line.

Approximation An approximation is an inexact result, which is accurate enough for some specific purpose.

An example, of the square root of 2, to the nearest tenth is 1.4. To the nearest hundredth it is 1.41.

So these are inexact results, but accurate enough for the specific requirement.

Other examples are the study of processes for approximating various forms in mathematics which is an important branch of the subject.

Taylor's Theorem provides a way of approximating to certain functions by polynomials.

Newton's Method gives an iterative way of obtaining approximations to the roots of certain equations.

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Apse When a something moves in a central orbit (for example, an ellipse), any point on the orbit where the motion of the thing is at right angles to the central radius vector is called an apse.

The distance from an apse point to the centre of the motion (apsidal distance) is a maximum or minimum value for the radius vector.

Arc An arc is a part of a curve, for example on a circle.

In particular for the circle, the points A and B define both minor (smaller part of the curve) and major (larger part of the curve) arcs of the circle.

Formulas for the minor arc AB (in degrees and radians):

Arc length=2πr×θ/360, if θ is in degrees.

or

Arc length=rθ, if θ is in radians.

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Area The measure of the size of a surface. The areas of certain shapes can be calculated. Examples of formulas of simple shapes are shown.

Argand diagram A grid used for diagrammatical representation of complex numbers. The horizontal axis is called the real axis, and the vertical axis is called the imaginary axis.

The complex number 6+2i is represented by the point shown on the diagram (6,2).

Argument

• 1- The argument of a complex number is the angle between the position of the coordinate number on the Argand diagram, and the positive real axis. So, in the diagram above, the argument of the complex number 6+2i is the angle between that line and positive real axis

• 2- The independent variable of a function is sometimes called the argument of the function. Examples of arguments of some functions:x is the argument of the function y=2x²+5x-1.t is the argument of the function R=2t-3.f is the argument of the function c=[(f-32)×5]/9

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Arithmetic This is the study of numbers and using them to solve problems using simple operations such as addition, subtraction, multiplication and division.

• 1- Arithmetic mean Is commonly referred to as the average of a set of numbers. The arithmetic mean of calculating n amount of numbers a1, a2...an is calculated by (a1+a2+a3+...an)÷n

Example of finding the arithmetic mean of: 8, 1, 5, 4, 2 is (8+1+5+4+2)÷5=4

• 2- Arithmetic progression A sequence of numbers where there is a common difference between the numbers that come before it and after it.

For example this sequence of numbers, 4, 7, 10, 13, 16 is an arithmetic progression with common difference 3.

Associative When this operation ❊ (+ − × or ÷) has the following property a❊(b❊c)=(a❊b)❊c for a, b and c it is called associative. In other words, if the right side of the equation is equal to the left side then it is associative. Examples include:

• Multiplication of real numbers is associative 4×(5×2)=(4×5)×2

• Division is not associative, 40÷(8÷2)≠(40÷8)÷2

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Asymptote A line on a curve never ending or an asymptote is the end part of the line on a curve tending to infinity.

The curve of this graph y=3-1/x has the line y=3 as an asymptote.

Average An average is a single number that represents a collection of numbers. Types of averages The mode, mean, median and range are the most commonly used averages.

• Mode This is the most common or the most popular data value. Sometimes it's called modal value.

• Mean The mean is the most typical way to work out averages. To find the mean of a set of data:1 Find the total of all the data values2 Divide by the number of data vaues

Example to find the mean

• Median This is the middle value of the data. First the data needs to be put into order from smallest to largest. The median is the middle value.

• Range The range of a set of data is the largest value minus the smallest value.

Axiom An axiom is a principle or a proposition taken to be self evident and not requiring any proof. An axiom is not a theory and does not need proving, simply because it is a starting point and is universally true.

An example of an axiom attributed to Euclid, 'Things equal to the same thing are equal to each other'.

Axis (Plural, axes). An axis is a line which a graph, plane shape or solid can be referenced to. For example, the coordinate axis, axis of symmetry, axis of rotation etc.

We mark positions of things at points. We use numbers called co-ordinates.

The horizontal line is called the x axis and the vertical line is called the y axis.

The x coordinate is given first, the y value given second. In the diagram, A has co-ordinates (0,2). B has co-ordinates (5,3).

Bar Chart A graph using parallel bars to illustrate information.

The lengths of the bars are proportional to the quantity represented. Its often used to represent statistical information.

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Base This can mean a number of things in mathematics.

• 1. It can be the side or face of a geometric shape. Usually the base is the bottom part of a shape.

• 2. Number base is the method of grouping, used in a number system that relies on place value. In other words it is the number that is raised to various powers, to achieve the units in a numbers system.

In the decimal number system grouping and place value is based on 10.

423 means 3+(2×10)+(4×10).

In base 5, 423 would mean 3+(2×5)+(4×25).

• 3. When a number is raised to a power of some number in an expression. In the expression, if xn, the x is called the base.

For example, in expressions such as 4⁵, 4 is called the base whilst 5 is called the exponent. This use of the term base is closely linked to its use in the context of logarithms.

In base 10 logarithms log 47 = 1.6721, this means that 10¹⋅⁶⁷²¹=47.

Bearing This is used for navigation and surveying. In maths it is used to indicate the direction something is going in.

One thing to remember about bearings is that it always starts at the origin O and points to start in the North direction. It also moves in a clockwise direction.

The bearing of a point A from someone standing at point O is the angle between the line OA and the North line through O, measured in a clockwise direction.

In the diagram the bearing of A from O is 220˚. The bearing from O to A is 40˚.

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Billion This is a very big number, infact it is a million millions (1 000 000 000 000). In USA it means a thousand millions.

Bimodal If we break the word up into two, 'bi' and 'modal'. 'Bi' means two and 'modal' means the most frequency (linked to averages, mode).

So, this term is used to describe distributions of data that show two modes or peaks in frequency.

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Binary This is a number system based on grouping in twos. The binary notation only uses two symbols, 0 and 1.

Its column headings or place values are: ...64 32 16 8 4 2 1.

Examples:

twenty nine in binary is : 1 1 1 0 1 (16+8+4+1=29)

ninety two in binary is : 1 0 1 1 1 0 0 (64+16+8+4=92)

Binomial This is the name given to a polynomial, which has two variables, for example 6x+2y.

Pascal's triangle are rows of numbers which are called binomial coefficients. Coefficients are the numbers in front of x and y.

The numbers in the rows are the coefficients corresponding to the xy terms when (x+y)n is expanded for various numbers of n.

For example: expanding (x+y)4=x4+4x3y+6x2y2+4xy3+y4

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Bisect This means to cut in half. This is usually used in a geometrical context.

For example the line BD bisects (angle) ∠ABC. The line DE bisects the line AB.

Brackets (Parentheses) are used to indicate the order in which a mathematical operation is carried out.

Brackets are usually the first thing to do in a sum, for example:

• 8+(2×3)=8+6=14
• (8+2)×3=10+6=16

Even though the numbers and operations are the same, the final answer is different. This is because the sum within the brackets must be done first. Basic algebra involves introducing or removing brackets. For example

• Putting them into brackets i.e. factorizing: 15x2y+10xy3=5xy(3x+2y2)
• Taking the brackets out i.e. expanding: 5xy(3x+2y2)=15x2y+10xy3

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Calculate To work out a sum or mathematical exercise, usually to get a numerical result.

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Calculus An important part of mathematics dealing with the study of the behavior of functions. Calculus is a huge thread of mathematics. It would be unusual for a scientist, engineer, or a quantitative economist not to have come across Calculus.

Historically it is associated with Isaac Newton and Gottfried Leibniz who pioneered it in the 17th century. However, their similar theories resulted in a dispute over who was the discoverer of Calculus. Both men came to the same conclusions independently and their methods were also quite different.

Integration and differentiation are 'two sides of the same coin' in that they are the inverses of each other.

• Differential calculus comes from the word Leibnis's differentialis which means taking differences or 'taking apart'. It is concerned with measuring change. The symbol for differentiation is dudx.

To work work out the derivative of an equation, it is formed by multiplying by the previous power and subtraction 1 from it to make the new power. There is a pattern to it:

 u du⁄dx x2 2x x3 3x2 x4 4x3 x5 5x4 ... ... xn nxn-1

• Integration calculus comes from the word integralis which means the sum of parts or 'bringing together'. Integration is concerned with measuring area. The symbol for integration is ∫.

To work work out the integral of an equation, it is formed by dividing by the 'previous power + 1' and adding 1 to it to make the new power. There is a pattern to it:

 u x2 x3⁄3 x3 x4⁄4 x4 x5⁄5 x5 x6⁄6 ... ... xn xn+1⁄(n+1)

The end result If we differentiate the the integral A=x33 we actually get the original u=x2. If we integrate the derivative dudx=2x we also get the original u=x2. So, this is how differentiation is the inverse of integration. Which is particulary known as the Fundamental Theorem of Calculus.

Without Calculus we would have no satellites in orbit, no economic theory and statistics would be a completely different ball game. So, where ever change is involved, there we find Calculus!

Cancellation To make something smaller, whether its numerical, fractional or algebra.

With regards to fractions this is the process of producing an equivalent fraction, by dividing the numerator and denominator by a common factor. For example:

3035=6×57×5=67

In this example 5 is a factor of 30 and 35. This cancels the numbers down into smaller numbers.

Cardinal number The characteristic of a number that reflects the manyness of a set. So the cardinal number 4 is the property shared by all sets containing four elements.

Cardioid The locus of a point on the circumference of a circle, which rolls on a fixed circle of equal radius.

Cartesian coordinates This is the method of locating a position of a point on a plane (2D) or in space (3D).

When locating the position of a point on a plane from two perpendicular axes, the coordinates are given in the order x first then y, (x,y). The cartesian coordinate of the point A are (5,3).

In space we have three perpendicular planes, which produce three perpendicular axes as shown in the diagram. The distance of the point are given by (x, y, z).

The cartesian coordinate of the point B are given by (2, 3, 5).

Catenary This is the name given to the shape of a curve, as if it were suspended from two points.

Centigrade The unit of temperature scale, shown on a thermometer where the freezing point of water is 0 degrees and the boiling point of water is 100 degrees.

Abbreviated to C as measured on a centigrade thermometer. An example of forty degrees centigrade would be written as 40°C.

(The standard term now is celcius, it is avoided due to the confusion over the prefix centi-. Which originally meant 100 but developed into hundredth of a grade, a unit of plane angle.)

Centimetre A linear unit of distance in metric terms. In relation to a metre, one centimetre is equal to one hundredth of a metre.

Characteristic With reference to a base 10 logarithm, the characteristic is the integer part of a number. For instance it is the number left to the decimal point. For example:

0.052=102×5.2
Taking logs of both sides to give:

log(0.052)=log(102×5.2)
=log(102)×log(5.2)
=2+0.716

written as 2.716
So, 2 is the characteristic of the logarithm.

Chord A line joining two points on a circe.

Circle This is a plane curve which is formed by sets of points from a fixed distance. The fixed point is the centre (British English) or center (American English). The distance from the centre to the edge is called the radius.

Properties of circles

• Circles have line and rotational symmetry of infinite order about the centre.

• The perimeter of a circle, is also known as the circumference, which is a measurement of the length around the cirlce.

There are 2 formulas to work out the perimeter/circumference

In short this is written as, c=2πr

or alternatively it can be found out by using this formula

• circumference=π×diameter

which is written as c=πd for short.

Example: Find the circumference/perimeter of a circle when the diameter is 8cm.

Answer: We can work it out using either equations.

Using c=2πr, half of the diameter will give the radius,

c= 2×3.14×4

c= 25.12

Using c=πd,

so c= π×diameter

c= 3.14×8

c= 25.12

• To find the area of a circle we use the formula

Area = π×r2

which is also written as A=πr2 for short.

Circumference This is the boundary line of a circle or the length of the boundary line, or distance around the circle.

The ratio of the circumference/diameter is a constant for all circles, and is given by a greek letter π. Numerically π is an irrational number which is approximately 3.14 (to 3 significant figures).

example to work out the circumference

Class Interval A method of grouping statistical data to be represented and interpreted in a much simpler way.

62, 45, 70, 40, 54, 39, 20, 77, 26, 31,

88, 15, 31, 48, 53, 60, 33, 29, 74, 70,

42, 12, 27, 55, 48, 43, 50, 79, 19, 44,

21, 60, 26, 69, 62, 11, 26, 31, 42, 52,

46, 39, 81, 62, 53, 16, 71, 49, 33, 29.

For example, the fifty examination percentages could be grouped into the following class intervals:

Clockwise The direction something is going in, for instance the way a clock hand rotates is in a clockwise direction.

ABC maps to A'B'C by moving in a clockwise direction.

• A set is closed when applying an operation on 2 numbers within the set results in the answer being from the original set.

For example, the set of Natural numbers {1, 2, 3, 4,...} is closed using the operations addition and multiplication, but is not closed under division or subtraction.

5÷2=2.5, which is not a natural number.

3-4=-1, which is not a natural number.

• A closed curve is one with no end points.

• A closed interval on the real number line is when the set of all numbers x, is defined by the inequalities of the type a≤x≤b.

The closed interval {x: 1≤x≤4} is the set of all real numbers between, and including, the 'end-points' 1 and 4.

• With regards to algebra, the coefficient of a term is the numerical part of the term. For example, in 3x2y+4xy2-6x+7y

the coefficient of x2y is 3

the coefficient of xy2 is 4

the coefficient of x is -6

the coefficient of x is 7

• In science and engineering the term can often be used to mean a specific numerical constant that is characteristic of a system. For instance, the coefficient of a linear expansion, the coefficient of friction etc.

Collinear Points are collinear if they lie on a straight line.

The points A, B and C are collinear.

Note: The vectors AB and BC are connected by AB=2BC.

Column Positioning elements in a vertical way.

For example, this matrix has 3 columns

Combination A combination of a set of n objects is any selection of n objects or less objects from the set, regardless of what order they are chosen.

For example, the full list of combinations of three letters from the set {a, b, c, d} is: abc, abd, acd, bcd.

The number of combinations of r objects that can be made from a set of n objects is usually denoted by: nCr or (nr)

It can be shown that:

nCr=n!r!(n-r)! (where n! means factorial n)

For example, 12C5=12!5!×7!=792

Notice that 12C5=5C12 and in general nCr=nCn-r.

The values of nCr for varying n and r can be found from the appropriate row of Pascal's Triangle.

Common Denominator In preparation for the addition or subtraction of fractions with unlike denominators, the fractions are usually converted into equivalent fractions with the same, or common denominator.

13+12=26+36

6 is the common denominator in this example.

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Common Difference The difference between successive terms in an arithmetic progression, for example:

8, 12, 16, 20, 24 common difference is 4

20, 18, 16, 14, 12 common difference is -2

Common Logarithm Logarithms to the base of ten are called logarithms.

Commutative Any operation * which has the property a*b=b*a for all numbers a and b of a given set, is called commutative.

For example:

Multiplication of real numbers is commutative, 4×3=3×4

Subtraction is not commutative, 4-3≠3-4

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Complement The complement of a set S is the set consisting of the elements within the universal set that are not in S.

The complement of S is usually denoted by S'.

The shaded portion of the diagram represents S'.

if ξ={letters of the alphabet}

and c={consonants}then c'={vowels}

Complementary angles Two angles whose sum is 90° are complementary angles.

In the right-angled triangle ABC, ∠BAC and ∠BCA are complementary angles.

∠BAC is the complement of ∠BCA.

Complete the square This is a method for solving quadratics by applying a suitable method to the equation to reduce it into the form (x+a)2=k, so that it can eventually be solved for x.

Example:

x2+10x+22=0

x2+10x=-22 First subtract both sides of the equation by 22

Taking the coefficient of x, which is 10, half it 102=5, then square it 52=25.

Then add 25 to both sides of the equation to complete the square, x2+10x+25=3

Factorising left hand side to give, (x+5)2=3

Square root both sides of the equation, x+5=±√3

Take 5 from both sides of the equation, x=-5±√3

So, x=-5+√3=-3.26

or x=-5-√3=-6.73 (to 2 d.p)

Complex numbers An extension of the real number system designed to overcome difficulties involving the solution of problems like x2+3=0, which have no solution in the real numbers.

Complex numbers are written in the form a+bi, where a and b are real numbers, and i2=-1: a and b are called the real and imaginary parts.

Complex numbers are usually written as pairs of real numbers (a, b) with a particular multiplication and addition defined, so that numbers of the form (a,0) behave as real numbers, and (0,1) behaves as √-1.

Complex numbers have important implications in the branches of pure and applied mathematics.

Component A component of a vector is one of a set of two or more often mutually perpendicular vectors that are equivalent in theory to the given vector.

If vector OA is of magnitude 5 and at 30° to the x-axis then:

Component in the direction of x-axis is vector OP.

Magnitude=5cos30°=4.33

Component in the direction of the y-axis is vector OQ.

Magnitude=5sin30°=2.5

Composite function The function f:x→(3x+1)2, can be thought of as the composition of two simpler functions:

g:x→(3x+1) and h:x→x2

x g→ (3x+1) h (3x+1)2

f=hg

Note the notation: f=hg means f is equivalent to the function g followed by h.

Any function that can be split into two or more simpler functions in this way is called a composite function. For example:

if f:x sin24x, then f=pqr, where r:x→4x, q:x→sinx and p:x→x2.

Compound interest This is a method of calculating interest on money where the interest earned during a period of time is calculated on the basis of the original sum together with any interest earned in previous periods.

Example: If £100 is invested at 10% per year for 2 years, it would appreciate in the following way:

• Original sum ➔ £100

• end of first year ➔ £100×10%=£10, £100+£10=£110

• End of second year ➔ £110×10%=£11,£110+£11=£121

After 2 years the amount has increased by £21, and the total amount is therefore £121.

In general, if £M is invested at i percent compound interest, then after n years it would be worth:

£M×(100+i100)n

Computer An electronic device to process large amounts of coded information quickly.

Computer systems commonly comprise of mainly of three elements: Input device, for example a keyboard, paper tape etc. Central Processor, output devices, for example television monitor, screen, printer, etc.

Concave A concave curve or surface is hollow towards a given point of reference.

Satellite dishes present a concave surface towards the sky.

Cone A cone is bounded by a plane (often circular) base, and pointing to a fixed point called the vertex.

A right cone is one in which the axis, or line joining the centre of symmetry of the base to the vertex, is perpendicular to the base.

Volume of a cone is: V=13πr2h

where r=radius of base and h=height.

Curved surface area of a right cone is: A=πrl

where l is the slant height

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Congruent Shapes, planes or solids are called congruent if they have the same shape and size.

Simple transformations such as rotations, reflections and translations map objects onto congruent images.

Conic A curve obtained by taking a plane section through a cone.

By varying the angle of cut, four main types of conic can be obtained.

Alternatively a conic can be thought of as the locus of a point that moves so that the ratio of its distance from a fixed point to its distance from a fixed line is constant. This ratio is called the eccentricity of the conic.

Conjugate Two angles whose sum is 360° are called conjugate.

The conjugate of the complex number a+bi is a-bi

A complex number and its conjugate are related by a reflection in the real axis of the Argand diagram.

The product of a complex number and its conjugate is equal to the square of its modulus

(a+bi)×(a-bi)=a²+b².

Constant A fixed quantity in an expression is called a constant.

In the expression y=3x+2, 3 and 2 are constants whilst x and y are variables.

Construction Additional points or lines to supplement a geometrical figure in order to prove some property of the figure are called constructions.

Continuity Quantities like the number of pupils in a class can be measured precisely, and vary in integer steps.

Class numbers do not vary continuously

When measuring height, the result can never be exact. In growing from 159cm to 160cm, we assume every possible value between the two numbers.

Quantities such as length, weight, temperature, speed, time, etc are all continuous variables.

A persons weight is a continuous variable.

It can go up or down and is not always the same number.

Converse The converse of the statement 'if a number ends in zero it is divisible by ten' is 'if a number is divisible by ten, then it ends in zero'.

Many important statements in mathematics are in the form 'if x then y'. The converse of such a statement is 'if y then x'.

The converse of a true statement need not necessarily be true. For example 'if n and m are even, then n+m is even'. This is a true statement. The converse 'if n+m is even, then n and m are even' is false.

Conversion The process of changing one system (often units) into another.

For example, to express a temperature, given in the Farenheit scale, in Centigrade, we use the conversion formula:

C=5/9(F-32)

Convex The opposite shape of a convex is that of a concave . A curve or surface that bulges towards a given point of reference is called convex.

A dome represents a convex shape.

A convex figure is when any line joining two points on a shape is contained within the shape

This is a convex shape. The two points from one side to the other are contained within it.

This is not a convex. The two points joining up have fallen outside the shape.

Coordinates A coordinate system is used to locate points on a plane (x,y) or in space (x,y,z). Cartesian and polar are two commonly used coordinate systems.

Coplanar Sets of points or lines lying in the same plane are called coplanar.

Correlation A statistical term used to describe the relationship between two variables. Changes in one variable are related to the changes in another variable.

A positive correlation is when an increase in one variable happens while the other variable decreases.

A negative correlation is when an increase in one variable is associated with the decrease of another.

The measurement of the correlation coefficient is between -1 and 1. This gives a measure of the two variables.

Correspondence A relationship matching pairs of elements from two sets.There are 4 types of correspondence are that are identified.

This is a one to one correspondence

This is a one to many correspondence

This is a many to one correspondence

This is a many to many correspondence

Cosecant This is a trigonometric function. In a right angled triangle, the cosecant of an angle is the ratio hypotenuse/opposite side.

In the triangle ABC:

Cosecθ=ACBC=54=1.25

The value of θ can now be determined by examining a table of values for the cosecant function θ=53.1°.

In the triangle PQR:

Cosec30°=RPQP

But from the table of values for the cosecant function cosec30°=2.

Therefore 2=RP3.4 and RP=6.8

Cosine A trigonometric function. In a right angled triangle the cosine of an angle is the ratio adjacent-side/hypotenuse.

In the triangle ABC: cosθ=CBCA=513=0.385

The number of θ can now be determined by examining a table for the cosine function. θ=67.4°

In the triangle PQR:

cos60°=PQPR

but from the table of values for the cosine function, cos60°=0.5

Hence 0.5=PR9.6 and PQ=4.8

Cotangent A trigonometric function.

In a right angled triangle the cotangent of an angle is the ratio adjacent side opposite side.

In the triangle ABC above:

cot α= BCBA=86=1.33

The value of α can now be determined by examining a table of values for the cotangent function. α=36.9.

In the triangle PQR above:

cot30°=QRQP

But from the table of values for the cotangent function cot30°=1.73.

Therefore 1.73= QR5 and QR=8.65.

Count To count the number of objects in a set is to match the objects in a one to one way using the names of the positive integers - one, two three...

The positive integers or natural numbers are often called counting numbers.

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