Math+Glossary

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Refers to both the process of //manipulating// variables and constants in a mathematical expression according to fixed laws, properties or rules; for example, simplifying an expression or solving an equation; and alternatively to a //mathematical structure// whose elements and operations satisfy a given //collection of laws//. The word algebra comes from the work of the Arabic scholar //Abu Abd-Allah ibn Musa al'Khwarizmi//, who was born about 790 AD near Baghdad, and died about 850 AD. Khwarizmi wrote one of the first books //Hisab al-jabr w'al-muqabala// on what is now called algebra in 830 AD. //Al-jabr// refers to the process of moving a subtracted quantity to the other side of an equation while //al-muqabala// involves subtracting equal quantities from both sides of an equation. In 1140 AD this text was translated into Latin as //Liber algebrae et almucabala//, from which the word algebra has become part of mathematical language. A process for computation that can be carried out mechanically; for example, the algorithm for subtraction of many-digit decimal numbers, or the algorithm for factorisation of a linear expression using the distributive rule. The word //algorithm// comes from the old English //algorisme//, from a Latin translation of the name of the ninth century AD Arabic scholar //al-Khwarizmi//, who investigated computation using the Hindu numeration system (leading to the Hindu-Arabic number system of today). An angle is formed at the point of intersection of two rays: [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/glossary_clip_image003.gif caption="diagram showing the angle formed at the point of intersection of two rays"]] Angle //measure// is commonly based on the amount of turn between two rays with a common point. There are three common measures of angle: //fraction of a full turn//, //degree// and //radian//. A full turn = 360 degrees = 2π radian. A half turn = 180 degrees = π radian.
 * Angles which are between two parallel lines, adjacent to a transversal cutting that pair of parallel lines and sum to 180 degrees are said to be //allied// or //co-interior// angles, for example angles //a// and //b// in the following diagram:

[[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/glossary_clip_image001.gif caption="diagram showing allied (co-interior) angles"]]


 * Angles which are symmetrically located under a half turn with respect to the midpoint of a transversal cutting a pair of parallel lines are said to be //alternate// angles, for example angles //a// and //b// in the following diagram:

> angles"]] > showing a transversal to a pair of parallel lines"]] To obtain a value to a particular accuracy. For example, the fraction 22 / 7 provides an approximate value for the irrational real number π. Rounded correct to 4 decimal places, 22 / 7 has the value 3.1429 while π has the value 3.1416. These values are themselves decimal approximations to 22 / 7 and π respectively. While it is not possible to trisect any angle //exactly// using only a compass and rules, it is possible to approximate such a trisection with reasonable accuracy. An operation is //associative// if the result of applying the operation to any three elements of an expression is the same regardless of which pair of elements (without changing their order) is combined first. Addition and multiplication //are// associative on the set of natural numbers, for example: 4 + (7 + 5) = 4 + 12 = 16 and (4 + 7) + 5 = 11 + 5 = 16 2 × (3 × 4) = 2 × 12 = 24 and (2 × 3) × 4 = 6 × 4 = 24 Subtraction and division are //not// associative on the set of natural numbers, for example: 10 − (4 − 2) = 10 − 2 = 8 but (10 − 4) − 2 = 6 − 2 = 4 24 ÷ (12 ÷ 2) = 24 ÷ 6 = 4 but (24 ÷ 12) ÷ 2 = 2 ÷ 2 = 1 In general the //associative// laws (properties) for addition and multiplication of real numbers state respectively that //for all// real numbers //a//, //b// and //c//: //a// + ( //b// + //c// ) = ( //a// + //b// ) + //c// and //a// × (//b// × //c//) = (//a// × //b//) × //c// A proposition which is taken as being true with respect to a given context. For example, in a modelling problem to design a seating arrangement in a theatre, it may be assumed that the height of the people watching the movie is no greater than 200 cm. [|Back to Top]
 * Angles which are in the same relative position with respect to a transversal cutting a pair of parallel lines are said to be corresponding angles, for example angles //a// and //b// in the following diagram: [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/glossary_clip_image015_0000.gif caption="diagram showing corresponding
 * Angles at a point on a line that sum to 180 degrees are said to be //supplementary// angles.
 * A line that cuts a pair of parallel line obliquely [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/glossary_clip_image002_0006.gif caption="diagram

[|Back to Top]  Data relating to the simultaneous measurement of two variables; for example, age and income. [|Back to Top]  The relative frequency of an event, this may be expressed qualitatively using terms such as: impossible, no chance, not likely, an even change, odds-on, likely, a certainty, or quantitatively using numbers on a scale from 0 (impossible) to 1 (certain). These numerical values are often expressed as fractions such as 1 / 2, ratios such as 2:3, decimals such as 0.87 or percentages such as 40%. The result of carrying out an operation on an element of a set, or elements of a set, is also an element of that set. For example, multiplication is closed on the set of natural numbers, because the result of multiplying any pair of natural numbers is also a natural number. Division is //not// closed on natural numbers, since 9 and 2 are both natural numbers, but the result of dividing 9 by 2 is not a natural number: 9 ÷ 2 = 9 / 2 = 4.5, and 4.5 is a decimal fraction, not a natural number. An operation is //commutative// if the result of applying the operation to any two elements of a set is the same, regardless of the order of the elements. Addition and multiplication //are// commutative on the set of natural numbers, for example: 6 + 12 = 18 = 12 + 6 and 6 × 12 = 72 = 12 × 6 but subtraction and division are //not// commutative for example: 6 − 12 = −6 but 12 − 6 = 6 and 6 ÷ 12 = 1 / 2 but 12 ÷ 6 = 2. In general the //commutative// laws (properties) for addition and multiplication of real numbers state that //for all// real numbers //a// and //b//, //a// + //b// = //b// + //a// and //ab// = //ba//, respectively. The set of all elements //not// in a given set with respect to the universal set for a particular context or situation. For example, if the universal set in a particular situation is taken to be the letters of the alphabet, the complement to the set of vowels is the rest of the alphabet. If the universal set in a particular situation is taken to be the set of numbers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then the complement to //A// = {4, 5, 6} is {1, 2, 3, 7, 8, 9, 10}. The complement of //A// is written as //A'// or // A //. A non-zero natural number which has //more than two distinct elements// in its factor set; for example, 8 is a composite number as it has four distinct elements in its factor set: {1, 2, 4, 8}. The number 2 is not a composite number since it has only two distinct elements in its factor set: {1, 2}. With the exception of 1, which has only one distinct element in its factor set: {1}, all non-zero natural numbers are either //composite// or //prime//. The property of being //identical// in shape and dimensions. Two shapes are congruent if one of them can be mapped onto the other by transformations that do not change the length of line segments or the angle between lines. A statement whose truth or otherwise is not yet determined, but is open to further investigation, for example, //Golbach's Conjecture: “every even natural number greater than 2 can be expressed as a sum of two prime numbers”//. First stated in 1742, the //Golbach conjecture// has not yet been either proven to be true or shown to be false, although many mathematicians intuitively believe that it is true. Two points in the plane are said to be connected if there is a line or curve (edge) that joins them. A set of points in the plane, such as a //network//, is said to be connected if there are no two points in the set which are not connected. A logical term that connects or qualifies other expressions, such as ‘and’, ‘or’, ‘not’, ‘if ... then ...’ and ‘is equivalent to’. For example, given a set of attribute blocks, specifying the blocks that are red //and// square involves two attributes ‘red’, ‘square’ which apply to some blocks but not to others. The use of the connective //and// to specify ‘red’ //and// ‘square’ required both attributes to apply. A number that has a fixed value in a given context. For example, in the calculation of //n// + 11 for different natural numbers //n//, the number 11 is a //constant//. In formulas such as //P// = 4 × //l//, 4 is a //constant// while //P// and //l// are //variables//. A condition which is applied in a given context; for example, in solving the equation 3//x// + 2//y// = 24 a //constraint// may be that only natural number solutions are required (there are an infinite number of integer solutions). Can in principle assume all possible values in a given interval. For example, height is a continuous data measurement. While the actual height of a person can only be physically measured to a given accuracy, it is possible in principle for a for person’s height to be any value within a typical range of heights for a human being. **correspondences:** An //instance// where a proposition or conjecture is //false//. The number 6 is a counter-example to the propostion that every even number is also a multiple of four. The process of listing a subset of the set of natural numbers //N// = {0, 1, 2, 3 ... } in consecutive order; for example {0, 1, 2, ...} or {11, 12, 13, ...}. [|Back to Top]  A number expressed using the base 10 place value system. For example 412.56 is a decimal number. Measure of angle where a full rotation around a fixed point corresponds to 360 degrees. Can only assume a countable number of data values. For example, shoe size is a discrete data measurement. An operation is said to be //distributive// over another operation if it can take priority over the operation used for combination within brackets (that is, its application can be distributed over the brackets). //Multiplication// is distributive over //addition// for real numbers, for example: 6 × 17 = 6 × (10 + 7) = (6 × 10) + (6 × 7) = 60 + 42 = 102 The distributive property underpins algorithms for multiplication and division that involve natural numbers of several digits. In general the //distributive// law (property) for (multiplication over addition) for real numbers states that //for all// real numbers //a, b// and //c//: //a// (//b// + //c//) = //ab// + //ac//. Addition is //not// distributive over multiplication, for example: 3 + (2 × 4) = 3 + 8 = 11 but (3 + 2) × (3 + 4) = 5 × 7 = 35 For a finite set this is the process of partitioning the set into subsets of equal size. For natural numbers division re-expresses a given natural number in terms of a multiple of a smaller natural number and a remainder. For example 68 = 7 × 9 + 5, so 68 //divided by// 9 is equal to 7 with 5 remainder. Using rational numbers in fraction form this is expressed exactly as: 68 ÷ 9 = 68 / 9 = 7 5 / 9 In general, for real numbers, if //xy// = //z// then //z// ÷ //y// = // z ///// y // = //x//, unless //y// = 0 in which case the process is not defined. A relation or function is a map between the elements of two sets. The set //from// which the mapping occurs is called the //domain// of the function or relation. The set onto which the elements of the domain are mapped is called the //co-domain//. For example, the domain of a relation (the favourite colour of) could be the students in a class, while the co-domain is a set of colours. Not all the colours may be selected as a favourite for some student in the class. The //range// of the relation is a subset of the co-domain, which corresponds to the set of favourite colours for that class. [|Back to Top]  A straight line or curve that forms the boundary of a region in the plane (such as the side of a triangle, or an edge in a network) or a boundary between two surfaces (such as the rim of a can or the edge of a box). Derived from observation, measurement or experiment. One shape is an enlargement of another shape if they are similar and the scale factor for dilation is greater than 1. The dilation transformation involved is also referred to as an enlargement. A mathematical expression that includes the ‘=’ symbol. Equations are used to assign a value to a pro-numeral; for example, //a// = 2; to define the rule of a function; for example //y// = 2//x// + 3, where whatever value //x// takes, //y// is two times //x// plus three; and to specify conditions that must be satisfied by the value of a variable; for example, if 2//x// + 3 = 10, then //x// = 4 for this statement to be //true//. Two statements or propositions are understood to be equivalent if they are //both// true or //both// false. That is, the conditions which make one true make the other true as well, and the conditions which make one false make the other false as well. Given any fraction, an equivalent fraction is one whose numerator and denominator is a common integer multiple of the numerator and denominator of the given fraction. For example, an equivalent fraction of 1 / 2 is 2 / 4, with a common integer multiple of 2 such that (1 × 2) / (2 × 2) = 2 / 4. For each fraction expressed in simplest form an //equivalence class// (or family of equivalent fractions) can be generated by successively multiplying its numerator and denominator by the natural numbers (excluding zero). For example, for the fraction 2 / 3, the corresponding family is: { 2 / 3, 4 / 6 , 6 / 9 , 8 / 12 , 10 / 15 , … } Is the difference between an actual value and its measured or estimated value and is defined as: error = measured or estimated value - actual value To form an approximate value for a quantity. An //instance// where a proposition or conjecture is //true//. The number 6 provides an example of a number which is even and a multiple of three. Working beyond known data to make predictions; for example, working past the last known point on a graph to predict a value beyond this point. [|Back to Top]  A bounded surface: a bounded region in a network, or on a three-dimensional shape or object. A natural number that divides exactly into a given natural number. For example, 2 is a factor of 12, since 2 x 6 = 12. The set of all factors of a given number is called its //factor set//. The //factor set// of 12 is {1, 2, 3, 4, 6, 12}. The elements of a factor set are often grouped in pairs. Thus, the set of //factor pairs// of 12 is {{1, 12}, {2, 6}, {3, 4}, {4, 3}, {6, 2}, {12, 1}}. This concept also applies to algebra, where, for example, the factors of the linear expression 6//x// + 9 are 3 and 2//x// + 3 since 3(2//x// + 3) = 6//x// + 9. The number formed by the product of a given natural number with all the natural numbers less than it down to 1. For example 4 //factorial// is 4! = 4 × 3 × 2 × 1 = 24. The set {//a//, //b// , //c// , //d// , //e//} is an example of a finite set. The set of all people alive on a given day is a very large, but finite set. The cardinal number of a finite set is a natural number, that is, the elements of any finite set can be put in a one-to-one correspondence with the elements of a set of the form {0, 1, 2, 3, ..., //n// } where //n// is a natural number. A unit whose value is fixed by agreement; for example, litre is a formal unit of capacity for fluids and hour is a formal unit of time. See also //rational number//. In a fraction, for example 3 / 4, the number 4 is called the //denominator// of the fraction (it specifies the number of equal sized partitions of a whole unit), the number 3 is called the //numerator// (this specifies how many of these parts), and the horizontal line indicating the part-whole relation of the fraction is called the //vinculum// (from the Latin meaning a line or stroke that connect quantities). A fraction is said to be expressed in //simplest form// if its numerator and denominator have no common factor, that is are co-prime. For example, 3 / 4 is expressed in simplest form, but 6 / 12 is not in simplest form, since 2 is a common factor of both 6 and 12 (as are 3 and 6). A correspondence (map or relation) between the elements of two sets where each element in the first set is mapped to exactly one corresponding element in the second set. A function is either a one-to-one correspondence or a many-to-one correspondence. [|Back to Top]  Is the irrational number whose value is given by the proportion AC : AB = AB : BC where A and C are the endpoints of a line segment and B is the point on the line segment between A and C such that AC : AB = AB : BC [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/glossary_clip_image009_0000.gif caption=" diagram showing line segment divided in golden ratio"]] It is called the golden ratio as it is believed to represent a proportion of lengths that is aesthetically attractive to the human eye in art and design contexts. The exact value of φ is (1 + √5) / 2 and its approximate value is 1.618 correct to 3 decimal places. The decimal expansion for //phi// to 100 significant figures is: 1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137. The digits in this decimal expansion do not display any recurring pattern, a property which distinguishes irrational numbers from rational numbers. A visual representation of data or functions. Cartesian graphs of functions and relations are plots of ordered pairs of values (//x//, //y//) that represent the function or relation relative to //x// and //y// coordinate axes and the fixed origin (0, 0). Statistical graphs include dot plots, box and whisker plots, bar graphs and histograms. [|Back to Top]  Also called the //greatest common divisor// (//gcd//). Given any two natural numbers, this is the largest natural number that divides both of them exactly, that is, the highest common number in their factor sets. For example, the //hcf// (//gcd//) of 18 and 24 is 6. Any fraction can be expressed in simplest terms by dividing its numerator and denominator by their //hcf// (//gcd//). [|Back to Top]  An element of a set which when combined (using a given operation) with any other element of the set leaves that element unchanged. For example, 0 is the identity element for addition of natural numbers, since for any natural number //n// it is the case that 0 + //n// = //n// and //n// + 0 = //n//. Similarly, 1 is the identity element for multiplication of natural numbers, since for any natural number //n// it is the case that 1 × //n// = //n// and //n// × 1 = //n//. A statement of the form //if ... then .//.. An implication is understood to be //true// unless the //first// part of the statement is //true// but the //second// part of the statement is //false//. A set //A// is a subset of another set //B// if all of the elements of //A// are also elements of //B//. For example, if //A// = {vowels} and //B// ={letters of the alphabet} then //A// is a (proper) subset of //B//, written symbolically as //A// ⊂ //B//. In the case where //A// is required to be a subset of //B//, but may include //all// of the elements of //B// then this is represented symbolically by //A// ⊆ //B//. A mathematical expression containing the terms ‘less than’, ‘less than or equal to’, ‘greater than’, or ‘greater than or equal to’ their respective symbolic representations ‘<’, ‘≤’, ‘>’ and ‘≥’. For example ‘the set of prime numbers less than or equal to 29’, is an inequality as is the expression 2 //y// ≥ //x//2 where //x// and //y// are real numbers. An assertion made on the basis of analysis from given data or propositions; for example, on the basis of the weather patterns observed over several years, a farmer might infer that it is likely to be a hot summer. The set of natural numbers //N// = {0, 1, 2, 3 ...} is an example of an infinite set. There are many examples of infinite sets, the set of all prime numbers is an infinite set (there is no largest prime number). The set of natural numbers, //N//, is an example of an infinite set which has a smallest element, 0, but no largest element. The set of integers //Z// = {... −3, −2, −1, 0, 1, 2, 3 ...} is an example of an infinite set which has no smallest or largest element. The set {0.9, 0.99, 0.999, 0.9999, ..., 1} is an example of an infinite set which has both a smallest element, 0.9, and a largest element, 1. It is //not// possible for the elements of any infinite set to be put in a one-to-one correspondence with the elements of a set of the form {0, 1, 2, 3, ..., //n// } where //n// is a natural number. A unit whose value is decided on in a given context, for example, the use of a pace to measure distance or the use of a cupped hand to measure capacity of rice for a meal (irregular informal units). An informal unit may also be regular, such as the use of paperclips to measure length or a drinking glass to measure a small amount of a substance (capacity). An element of the infinite set of numbers Z = {... −3, −2, −1, 0, 1, 2, 3 ...}, sometimes also referred to as a positive or negative whole number. Working within known data to make predictions, for example working between two known points on a graph to predict a value in between these points. Given two sets //A// and //B//, their intersection, written //A// ∩ //B,// is the set of all elements common to both sets. If //A// and //B// have no elements in common then their intersection is the empty set { }. For example, if //A// = { //a//, //b// , //d// , //z// } and //B// = { //a// , //c// , //x// , //y// , //z// } then //A// ∩ //B// = { //a// , //z// }; however, if //C// = { //m// , //n// } then //A// ∩ //C// = { }. The property of not changing under a process such as transformation; for example, the points on a mirror line are invariant under the transformation of reflection in that mirror line. If a person touches a mirror with their finger, then the point of contact will be invariant under reflection in the mirror, all other points their image will have left- and right-hand senses reversed. For each element of a set its inverse with respect to a given operation defined on the set is the element in the set which, when they are combined using the operation result in the identity element. For example, the inverse of the integer + 4 with respect to the operation of addition is the integer −4 since + 4 + (−4) = 0 and −4 + (+ 4) = 0. The inverse of the rational number 2 / 3 with respect to the operation of multiplication is the rational number 3 / 2 since 2 / 3 × 3 / 2 = 1 / 1 = 1. Exploration of a situation or context. A number that cannot be expressed as a fraction in the form // m ///// n //, where //m// and //n// are integers and //n// is non-zero. The decimal form of such numbers does not terminate, and is non-recurring, that is, there is no finite sequence of digits that repeats itself. For example, //r// = 0.12345678910111213 is part of the decimal expansion of an irrational real number. Numbers such as √2, the golden ratio //φ,// and //π// are examples of irrational numbers. [|Back to Top]  A diagram consisting of a small number of non-overlapping (mutually exclusive) rectangles used to indicate the relationship between elements of a set and given properties or attributes. When two properties or attributes are involved, the corresponding karnaugh map is also called a //two-way table//. Suppose a set of //attribute blocks// has several shapes (squares, circles, triangles and hexagons) of various colours (red, blue, yellow), size (small, large) and thickness (thin, thick). If we consider the properties //red// (colour) and //square// (shape), then any of the blocks from the set will satisfy //exactly one// of the following four combinations of these //two// properties: it is red and square; it is red and not square; it is not red and square; it is not red and not square. This can be represented diagrammatically using a karnaugh map, where each attribute block is place in exactly one of the four regions shown below, for example: Note that the properties of size and thickness are not represented in this diagram. Each of the shapes shown could be small or large, thick or thin. If, for example, a //third// property such as thickness (thin, thick) were to be considered then previous karnaugh map could be modified to have eight regions as follows:  A description of position with respect to some fixed reference. Principles of reasoning where one proposition is deduced from other propositions. Given any two natural numbers, their lowest common multiple is the smallest natural number which they both divide exactly. This is //not// necessarily their product. For example, the //lcm// of 6 and 9 is 18, since 3 × 6 = 18 and 2 × 9 = 18, but 6 × 9 = 54. The //lcm// is used in the operation of addition and subtraction of fractions, to identify the equivalent fractions with the same denominator. [|Back to Top]  The size, or absolute value of a number; for example, both +5 and -5 have magnitude 5. The magnitude of certain numbers can only be approximated to a given accuracy, for example the magnitude of the number //pi//, correct to two decimal places, is 3.14. The sum of values in a data set divided by the total number of values in the data set. For example, is a data set consists of the values { //x//1, //x//2 , //x//3 , ... , //x//n }, then the mean is defined as: . A measure is a record of the magnitude of an attribute (such as weight, length, time, and likelihood) associated with an object or event. Also called an //average//. This is a statistic that is used to represent a data set. There are three common measure of centre for a data set: //mode// (the most common value), //median// (the middle value) and the //mean// (the sum of all the values divided by the number of values). The middle value of a data set when its elements are arranged in numerical order. For an even set of elements the mode is taken to be the value half way between the two middle values. Define the measure of one quantity as a function of other quantities using an algebraic formula. For example, the area, //A//, of a circle radius, //r//, is defined by the mensuration formula //A// = //πr//2and the average speed, //s//, of a moving object which travels a distance //d// in time //t// is defined by the formula //s// = // d ///// t //. The //most common// value in a data set. If there are two such common values the data set is sometimes said to be bi-modal. In some cases it is not useful to consider the mode as a measure of centre for a data set. Using mathematical concepts, structures and relationships to describe and characterise, or model, a situation in a way that captures its essential features. [|Back to Top]  An element of the infinite set of numbers //N// = {0, 1, 2, 3 ...}, sometimes also referred to as a counting number. In some references the number 0 is not included in the set of natural numbers; however, it is useful to include as it corresponds to the number of elements in an empty set. A real number //x// is //negative// if //x// < 0. The set of negative integers (regular whole numbers) //Z−// = {−1, −2, −3, ...} is sometimes referred to as ‘the negative counting numbers’. A two-dimensional representation of a three-dimensional shape in such a way that it can be folded to construct the three dimensional shape. A set of points (//vertices// or //nodes//) some of which are joined by lines or curves (//edges//) which sometimes enclose regions (//faces)//. Networks are used to represent relationships involving connectedness; for example, road networks, a family tree or the edges lining a tennis court. The //designation// of a number in a given language; for example, the number ‘three’ is designated by the Hindu-Arabic numeral **3**, the Roman numeral **III**, and the Chinese numeral **三**. [|Back to Top]  Is a relation that describes the location of elements in a set with respect to each other. These elements may be totally ordered or partially ordered. For example, the set of natural numbers is totally ordered by the relation less than or equal to since for any two natural numbers //m// and //n// exactly one of the following is true: //m// < //n// or //m// = //n// or //m// > //n//. Similarly, the set of students in a class can be totally ordered with respect to their height using the relation less than or equal to. However, the set of people at a school fair is only partially ordered by the relation ‘is a parent of’ since there will likely be many pairs of people who are not each others parent, such as siblings. A special type of set of two elements for which order is significant. For example, the grid reference used on a map is an ordered pair such as (//K//, 7) where //K// is the horizontal grid reference on the directory page and 7 is the vertical grid reference on the directory page. [|Back to Top]  A ratio expressed as a proportion to 100. The boundary of a closed shape or curve, also the //length// of this boundary. Repeats at regular intervals; for example, if the elements of the set of natural numbers are divided by 3, in order, and the remainder on division recorded, the following periodic pattern of remainders occurs: 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2 ... The graph of a function that describes the rise and fall of tides in a given location from high tide to low tide over several days will also be periodic. Represented by the symbol //π//, is the irrational number defined by the ratio of the circumference //C// of a circle to its diameter, //d//: [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/glossary_clip_image012_0002.gif caption=" diagram showing circle circumference, centre and diameter"]] Its approximate value, correct to 2 decimal places is 3.14, and 22 / 7 is a reasonably accurate fraction approximation to //π//. The decimal expansion for //pi// to 100 significant figures is: 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068. The digits in the continued decimal expansion of //π// do not have any recurring pattern, a property which distinguishes irrational numbers from rational numbers. The five platonic solids are the //tetrahedron// (4 equilateral triangles as faces), the //cube// (six squares as faces), the //octahedron// (8 equilateral triangles as faces), the //dodecahedron// (12 regular pentagons as faces), and the //icosahedron// (20 equilateral triangles as faces). They are solid shapes with faces that are made of regular polygons which tessellate with an equal number of faces at each vertex. Literally ‘many-sides’; a closed plane figure with sides formed by straight lines; for example, triangles, quadrilateral, pentagons, hexagons and the like. Polygons with all sides of equal length and all angles between adjacent sides equal are said to be //regular// polygons. A three-dimensional shape whose faces are adjacent polygons; for example, a pyramid is a polyhedron but a cone is not a polyhedron (part of a cone is a curved surface which is not a polygon). The complete or universal set for a given context or situation. This may refer to the Australian population with respect to an election, or the population of wombats in Victoria. **(exponent or index):** The number 43 = 4 × 4 × 4 = 64 is read as four to the power (also called the //exponent// or //index//) 3. A function //ƒ// of the variable //x//, with rule of the form //ƒ//(//x// ) = //ax// where //a// is a fixed constant and the variable //x// is the //exponent// is called an //exponential// function. A natural number that has //exactly two distinct factors//, 1 and itself. The number 1 is //not// a prime number (it has only one distinct factor), nor is the number 8, as it has four distinct factors {1, 2, 4, 8}. The number 2 is the only even prime number. The set of the first 100 prime numbers is: code {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,       59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109,        113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173,	179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239,	241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311,	313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383,	389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457,	461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541}
 * A function between two sets where each element in one set (domain) corresponds to exactly one element in the other set (range) and vice versa. Thus, in a ballroom dancing class, there will be a one-to-one correspondence between male and female partners during a given dance.
 * A function between two sets where each element in one set (domain) corresponds to exactly one element in the other set (range); however, an element in the range may be mapped onto by more than one element in the domain. For example, each student in a class has exactly one height measure (to the nearest centimetre) at a given instant (so the relation 'the height of' is a function) but it may be the case that two students are the same height.
 * A relation is a correspondence between two sets. When one of the sets is specified as the set of elements //to// which the correspondence is made, this set is called the //co-domain// of the relation. The //subset// of the //co-domain// to which elements of the domain are actually matched up, is called the //range// of the relation. For example, if a correspondence is formed between the //students// in a class and their favourite colour, the //students// would be a natural choice for the domain of relation. The //range// corresponds to the actual set of favourite colours of the students. The //co-domain// is the set of all possible colours, not just the favourites of the class of students. Thus, the //range// of a relation is, in general, a //subset// of its //co-domain//.
 * To divide into separate parts which together constitute the whole. For example, the letters of the alphabet can be partitioned into vowels and consonants, the set of natural numbers can be partitioned into those with remainder 0, 1 or 2 on division by 3.
 * Two edges in a shape are said to be adjacent if they meet at a common vertex; similarly, two faces in a shape are said to be adjacent if they meet at a common edge.
 * Two edges in a shape are said to be adjacent if they meet at a common vertex; similarly, two faces in a shape are said to be adjacent if they meet at a common edge.
 * The variable associated with the //range// of a relation. For a function //ƒ// with rule //y// = //ƒ//(//x//), //y// is the dependent variable.
 * The variable associated with the //domain// of a relation. For a function //ƒ// with rule //y// = //ƒ//(//x//), //x// is the //independent// variable.
 * An element of the set {0, 2, 4, 6 ...}
 * An element of the set {1, 3, 5, 7 ...}
 * A Chinese puzzle formed by a square cut into several pieces that are then rearranged to create other shapes.

code There is no known function for generating the sequence of prime numbers, although there are algorithms for identifying whether a number is prime or not. A three-dimensional shape that has a polygonal cross section, formed by having its edge points translated parallel to a given direction. For example, the following shape is a //triangular prism//: Formulating a problem in such a way that it is amenable to mathematical analysis. The application of mathematical reasoning to the development of a solution or solutions to a given problem. Two ratios are said to be in proportion if they are equivalent ratios. That is //a//://b// is in proportion to //c//://d// if there is a non-zero real number //k// such that //a//://b// = //kc//://kd.// A three-dimensional shape that has a square base and edges from the corners of the base to a point (the //vertex//) above the square base. If the vertex is vertically above the centre of the square base, the pyramid is said to be a //right// pyramid (the line segment connecting the centre of the square base to the vertex is at 90 degrees – a //right// angle – to the plane of the square base). If //a//, //b// and //c// are the lengths of the sides of a right angled triangle, such that //c// is the length of the side opposite the right angle, then //a2// + //b2// = //c2// :

[[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/glossary_clip_image001_0001.gif caption="right angled triangle showing sides a, b and c"]] [|Back to Top]  Measure of angle where one cycle around the circumference of a unit circle from a fixed point corresponds to //2 π// radians. Thus, as a measure of angle, 180 degrees is equivalent to //π// radian. Not able to be predicted, not regular, a chance event. A number which is generated at random. The //n//th digit of the decimal expansion of a rational number is //not// random since these numbers can be predicted in advance. Selecting an arbitrary sequence of digits in the decimal expansion of an irrational real number can be used as a random number generator since the decimal expansion have no repetition. For example, the decimal expansion of //pi, π//, could be used for this purpose. The difference between the largest value and the smallest value in a data set. A comparison //a// : //b// of the size of two (or more) quantities relative to each other. An element of the infinite set of numbers: //Q// = { // m ///// n //, where //m// and //n// are integers and //n// is not equal to zero}, sometimes also referred to as a fraction. Rational numbers can be expressed in fraction form, with corresponding terminating decimal expansion, for example, 1 / 8 = 0.125, or with infinite recurring decimal expansion, for example, 4 / 9 = 0.444… Rational numbers may be positive or negative; for example, −17 / 5 is also a rational number. an element of the infinite set of real numbers, //R//, which comprises the union of the set of rational numbers with the set of irrational numbers. A rational real number has either a terminating (finite) decimal expansion, for example 5 3 / 4 = 5.75, or an infinite recurring decimal expansion, for example, 23 / 99 = 0. 23 = 0.232323… An irrational real number has an infinite non-recurring decimal expansion, for example //x// = 0.123456789101112… ; //z// = 1.22333444455555… ; //y// = 2.35711131719… Some well known irrational real numbers are the square root of 2, √2, pi, π, and the golden ratio, phi, φ. These can be used in exact calculations, for example √2 + √8 = √2 + 2√2 = 3√2 ; π / 3 + π / 4 = 7π / 12 ; 1 / φ = φ − 1. Often //rational approximations// to irrational real numbers are used in arithmetic calculations, with a suitable accuracy for the purpose of computation, for example √2 + √8 ≈ 4.24 ; π / 3 + π / 4 = 1.833, correct to 3 decimal places; 1 / φ = 0.6180, correct to 4 significant figures. All real numbers can be expressed using an infinite decimal expansion, for example, the natural number two can be written as either: 2 = 1.9999999999 … or 2 = 2.0000000000 … The set of real numbers, //R//, includes the natural numbers, //N//, integers, //Z//, rational numbers, //Q// and irrational numbers, //Q′// as proper subsets, illustrated in the venn diagram below: [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/realnumb_gloss.gif caption="Venn diagram showing relation between R, N, Z, Q, and Q'"]] The real numbers are often represented using the infinite set of points on a continuous geometric line. This representation is referred to as the real number line. A one to one correspondence between the set of real numbers and the infinite set of points on the line is taken as given, with a specific point, //O// (the origin) selected to correspond to zero. A point //X// to the right of //O// is taken to correspond to the positive real number //x//, where the magnitude of //x// measures the length of the line segment //OX//. The point //X′// to the left of //O// located by the endpoint of the line segment //OX// after a half-turn rotation about //O//, corresponds to the real number //-x//: [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/realnumb_gloss1.gif caption="diagram showing number line between o, O, X', x, -x"]] Compass and unmarked straight-edge constructions can be used to determine the exact location of any rational number and some irrational numbers, for example, √2, on the real number line.

The process of carrying out the current step of a process using the results of the previous step (or steps) of the same process. For example, the sequence of numbers: {3, 6, 12, 24. } can be described using recursion as ‘start at 3 and make the next term in the sequence twice the previous term in the sequence’. Students often intuitively define sequences using recursion. Skip-counting, for example, ‘counting by fives starting from 12’ is another example of a recursive process. A transformation where each point in the plane is reflected in a given mirror line. A correspondence (map) between the elements of two sets, for example, the relation ‘has the favourite colour’ between the set of students in a class (the //domain// of the relation) and the set of colours (the //co-domain// of the relation). A relation can be represented as a set of ordered pairs, a map (arrow diagram) or a graph. If //m// and //n// are two natural numbers with //m// greater than //n//, such that //m// = //p// × //n// + //r// where //p// and //r// are also natural numbers, then //r// is said to be the //remainder// of //m// on division by //n//. For example, if //m// = 29 and //n// = 6, then //m// = 4 × //n// + 5, so the remainder of 29 on division by 6 is 5. A transformation where each point in the plane is rotated through a given angle about a fixed point (the point of rotation). The process for //approximating// a value that lies between two known values by one of these values. In particular, when a measure lies between two of the smallest marks on a scale it is rounded to the nearest value represented by either one of these marks. Rounding is used to specify a number correct to a given accuracy. With //measurements// this often means rounding a decimal number. For example, 16.29 rounded to the nearest //tenth// of a unit is 16.3, rounded to the nearest //unit// is 16, rounded to the nearest //five// is 15, and rounded to the nearest //ten// is 20. The decimal number 57.139 rounded to the nearest //hundredth// is 57.14 and rounded to the nearest //tenth// is 57.1. However, 57.199 rounded to the nearest //hundredth// is 57.20. Some general principles for rounding are: It should be noted that several different conventions for rounding a last digit of 5 can be found in the literature, and these relate to different contexts for number computation. [|Back to Top]  A subset of a population; for example, a set of people used for a newspaper survey is a sample of the population. A //random// sample is one which is obtained by using a random process for selecting a sample. Scale specifies the proportion between two measures. For example, a model of a house may be made on a 1: 10 //scale// of length. A measuring //scale// for weight could be based on the extension of a spring in proportion to the mass of an object (each 500 grams could cause an extension 5 cm). The tick marks on the axes of a graph are specified according to some scale, for example each mark along the horizontal axis might correspond to 5 units, while each mark along the vertical axis might correspond to 2 units. These are then referred to as axes //scales//. A notation used in particular to express very small or very large numbers in the form of the product of a decimal number between 1 and 10 (not inclusive) with an integer power of 10 expressed in exponential form. For example, in scientific notation: 1567000 = 1.567 × 106 and 0.000034 = 3.4 × 10−5 The interior part of a circle formed by two radiuses: The interior part of a circle formed by a chord: ‘set’ is an undefined term that informally corresponds to the notion of a collection of objects or elements. Sets are usually specified by listing their elements, for example: { a, e , i , o , u }, describing them in words; for example, ‘the set of Australian citizens’, by or using a mathematical rule: {( //x//, //y// ): //y// = 2//x// + 1, //x// ∈ //N// } = {(0, 1), (1, 3), (2, 5), (3, 7) ... }. > element of"]] is a shorthand for ‘is **not** an element of’. A geometric object or representation of a common real-life object, in two-dimensional space, such as a free-hand closed curve, a triangle, circle, square; or in three-dimensional space (also called solids) such as a blob of play-dough, a cube, sphere or pyramid.  If a numerical value is expressed in scientific notation (standard form) //a// × 10//n//, where 1 < //a// < 10 and //n// is an integer, then all the digits in //a// are //significant//. For example, 1567000 × 106 has four significant figures and 0.000034 = 3.4 × 10−5 has two significant figures.  The property of one shape being an exact enlargement or reduction of another shape.  The process of modelling an event using various devices or technology. For example, if two players are equally likely to win a game of tennis on past performance, then a sequence of games between the two players could be simulated by successive tossing of a fair coin (heads player //A// wins, tails player //B// wins) or randomly selecting numbers from the list of natural numbers and noting whether the result is even (player //A// wins) or odd (player //B// wins). Counting from a given starting value using multiples of a fixed natural number; for example, {2, 4, 6, ...} or {7, 12, 17...}. This is a statistic that indicates how widely the values of a data set are distributed. Common measures of spread include //range//, //inter-quartile range//, //quantiles// and //percentiles// and //mean absolute difference//. > percentiles"]] > || 15 || 2 8 9 || > || 16 || 4 4 6 9 || > || 17 || 0 2 || > || 18 || 8 || An element of the set {1, 4, 9, 16, 25 ...}. A square number has an //odd// number of distinct elements in its factor set; for example, the factor set of 16 has five distinct elements: {1, 2, 4, 8, 16}.  The positive square root of a given real number //x// is the real number //y// such that //y//2 = //x//. For example, the square root of 9 is 3. This is written symbolically as √9 = 3. Originally, the square root was taken to refer to the //side length// (root) of a square whose //area// was a given positive number. Thus, a square of area 9 square units has a side length (square root) of 3 units. Most square roots are //not// rational numbers but irrational real numbers. For example, a square of area 2 has an exact side length of the square root of 2, or √2. This is //approximately// 1.4 units in length. Every positive real number has two square roots, one positive and one negative, for example, the square roots of 9 are 3 and −3. This is written symbolically as ±√9 = −3 and 3. Is the irrational number, √2, whose value corresponds to the length of the diagonal of a unit square. [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/glossary_clip_image024_0000.gif caption=" diagram showing unit square with diagonal length square root of 2"]] Its approximate value is 1.414 correct to 3 decimal places. The decimal expansion for the square root of two correct to 100 significant figures is: 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573. The digits in this decimal expansion do not display any recurring pattern, a property which distinguishes irrational numbers from rational numbers. A formal unit from a system of units which is comprehensive and is used to define other units or combinations of units. For example, in the metric system, the standard units for //length//, //mass// and //time// are respectively, //metre//, //kilogram// and //second//. The standard units are described in the International System of Units (SI). Related //formal//, units are:
 * if the last digit is a 1, 2, 3 or 4 then the previous digit is left unchanged and the number is said to be //rounded down;// for example, 296.2 rounded to the nearest //whole number// is 296
 * if the last digit is a 6, 7, 8 or 9 then the previous digit is increased by 1 and the number is said to be //rounded up//; for example, 296.8 rounded to the nearest //whole number// is 297
 * if the last digit is a 5 then the previous digit can be //randomly// rounded up or down, especially where several measurements are taken. This avoids cumulative error that would arise from either //always// rounding up or //always// rounding down; for example, 296.5 would be randomly rounded to either 296 or 297
 * if rounding to a given accuracy has a cumulative effect, zeroes should be used to indicate the accuracy; for example, 299.97 rounded to the nearest //tenth// is 300.0.
 * ‘element’ is an undefined term that informally corresponds to the notion of //belonging// or //membership// of a set. For example, 3 //is a member// of the set of natural numbers //N// = {0, 1, 2, 3, ... }. This relation can be written more concisely as 3 ∈ //N//. The symbol ‘∈’ is a short-hand for ‘is an element of’. The number 1 / 2 is //not// a natural number, and this can be written as 1 / 2 [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/notelement.gif width="9" height="11" caption="symbol not an element of"]] //N//, where [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/notelement.gif width="9" height="11" caption="symbol not an
 * The power set of a given set is the set of all possible subsets of the given set, including the empty set and the given set itself. For example, if //A// = { //a//, //b// , //c// } then the power set of //A// , written //P// (//A//) is the set {{ }, { //a// }, { //b// }, { //c// }, { //a// , //b// }, { //a// , //c// }, { //b// , //c// }, { //a// , //b// , //c// }}. If there are //n// elements in the set //A// then there are 2 //n// elements in its power set, in this example //A// has 3 element and its power set has 23 = 8 elements.
 * In a given context, the //universal set// is the set of all objects under consideration. For example, the set of natural numbers is often the universal set for basic arithmetic computation in the early years of schooling. When students conduct a survey about students in their school, the set of all students in the school is the universal set (often called the population in this situation) for the survey.
 * A line segment joining two points on a curve: [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/glossary_clip_image004.gif caption="diagram showing a chord on a curve"]]
 * The shape produced by cutting a three-dimensional shape completely through by a plane.
 * The two-dimensional image formed on a plane surface by the shadow of a three-dimensional object illuminated by a light source; for example, a person’s shadow on the ground on a sunny day. In geometry this usually corresponds to the projections of a shape onto three plane surface at right angles to each other, such as front view, side view, top view.
 * A line that touches but does not cut a curve at a point: [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/tangent.gif width="453" height="156" caption="diagram showing tangent to a curve at a point P"]] In the case of a circle, a tangent is always at right angles to the radius which meets the circumference of the circle at the point of contact.
 * A form of data representation where the ends of a rectangular box are aligned on a numerical scale with a given proportion of a //sample// of uni-variate data. For example, the resting heartbeats of a group of athletes may be measured and a box-plot constructed to correspond to the middle 50% of values. Lines (whiskers) are added to show the lower and upper 25% of the data. The //median// value (the middle value of the sample) is also indicated by a vertical line parallel to the ends of the box: Box-plot of proportions of the sample [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/glossary_clip_image001_0000.gif caption="diagram showing a box-plot with 25%, 50% and 75% data
 * When a data set is ordered, inter-quartile range is the difference between the upper quarter value and the lower quarter value (that is the difference between the 75th percentile and the 25th percentile).
 * Informally an outlier is a value which lies outside the distribution of most of the values in a data set. For example, the height of a very tall or very short person would likely be an outlier in a data set of heights of randomly selected people. A characterisation of a likely outlier is a value that lies more than 1.5 times the inter-quartile range outside of the upper or lower quartiles.
 * Also called a stem and leaf plot, a table where discrete data is represented (usually in order) by distinguishing values (the leaf) within set intervals (the stem). For example, the set of students heights in cms {152, 158, 159, 164, 164, 166, 169, 170, 172, 188} can be represented using a stem and leaf plot as:
 * **Stem** || **Leaf** ||
 * Key** : 15|2 = 152 cms Stem plots provide a visual indication of spread.
 * A number is a perfect square if it is the square of an integer or rational number; for example, 169 is a perfect square as 132 = 169; similarly 0.81 is a perfect square as ( 9 / 10 )2 = 0.81.

//centimetre// = metre × 1 / 100 //tonne// = kilogram × 1 000 //minute =// second × 60.

Other non-standard formal units are, for example, carat, gallon, hour and knot. An irrational number such as √2 or (5 + √3) / 4. An older term of reference for a particular type of irrational number. Area of the surfaces of a three-dimensional shape or object. Using marks or symbols that have a meaning particular to mathematical language, for example, the written statement ‘two is less than 3’ can be written symbolically as ‘2 < 3’. Property of regularity in shape by, for example, reflection or rotation. Thus the letter **T** is symmetrical by reflection, the letter **Z** is symmetrical by rotation, the letter **H** is symmetrical by both reflection and rotation, the letter **R** is //not// symmetrical. [|Back to Top]  A repeated pattern in the plane or on a surface where shapes completely fill all of the space around a given point where their boundaries meet. For example, a honeycomb is a tessellation using hexagons Tiling patterns are tessellations using rectangular tiles or brick pavers in paths, mosaics in buildings, quilts and art. A one-to-one correspondence of points in the plane. //Reflections//, //rotations// and //dilations// are examples of transformations. A diagram consisting of line segments connected like the branches and twigs of a tree used to indicate the relationship between sets or events, for example a family tree. In the following right angled triangle, with long side length 1 unit, the vertical side length is sin(//t)// and the horizontal side length is cos(//t//)
 * Irregular, does not display symmetry. The human body is asymmetrical with respect to an imaginary line down the middle.
 * Literally ‘the same measure’, an isometry is a transformation that leaves lengths, area and angles unchanged.

[[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/glossary_clip_image010_0005.gif caption="diagram showing right-angled triangle with unit hypotenuse sin(t) and cos(t)"]] [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/glossary_clip_image011_0001.gif caption="diagram showing right-angled triangle with hypotenuse, a=rsin(t) and b=rcos(t0"]] if the triangle is dilated by a factor //r// from the point at which the angle //t// is formed, then, by similarity: [|Back to Top]  A term or expression taken as accepted without definition. These are the basic building blocks of mathematics; for example, element and set are undefined terms in the //Structure// dimension, while point and line are undefined terms in the //Space// dimension. Undefined terms may be characterised by informal description, or illustrated by examples. Other mathematical terms and expressions are defined using undefined terms and relations on them.  Given two sets //A// and //B//, their union, written //A// ∪ //B// is the set of all elements which occur in either set, listed without repetition. For example, if //A// = { //a//, //b// , //d// , //z// } and //B// = { //a// , //c// , //x// , //y// , //z// } then //A// ∪ //B// = { //a// , //b// , //c// , //d// , //x// , //y// , //z// }. A basic or fundamental construct for counting and/or measurement. For example, the number 1 is the unit for counting (from the Latin //unus// for one). The metre is the standard unit for measurement of length in the metric system. Data relating to measurement of a single variable, for example, shoe size. [|Back to Top]  A term used to designate an //arbitrary// element of a set. For example, if //n// is any natural number, then //m// = 2//n// + 1 is an odd natural number. The terms //n// and //m// are called variables. A diagram consisting of a small number of possibly overlapping circles used to indicate the relationship between elements of a set and given properties or attributes. A vertex is a point in the plane or in space where several edges meet, but do not extend beyond, for example the corners of a triangle or the point of a cone: [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/vertex.gif width="446" height="179" caption="Left diagram showing the vertices of a triangle; right diagram showing the vertex of a cone"]] [|Back to Top] The word //zero//, comes from the Arabic word //sifr//, or //cipher// in English, which means a secret or disguised writing, or a symbol for a vacant place. Leonardo Fibonacci, the Venetian merchant and mathematician who played an important role in the introduction and popularisation of the Hindu-Arabic number system in Europe in the latter part of the 12th century and early 13th century, had travelled and worked in the Orient and used the term //zephyrum// for zero. Zero is represented by the numeral 0, and plays two important roles in mathematics: as a //number// and as an empty place holder //digit// in the decimal expansion of numbers. For example, the digit 0 in the number 2057 indicates ‘no hundreds’ in the place value expansion of the number //two thousand and fifty seven// or equivalently 2 × 1000 + 0 × 100 + 5 × 10 + 7 × 1. Zero specifies the //number of elements// in an //empty set// (none). Although closely related, the //number// zero, 0, is not the same as the empty set, { }, which is sometimes represented by the special symbol, Ø to distinguish the set from the number. The empty set is a collection that has no elements. Zero indicates the number of elements in this set - none. Zero also corresponds to the //origin// on the real number line: [[image:http://vels.vcaa.vic.edu.au/images/content/maths/glossary/zero_gloss.gif caption="Zero also corresponds to the origin on the real number line"]] The point of intersection of the vertical and horizontal axes of the cartesian coordinate system is also called the origin and designated by the letter //O//. This origin is specified by the coordinates (0, 0) and plays an important role in work on graphs of functions and other relations.
 * A variable whose scope is not limited by a logical quantifier. Free variables frequently are used in proofs to represent an arbitrary element of a set.

Zero has several important number properties. For any real number //r//, it is the case that 0 + //r// = //r// = //r// + 0 and 0 × //r// = 0 = //r// × 0. Zero is not a factor of any real number other than itself, and any real number is a factor of zero. The arithmetic operation of division by zero is //not well defined//, and results in an error statement when this computation is attempted using technology. The expression 0 / 1 is a fraction representation of the integer zero, as are the equivalent fractions 0 / 1 = 0 / 2 = 0 / 3 = 0 / 4 = … However, the expression 0 / 0 is said to be //indeterminate//, since an assumption that 0 / 0 = //r//, where //r// is some real number, would imply 0 = 0 × //r//, which is true for any real number. The expression 1 / 0 is said to be //inconsistent// or //undefined//, since an assumption that 1 / 0 = //r//, where //r// is some real number, would imply 1 = 0 × //r//, which is false for all real numbers.