The Math Book

by DK

  Mean An average found by adding up the values of a set of data and dividing by the number of values. For example, the mean of the four numbers 1, 4, 6, and 13 is 1 + 4 + 6 + 13 = 24 divided by 4 = 6.

  Median The middle value of a set of data, when the values are put in order from lowest to highest.

  Meridian An imaginary line on Earth’s surface joining the North Pole and South Pole through any given locality. Lines of longitude are meridians.

  Mode The value that occurs most often in a set of data.

  Modular arithmetic Also called clock arithmetic, a form of arithmetic where, after counting up to a certain point, 0 is reached, and the process is repeated.

  Natural logarithm See logarithm.

  Natural number Any of the positive whole numbers. See also integer.

  Non-Euclidean geometries A key postulate of traditional geometry, as described by Euclid in ancient times, is that parallel lines never meet (often expressed as meeting at infinity). Geometries in which this and other Euclidean postulates are not valid are called non-Euclidean.

  Number line A horizontal line with numbers written on it that is used for counting and calculating. The lowest numbers are on the left, the highest on the right. All real numbers can be placed on a number line.

  Number system Any system of writing down and expressing numbers. The Hindu–Arabic system used today is based on the ten numerals 0 to 9: when 10 is reached, 1 is written again, but with a 0 after it. This system is both a place value system and a base-10 or decimal system.

  Number theory A branch of mathematics that studies the properties of numbers (especially whole numbers), their patterns, and their relationships. It includes the study of prime numbers.

  Numerator The upper number in a fraction, such as the 3 in 3⁄4.

  Obtuse angle An angle between 90 and 180 degrees.

  Octahedron A 3-D polyhedron made up of eight triangular faces. A regular octahedron is one of the five Platonic solids.

  Operation Any standard mathematical procedure such as addition or multiplication. The symbols used for such operations are called operators.

  Order See degree.

  Ordinal numbers Numbers that denote a position, such as 1st, 2nd, or 3rd. See also cardinal numbers.

  Origin The point at which the x and y axes of a graph intersect.

  Oscillation A regular to-and-fro movement between one position or value to another and back again.

  Output The result when an input is combined with a function.

  Parabola A curve that is similar to one end of an ellipse, except that the arms of a parabola diverge.

  Parabolic Relating to a parabola, or to a function based on it, such as a quadratic function, which produces a parabola-shaped graph.

  Parallel Of a line, going in exactly the same direction as another line.

  Parallelogram A quadrilateral where each side has the same length as the side opposite to it and the two sides are also parallel. A square, rectangle, and rhombus are types of parallelogram.

  Partial differential equation A differential equation containing several variables, in which the differentiation is applied to only one of the variables at a time.

  Periodic function A function whose value repeats periodically, as seen, for example, in the graph of a sine function, which is in the form of a repeating series of waves.

  Perpendicular At right angles to something else.

  Pi (π) The ratio of a circle’s circumference to its diameter, approximately 22⁄7, or 3.14159. It is a fundamental transcendental number that appears in many branches of mathematics.

  Place value system The standard system for writing numbers, where the value of a digit depends on its place in a larger number. The 2 in 120, for example, has a place value of 20, but in 210 it stands for 200.

  Placeholder A numeral, usually zero, used in a place value system to differentiate 1 from 100, for example, but that does not necessarily imply an exact measurement as in phrases such as “about 100 miles away.”

  Plane A flat surface.

  Plane geometry The geometry of 2-D figures on a flat surface.

  Platonic solid One of the five polyhedra that form completely regular and symmetrical shapes: each face is an identical polygon and all the angles between the faces are the same. The five Platonic solids are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

  Polygon Any flat shape with three or more straight sides, such as a triangle or pentagon.

  Polyhedron Any 3-D shape whose faces are polygons.

  Polynomial A mathematical expression made up of two or more terms added together. A polynomial expression usually includes different powers of a variable, together with constants, for example, x3 + 2x + 4.

  Positional number An individual numeral whose value depends on its position within a larger number. See place value system.

  Postulate In mathematics, a statement whose truth is taken for granted or thought to be obvious, but is not supported by a proof.

  Power The number of times a quantity or number has been multiplied by itself. For example, four ys multiplied together (y × y × y × y) is called “y raised to the power of 4” and written y4.

  Power series A mathematical series where each term has a greater power than the previous one, such as x + x2 + x3 + x4 +… .

  Prime number A natural number that can be divided exactly only by itself and 1.

  Probability The branch of mathematics that studies the likelihood of different outcomes occurring in the future.

  Product The result of one number or quantity being multiplied by another.

  Proof Any method of showing beyond doubt that a mathematical statement or result is true. There are different kinds, including proof by induction and existence proofs.

  Proportion The relative size of something compared with something else. For example, if two quantities are in inverse proportion, the larger one of them becomes, the smaller the other one will become; for example, if one quantity is multiplied by 3, the other is divided by 3.

  Pure mathematics Topics in mathematics that are studied for their own sake rather than for any practical application. See also applied mathematics.

  Quadratic equation An equation containing at least one variable multiplied by itself once (for example y × y, also written y2), but containing no variables raised to higher powers.

  Quadrilateral Any flat 2-D shape with four straight sides.

  Quartic Referring to equations or expressions of the fourth degree, where the highest power contained in them is 4—for example, x4.

  Quaternion A mathematical object that is a development of the idea of a complex number, but uses four components added together, rather than just two.

  Quintic Referring to equations or expressions of the fifth degree, where the highest power contained in them is 5, for example, x5.

  Quotient The result that is obtained when one number is divided by another.

  Radian A measure of angles that is an alternative to degrees and is based on the length of the radius and circumference of a circle. Turning around by 2 × pi (2π) radians is the same as turning 360 degrees (that is, in a complete circle).

  Radius Any straight line from the center of a circle or sphere to its circumference.

  Rational number A number that can be expressed as a fraction of one whole number over another. See also irrational number.

  Real number Any number that is either a rational number or an irrational number. Real numbers include fractions and negative numbers, but not imaginary or complex numbers.

  Reciprocal A number or expression that is the inverse of another one, meaning that the result of multiplying them together is 1. For example, the reciprocal of 3 is 1⁄3.

  Recurring Any number that is repeated without limit. For example, 1⁄3 expressed in decimals is 0.333333…, which can also be described in words as “zero point three recurring.”

  Rhombus A quadrilateral with all four sides the same length; informally, a diamond shape. A s
quare is a special kind of rhombus, with all angles 90 degrees.

  Right angle An angle that is 90 degrees (a quarter turn), such as the angle between vertical and horizontal lines.

  Ring A mathematical structure that is like a group except that it includes two operations rather than one. For example, the set of all integers forms a ring when taken together with the operations addition and multiplication, because performing these operations on members of the set produces an answer that is still a member of the set.

  Root (1) The root of a number, which is another number that when multiplied gives the original number. For example, 4 and 8 are roots of 64, with 8 being the square root (8 × 8 = 64) and 4 the cube root (4 × 4 × 4 = 64). (2) The root of an equation is its solution.

  Scalar A quantity that has magnitude (size), but not direction, in contrast to a vector.

  Scalene triangle A triangle where none of the sides and none of the angles are the same size.

  Segment (1) Part of a line, with definite end points. (2) In a circle, the area between a chord and the outside edge (circumference).

  Sequence An arrangement of numbers or mathematical terms placed one after the other and usually following a set pattern.

  Series A list of mathematical terms added together. Series usually follow a mathematical rule, and even if the series is infinite, it may add up to a finite number. See also sequence.

  Set Any collection of numbers, or mathematical structures based on numbers. Sets can be finite or infinite (for example, the set of all integers).

  Set theory The theory of sets and a branch of mathematics which now forms the underlying basis of many other branches of mathematics.

  Sexagesimal A number system used by the ancient Babylonians based on the number 60, and still used in a modified form for time, angles, and geographic coordinates.

  Simultaneous equations A set of several equations that include the same unknown quantities, such as x, y, and z. Usually, the equations must be calculated together to solve the value of the unknowns.

  Sine (abbreviation sin) An important function in trigonometry, and defined as the ratio of the length of the side opposite a given angle in a right-angled triangle to the length of the triangle’s hypotenuse. This ratio starts at 0 and varies with the size of the angle, repeating its pattern after 360 degrees. The graph of a sine function is also the shape of many waves, including light waves.

  Slope The angle of a line to the horizontal, or an angle of a tangent to a curve to the horizontal.

  Square number A whole number that can be formed by multiplying a smaller whole number by itself once. For example, 25 is a square number as it is 5 × 5 (52).

  Statistics (1) Measurable data collected in an orderly way for any purpose. (2) The branch of mathematics that develops and applies methods for analyzing and studying such data.

  Surd An expression that includes a root that is an irrational number such as . It is left in root form as it cannot be simplified or written exactly as a decimal.

  Surface area The area of a flat or curved surface, or of the outside of a 3-D object.

  Tangent (1) A line which grazes the outside of a curve, just touching it at one point. (2) In trigonometry, the tangent function, abbreviated as tan, is defined as the ratio of the side length opposite a given angle to the side length adjacent to that angle, in a right-angled triangle.

  Term In an algebraic expression, one or more numbers or variables, usually separated by a plus (+) or minus (˗) sign, or in a sequence, by a comma. In x + 4y ˗ 2, for example, x, 4y, and 2 are all terms.

  Tessellation A pattern that is formed on a flat 2-D surface by repeated copies of one or more regular geometrical shapes that cover the surface without any gaps in between. This is also called a tiling.

  Tesseract A 4-D shape with four edges at every vertex, whereas a cube has three edges at every vertex, and a square has two.

  Tetrahedron A 3-D polyhedron that is made up of four triangular faces. A regular tetrahedron is one of the five Platonic solids.

  Theorem A significant proven result on a mathematical topic, especially one that is not self-evident. An unproved statement is called a conjecture.

  Topology The branch of mathematics that studies surfaces and objects by examining how their parts are connected rather than according to their exact geometrical shapes. For example, a doughnut and a teacup are topologically similar because they are both shapes that have one hole going through them (going through the handle, in the case of the teacup).

  Transcendental number Any irrational number that is not an algebraic number. The number pi (π) and Euler’s number e are both transcendental numbers.

  Transfinite number Another term for an infinite number. It is used particularly when infinities of different sizes or infinite collections of objects are compared.

  Transformation The conversion of a given shape or mathematical expression into another related one, using a particular rule.

  Translation A function that moves an object a certain distance in a direction without affecting its shape, size, or orientation.

  Trigonometry Originally, the study of the way the ratios between different sides of a right-angled triangle change when other angles in the triangle change, and later extended to all triangles. The way the ratios change is described by trigonometric functions, which are now fundamental to many branches of mathematics.

  Variable A mathematical quantity that can take on different values, often symbolized by a letter such as x or y.

  Vector A mathematical or physical quantity that has both magnitude and direction. In diagrams, vectors are often represented by bold arrows.

  Vector space A complex abstract mathematical structure that involves the multiplication of vectors by each other and by scalars.

  Venn diagram A diagram that shows sets of data as overlapping circles. The overlaps show what the sets have in common.

  Vertex (plural vertices) A corner or angle, where two or more lines, curves, or edges meet.

  Volume The amount of space inside a 3-D object.

  Whole number Any of the negative and positive counting numbers. For example, –1, 0, 19, 55, and so on. It is another term for integer.

  CONTRIBUTORS

  KARL WARSI, CONSULTANT EDITOR

  Karl Warsi taught mathematics in UK schools and colleges for many years. In 2000, he began publishing books on mathematics, creating bestselling textbook series for secondary-level students, both in the UK and worldwide. He is committed to inclusion in education, and the idea that people of all ages learn in different ways.

  JAN DANGERFIELD

  A lecturer and senior examiner in Further Mathematics, Jan Dangerfield is also a fellow of the UK’s Chartered Institute of Educational Assessors and a Fellow of the Royal Statistical Society. She has been a member of the British Society for the History of Mathematics for more than 30 years.

  HEATHER DAVIS

  British author and educator Heather Davis has taught mathematics for 30 years. She has published textbooks for Hodder Education and managed publications for the UK’s Association of Teachers of Mathematics. She presents courses for examination boards both in the UK and internationally and writes and presents enrichment activities for students.

  JOHN FARNDON

  A widely published author of popular books on science and nature, John Farndon has been shortlisted five times for the Royal Society’s Young People’s Science Book Prize, among other awards. He has written around 1,000 books on a range of subjects, including internationally acclaimed titles such as The Oceans Atlas, Do You Think You’re Clever? and Do Not Open, and contributed to major books such as Science and Science Year by Year.

  JONNY GRIFFITHS

  After studying mathematics and education at Cambridge University, the Open University, and the University of East Anglia, Jonny Griffiths taught math at Paston Sixth Form College in Norfolk, UK, for over 20 years. In 2005–06, he was made a Gatsby Teacher Fellow for creating the popular mathematics website Risps. In 2
016, he founded the competition Ritangle for students of mathematics.

  TOM JACKSON

  A writer for 25 years, Tom Jackson has written about 200 non-fiction books for adults and children and contributed to many more on a wide range of science and technology topics. They include Numbers: How Counting Changed the World; Everything is Mathematical, a book series with Marcus du Sautoy; and Help Your Kids with Science with Carol Vorderman.

  MUKUL PATEL

  Mukul Patel, who studied mathematics at Imperial College, London, writes and collaborates across many disciplines. He is the author of We’ve Got Your Number, a book on mathematics for children, and film scripts voiced by Tilda Swinton. He has also composed extensively for contemporary choreographers and designed sound installations for architects. He is currently investigating ethical issues in AI.

  SUE POPE

  A mathematics educator, Sue Pope is a long-standing member of the Association of Teachers of Mathematics and co-runs workshops on the history of mathematics in teaching at their conferences. Published widely, she recently co-edited Enriching Mathematics in the Primary Curriculum.

  MATT PARKER, FOREWORD

  Originally a math teacher from Australia, Matt Parker is a now a stand-up comedian, mathematics communicator, and a prominent math YouTuber on the Numberphile and Stand-up Maths channels, where his videos have had more than 100 million views. Matt performs live comedy with Festival of the Spoken Nerd and once calculated pi live in front of a sold-out Royal Albert Hall. He also presents television and radio programs for Discovery Channel and the BBC, and his 2019 book Humble Pi: A Comedy of Maths Errors topped the Sunday Times best-seller chart.