by DK
See also: Diophantine equations • 23 problems for the 20th century
RADIA PERLMAN
1951–
Virginia-born Perlman has been described as the “mother of the internet.” While a student at the Massachusetts Institute of Technology (MIT), she worked on a program that introduced children as young as three to computer programming. After graduating with a masters degree in mathematics in 1976, Perlman worked for a government contractor that developed software. Then, in 1984, while working for the Digital Equipment Corporation (DEC), she invented the Spanning Tree Protocol (STP), which ensures there is only one active path between two network devices; this would later prove crucial for the development of the internet. Perlman has taught at MIT and the universities of Washington and Harvard, and continues to work on computer network and security protocols.
See also: The mechanical computer • The Turing machine
MARYAM MIRZAKANI
1977–2017
At the age of 17, Mirzakani became the first Iranian woman to win a gold medal in the International Mathematical Olympiad. She graduated from Tehran’s Sharif University of Technology, before earning a doctorate from Harvard University in 2004 and taking up a professorship at Princeton University. Ten years later, Mirzakani was both the first woman and the first Iranian to receive the Fields Medal—for her contribution to the study of Riemann surfaces. She was working at Stanford University when she died of breast cancer, aged 40.
See also: Non-Euclidean geometries • The Riemann hypothesis • Topology
GLOSSARY
In this glossary, terms defined within another entry are identified with italic type.
Abstract algebra The branch of algebra, developed mainly in the 1900s, that investigates abstract mathematical structures such as groups and rings.
Acute angle An angle that is less than 90 degrees.
Algebra A branch of mathematics that involves the use of letters to stand for unknown or variable numbers in calculations.
Algebraic geometry The use of graphs to plot lines and curves that represent algebraic functions, such as y = x2.
Algebraic numbers All the rational numbers and those irrational numbers that can be obtained by calculating the roots of a rational number. An irrational number that is not algebraic (such as pi or e) is called a transcendental number.
Algorithm A defined sequence of mathematical or logical instructions, or rules, devised to solve a class of problems. Algorithms are widely used in mathematics and computer science for calculation, organizing data, and a multitude of other tasks.
Amicable numbers Any pair of whole numbers, where the factors of each one add up to form the other. The smallest pair are 220 and 284.
Analysis The branch of mathematics that studies limits and handles infinitely large and small quantities, especially to solve problems in calculus.
Analytic geometry See algebraic geometry.
Apex The vertex that is furthest from the base in a 3-D shape.
Applied mathematics The use of mathematics to solve problems in science and technology. It includes techniques for solving particular kinds of equations.
Arc A curved line that forms part of the circumference of a circle.
Area The amount of space inside any 2-D shape. Area is measured in square units, such as square inches (in2).
Associative law This states that if you add, for example, 1 + 2 + 3, the numbers can be added in any order. The law works for ordinary addition and multiplication, but not for subtraction or division.
Average The typical or middle value of a set of data. For the different kinds of averages, see mean, median, and mode.
Axiom A rule, especially one that is fundamental to an area of mathematics.
Axis (plural axes) A fixed reference line, such as the vertical y-axis and horizontal x-axis on a graph.
Base (1) In a number system, the base is the number around which the system is organized. The main number system we use today is the base-10 or decimal system, where the numerals 0 to 9 are used and the next number is written 10, indicating one ten and no units. See also place value system. (2) In logarithms, a fixed base (usually 10 or Euler’s number e) is used; the logarithm of any given number x is the power to which that base must be raised to produce x.
Binary notation Writing numbers using the binary system, in which the only numerals used are 0 and 1. For example, the number 6 is written as 110 in the binary system. Here, the leftmost 1 has the value of 4 (2 × 2), the middle 1 means one 2, and the zero means no single units: 4 + 2 + 0 makes 6.
Binomial An expression consisting of two terms added together, e.g. x + y. When a binomial expression is raised to a power, for example (x + y)3, the result when multiplied out gives (in this case) x3 + 3x2y + 3xy2 + y3. This process is called binomial expansion, and the numbers multiplying the terms (3s in this case) are called binomial coefficients. The binomial theorem is a rule for working out binomial coefficients in complex cases. See also polynomial.
Calculus A branch of mathematics that deals with continuously changing quantities. It includes differential calculus, which is concerned with rates of change, and integral calculus, which calculates areas and volumes under curves or curved surfaces.
Cardinal numbers Numbers that denote a quantity, such as 1, 2, 3 (in contrast to ordinal numbers).
Chord A straight line that cuts across a circle, but does not go through its center.
Cipher Any systematic method of coding messages so that they cannot be understood without being deciphered first.
Circumference The distance all the way around the outside edge of a circle.
Coefficient A number or expression, usually a constant, that is placed before another number (especially a variable) and multiplies it. For example, in the expressions ax2 and 3x, a and 3 are coefficients.
Coincident In geometry, two or more lines or figures that, when superimposed on each other, share all points and occupy exactly the same space.
Combinatorics A branch of mathematics that studies the ways in which sets of numbers, shapes, or other mathematical objects can be combined.
Commutative law The law that states that 1 + 2 = 2 + 1, for example, and that the order in which the numbers are set down doesn’t matter. It works for ordinary addition and multiplication, but not for subtraction and division.
Complex number A number that is a combination of a real number and an imaginary number.
Complex plane The infinite 2-D plane on which complex numbers can be plotted.
Composite number A whole number that is not prime, but can be created by multiplying together smaller numbers.
Cone A 3-D shape with a circular base and a side that narrows upward toward a point (apex).
Congruent Having the same size and shape. (Used when comparing geometrical shapes.)
Conjecture A mathematical statement or claim that has not yet been proved or disproved. A pair of related conjectures can be strong or weak: if the strong conjecture is proved, then the weak conjecture is also proved, but not vice versa.
Constant A quantity in a mathematical expression that does not vary—often symbolized by a letter such as a, b, or c.
Convergence A property of some infinite mathematical series where not only is each term smaller than the last, but the terms, when added up, approach a finite answer. The value of numbers such as pi can be estimated using convergent series.
Coordinates Combinations of numbers that describe the position of a point, line, or shape on a graph or a geographical position on a map. In mathematical contexts, they are written (for a 2-D case) in the form (x,y), where x is the horizontal position and y the vertical position.
Cosine (abbreviation cos) A function in trigonometry similar to a sine, except that it is defined as the ratio of the length of the side of a right-angled triangle adjacent to a given angle to the length of the triangle’s hypotenuse.
Cube A 3-D geometrical figure whose faces are six identical squares. A cube number is one that
is obtainable by multiplying a smaller number together twice— for example 8, which is 2 × 2 × 2 (23). This multiplication resembles the way the volume of a cube is calculated, by multiplying its length × height × depth.
Cubic equation An equation containing at least one variable multiplied by itself twice (for example, y × y × y, also written as y3), but no variable multiplied more times than this.
Cubit A measure of length used in the ancient world, based on the length of the human forearm.
Cylinder A 3-D shape, such as a tin can, with two identical circular ends joined by one curved surface.
Deduction A process by which a problem is solved by drawing on known or assumed mathematical principles. See also induction.
Degree (1) A measure of angle in geometry: rotating a full circle involves turning 360 degrees. (2) The degree or order of a polynomial is the highest-power term within it: for example, a polynomial is “of degree 3” or “of order 3” if it contains a cubed term, such as x3, as its highest power. Similarly, in differential equations, the term that has been differentiated most times in a given equation determines its degree or order.
Denominator The lower number in a fraction, such as the 4 in 3⁄4.
Derivative See differentiation.
Diameter A straight line touching two points on the edge of a circle and passing through the center.
Differential equation An equation that represents a function including the derivative(s) of a given variable.
Differentiation In calculus, the process of working out the rate of change of a given mathematical function. The result of the calculation is another function called the differential or derivative of the first function.
Divergence A term applied mainly to infinite series that do not approach closer and closer to an end-number. See also convergence.
Divisor The number or quantity by which another number or quantity is being divided.
Dodecahedron A 3-D polyhedron made up of 12 pentagonal (5-sided) faces. A regular dodecahedron is one of the five Platonic solids.
Ellipse A shape like a circle, but stretched out symmetrically in one direction.
Encryption The process of converting data or a message to a secure, coded form.
Equation A statement that two mathematical expressions or quantities are equal to each other. An equation is the usual way of expressing a mathematical function. When an equation is true of all the values of a variable (for example, the equation y × y × y = y3), it is called an identity.
Equilateral triangle A triangle in which all three sides are the same length and all three angles the same size.
Existence proof A mathematical proof that something exists, obtained either by constructing an example or by general deduction.
Expansion In algebra, the expansion of an expression is the opposite of factorization. For example, (x + 2)(x + 3) can be expanded to x2 + 5x + 6, by multiplying each term in the first pair of parentheses by each term in the second pair of parentheses.
Exponent The superscript number that indicates the power to which a number or quantity has been raised, such as the 2 in x2 (x × x). Also called an index.
Exponential function A mathematical function where, as a quantity gets larger, its rate of increase also gets faster. The result is often called exponential growth.
Expression Any meaningful combination of mathematical symbols, such as 2x + 5.
Face A flat surface of any 3-D shape.
Factor A number or expression that divides exactly into another number or expression. For example, 1, 2, 3, 4, 6, and 12 are all factors of 12.
Factorial The product of any positive integer and all the positive integers that are smaller than it. For example, factorial 5, also written 5! (with an exclamation mark) is 5 × 4 × 3 × 2 × 1 = 120.
Factorization Expressing a number or mathematical expression in terms of factors that when multiplied together give the original number or expression.
Formula A mathematical rule that describes a relationship between quantities.
Fractals Self-similar curves or shapes of different sizes that form complex patterns that have the same general appearance at any magnification. Many natural phenomena, such as clouds and rock formations, approximate to fractals.
Function A mathematical relationship where the value of one variable is worked out uniquely from the value of other numbers, using a particular rule. For example, in the function y = x2 + 3, the value of y is calculated by squaring x and then adding 3. The same function can also be written f(x) = x2 + 3, where f(x) stands for “function of x.”
Geometry The branch of mathematics that studies shapes, lines, points, and their relationships. See also non-Euclidean geometries.
Gradient The slope of a line.
Graph (1) A chart on which data is plotted using, for instance, lines, points, curves, or bars. (2) In graph theory, a graph is a collection of points, called vertices, and lines, called edges, that can be used to model theoretical and real networks, relations, and processes in a range of scientific and social fields.
Graph theory A branch of mathematics that studies how graphs made up of points and lines are connected.
Group A mathematical set, together with an operation which, when performed on members of the set, yields an answer that is still a member of the set. For example, the set of integers forms a group when addition is the operation. Groups can be finite or infinite, and their study is called group theory.
Harmonic series The mathematical series 1 + 1⁄2 + 1⁄3 + 1⁄4 + 1⁄5 +… . The individual terms in the series define the different ways that a stretched string, for example, or air in a tube, can vibrate to produce sound. The resulting series of musical pitches forms the basis of the musical scale.
Hyperbola A mathematical curve that looks something like a parabola, but in which the two extensions of the curve approach two imaginary straight lines at angles to each other without ever touching or crossing the lines.
Hypotenuse The longest side of a right-angled triangle, located on the opposite side from the right angle.
Icosahedron A 3-D polyhedron made up of 20 triangular faces. A regular icosahedron is one of the five Platonic solids.
Ideal In abstract algebra, a mathematical ring that is a component of a larger ring.
Identity element In a set of numbers or other mathematical objects, an operation carried out on the set, such as multiplication or addition, always has an identity element—a number or expression that leaves other terms unchanged after the operation has been carried out. The identity element in ordinary multiplication, for example, is 1, as 1 × x = x, and in the addition of real numbers, it is 0, as 0 + x = x.
Imaginary number Any number that is a multiple of , which does not exist as a real number. It is expressed as the symbol i.
Incommensurable Something that cannot be measured exactly in terms of something else.
Index (plural indices) Another word for an exponent.
Induction A way of obtaining a general conclusion in mathematics by establishing that if a statement is true for one step in a process, it is also true for the next step in a process and all those that follow. See also deduction.
Infinite Indefinitely large and without limit. In mathematics, there are different kinds of infinity: the set of natural numbers, for example, is countably infinite (countable one by one, even though the end is never reached), while the real numbers are uncountably infinite.
Infinite series A mathematical series with an infinite number of terms: see series.
Infinitesimal calculus Another term for calculus, generally used in the past when calculus was viewed as involving the adding up of infinitesimals (infinitely small but nonzero quantities).
Input Any variable, which when combined with a function, produces an output.
Integer Any of the negative or positive whole numbers. (Fractions are not integers.)
Integral (1) Relating to integers. (2) A mathematical expression used in integral calculus, or the result of p
erforming an integration.
Integration The process of performing a calculation in integral calculus.
Inverse A mathematical expression or operation that is the opposite of another one and undoes it. For example, division is the inverse of multiplication.
Irrational number Any number that cannot be expressed as one whole number divided by another and is not an imaginary number.
Isosceles triangle A triangle with two sides the same length and two angles the same size.
Iteration Performing the same operation again and again to achieve a desired result.
Limit The end number that is approached as certain calculations are iterated to infinity.
Linear equation An equation that contains no variable multiplied by itself (for example, no x2 or x3). Linear equations result in straight lines when they are plotted as graphs.
Linear transformation Also called linear mapping, a mapping between vector spaces.
Logarithm The logarithm of a number is the power to which another number (called the base—usually either 10 or Euler’s number e)—must be raised to give the original number. For example, 100.301 = 2, and so 0.301 is the logarithm (to the base 10) of 2. A logarithm to the base e (2.71828…) is called a natural logarithm and is indicated by the prefix ln or loge. The advantage of logarithms is that when numbers need to be multiplied, the calculation can be simplified by adding their logarithms instead.
Logic The study of reasoning, that is, how conclusions can be deduced correctly from given starting information (premises) by following valid rules.
Manifold A kind of abstract mathematical space that in any particular small region resembles ordinary 3-D space. It is a concept within topology.
Mapping Establishing a relationship between members of one mathematical set and another. It is often but not always used to mean a one-to-one mapping, where each member of one set is associated with one member of the other set, and vice versa.
Matrix (plural matrices) A square or rectangular array of numbers or other mathematical quantities that can be treated as a single object in calculations. Matrices have special rules for addition and multiplication. Their many uses include solving several equations simultaneously, describing vectors, calculating transformations in the shape and position of geometrical figures, and representing real-world data.