Definition of the word of in math

This is a glossary of common mathematical terms used in arithmetic, geometry, algebra, and statistics.

Abacus: An early counting tool used for basic arithmetic.

Absolute Value: Always a positive number, absolute value refers to the distance of a number from 0.

Acute Angle: An angle whose measure is between 0° and 90° or with less than 90° (or pi/2) radians.

Addend: A number involved in an addition problem; numbers being added are called addends.

Algebra: The branch of mathematics that substitutes letters for numbers to solve for unknown values.

Algorithm: A procedure or set of steps used to solve a mathematical computation.

Angle: Two rays sharing the same endpoint (called the angle vertex).

Angle Bisector: The line dividing an angle into two equal angles.

Area: The two-dimensional space taken up by an object or shape, given in square units.

Array: A set of numbers or objects that follow a specific pattern.

Attribute: A characteristic or feature of an object—such as size, shape, color, etc.—that allows it to be grouped.

Average: The average is the same as the mean. Add up a series of numbers and divide the sum by the total number of values to find the average.

Base: The bottom of a shape or three-dimensional object, what an object rests on.

Base 10: Number system that assigns place value to numbers.

Bar Graph: A graph that represents data visually using bars of different heights or lengths.

BEDMAS or PEMDAS Definition: An acronym used to help people remember the correct order of operations for solving algebraic equations. BEDMAS stands for «Brackets, Exponents, Division, Multiplication, Addition, and Subtraction» and PEMDAS stands for «Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction».

Bell Curve: The bell shape created when a line is plotted using data points for an item that meets the criteria of normal distribution. The center of a bell curve contains the highest value points.

Binomial: A polynomial equation with two terms usually joined by a plus or minus sign.

Box and Whisker Plot/Chart: A graphical representation of data that shows differences in distributions and plots data set ranges.

Calculus: The branch of mathematics involving derivatives and integrals, Calculus is the study of motion in which changing values are studied.

Capacity: The volume of substance that a container will hold.

Centimeter: A metric unit of measurement for length, abbreviated as cm. 2.5 cm is approximately equal to an inch.

Circumference: The complete distance around a circle or a square.

Chord: A segment joining two points on a circle.

Coefficient: A letter or number representing a numerical quantity attached to a term (usually at the beginning). For example, x is the coefficient in the expression x(a + b) and 3 is the coefficient in the term 3y.

Common Factors: A factor shared by two or more numbers, common factors are numbers that divide exactly into two different numbers.

Complementary Angles: Two angles that together equal 90°.

Composite Number: A positive integer with at least one factor aside from its own. Composite numbers cannot be prime because they can be divided exactly.

Cone: A three-dimensional shape with only one vertex and a circular base.

Conic Section: The section formed by the intersection of a plane and cone.

Constant: A value that does not change.

Coordinate: The ordered pair that gives a precise location or position on a coordinate plane.

Congruent: Objects and figures that have the same size and shape. Congruent shapes can be turned into one another with a flip, rotation, or turn.

Cosine: In a right triangle, cosine is a ratio that represents the length of a side adjacent to an acute angle to the length of the hypotenuse.

Cylinder: A three-dimensional shape featuring two circle bases connected by a curved tube.

Decagon: A polygon/shape with ten angles and ten straight lines.

Decimal: A real number on the base ten standard numbering system.

Denominator: The bottom number of a fraction. The denominator is the total number of equal parts into which the numerator is being divided.

Degree: The unit of an angle’s measure represented with the symbol °.

Diagonal: A line segment that connects two vertices in a polygon.

Diameter: A line that passes through the center of a circle and divides it in half.

Difference: The difference is the answer to a subtraction problem, in which one number is taken away from another.

Digit: Digits are the numerals 0-9 found in all numbers. 176 is a 3-digit number featuring the digits 1, 7, and 6.

Dividend: A number being divided into equal parts (inside the bracket in long division).

Divisor: A number that divides another number into equal parts (outside of the bracket in long division).

Edge: A line is where two faces meet in a three-dimensional structure.

Ellipse: An ellipse looks like a slightly flattened circle and is also known as a plane curve. Planetary orbits take the form of ellipses.

End Point: The «point» at which a line or curve ends.

Equilateral: A term used to describe a shape whose sides are all of equal length.

Equation: A statement that shows the equality of two expressions by joining them with an equals sign.

Even Number: A number that can be divided or is divisible by 2.

Event: This term often refers to an outcome of probability; it may answers question about the probability of one scenario happening over another.

Evaluate: This word means «to calculate the numerical value».

Exponent: The number that denotes repeated multiplication of a term, shown as a superscript above that term. The exponent of 3⁴ is 4.

Expressions: Symbols that represent numbers or operations between numbers.

Face: The flat surfaces on a three-dimensional object.

Factor: A number that divides into another number exactly. The factors of 10 are 1, 2, 5, and 10 (1 x 10, 2 x 5, 5 x 2, 10 x 1).

Factoring: The process of breaking numbers down into all of their factors.

Factorial Notation: Often used in combinatorics, factorial notations requires that you multiply a number by every number smaller than it. The symbol used in factorial notation is ! When you see x!, the factorial of x is needed.

Factor Tree: A graphical representation showing the factors of a specific number.

Fibonacci Sequence: A sequence beginning with a 0 and 1 whereby each number is the sum of the two numbers preceding it. «0, 1, 1, 2, 3, 5, 8, 13, 21, 34…» is a Fibonacci sequence.

Figure: Two-dimensional shapes.

Finite: Not infinite; has an end.

Flip: A reflection or mirror image of a two-dimensional shape.

Formula: A rule that numerically describes the relationship between two or more variables.

Fraction: A quantity that is not whole that contains a numerator and denominator. The fraction representing half of 1 is written as 1/2.

Frequency: The number of times an event can happen in a given period of time; often used in probability calculations.

Furlong: A unit of measurement representing the side length of one square acre. One furlong is approximately 1/8 of a mile, 201.17 meters, or 220 yards.

Geometry: The study of lines, angles, shapes, and their properties. Geometry studies physical shapes and the object dimensions.

Graphing Calculator: A calculator with an advanced screen capable of showing and drawing graphs and other functions.

Graph Theory: A branch of mathematics focused on the properties of graphs.

Greatest Common Factor: The largest number common to each set of factors that divides both numbers exactly. The greatest common factor of 10 and 20 is 10.

Hexagon: A six-sided and six-angled polygon.

Histogram: A graph that uses bars that equal ranges of values.

Hyperbola: A type of conic section or symmetrical open curve. The hyperbola is the set of all points in a plane, the difference of whose distance from two fixed points in the plane is a positive constant.

Hypotenuse: The longest side of a right-angled triangle, always opposite to the right angle itself.

Identity: An equation that is true for variables of any value.

Improper Fraction: A fraction whose numerator is equal to or greater than the denominator, such as 6/4.

Inequality: A mathematical equation expressing inequality and containing a greater than (>), less than (<), or not equal to (≠) symbol.

Integers: All whole numbers, positive or negative, including zero.

Irrational: A number that cannot be represented as a decimal or fraction. A number like pi is irrational because it contains an infinite number of digits that keep repeating. Many square roots are also irrational numbers.

Isosceles: A polygon with two sides of equal length.

Kilometer: A unit of measure equal to 1000 meters.

Knot: A closed three-dimensional circle that is embedded and cannot be untangled.

Like Terms: Terms with the same variable and same exponents/powers.

Like Fractions: Fractions with the same denominator.

Line: A straight infinite path joining an infinite number of points in both directions.

Line Segment: A straight path that has two endpoints, a beginning and an end.

Linear Equation: An equation that contains two variables and can be plotted on a graph as a straight line.

Line of Symmetry: A line that divides a figure into two equal shapes.

Logic: Sound reasoning and the formal laws of reasoning.

Logarithm: The power to which a base must be raised to produce a given number. If nx = a, the logarithm of a, with n as the base, is x. Logarithm is the opposite of exponentiation.

Mean: The mean is the same as the average. Add up a series of numbers and divide the sum by the total number of values to find the mean.

Median: The median is the «middle value» in a series of numbers ordered from least to greatest. When the total number of values in a list is odd, the median is the middle entry. When the total number of values in a list is even, the median is equal to the sum of the two middle numbers divided by two.

Midpoint: A point that is exactly halfway between two locations.

Mixed Numbers: Mixed numbers refer to whole numbers combined with fractions or decimals. Example 3 ¹/₂ or 3.5.

Mode: The mode in a list of numbers are the values that occur most frequently.

Modular Arithmetic: A system of arithmetic for integers where numbers «wrap around» upon reaching a certain value of the modulus.

Monomial: An algebraic expression made up of one term.

Multiple: The multiple of a number is the product of that number and any other whole number. 2, 4, 6, and 8 are multiples of 2.

Multiplication: Multiplication is the repeated addition of the same number denoted with the symbol x. 4 x 3 is equal to 3 + 3 + 3 + 3.

Multiplicand: A quantity multiplied by another. A product is obtained by multiplying two or more multiplicands.

Natural Numbers: Regular counting numbers.

Negative Number: A number less than zero denoted with the symbol -. Negative 3 = -3.

Net: A two-dimensional shape that can be turned into a two-dimensional object by gluing/taping and folding.

Nth Root: The nth root of a number is how many times a number needs to be multiplied by itself to achieve the value specified. Example: the 4th root of 3 is 81 because 3 x 3 x 3 x 3 = 81.

Norm: The mean or average; an established pattern or form.

Normal Distribution: Also known as Gaussian distribution, normal distribution refers to a probability distribution that is reflected across the mean or center of a bell curve.

Numerator: The top number in a fraction. The numerator is divided into equal parts by the denominator.

Number Line: A line whose points correspond to numbers.

Numeral: A written symbol denoting a number value.

Obtuse Angle: An angle measuring between 90° and 180°.

Obtuse Triangle: A triangle with at least one obtuse angle.

Octagon: A polygon with eight sides.

Odds: The ratio/likelihood of a probability event happening. The odds of flipping a coin and having it land on heads are one in two.

Odd Number: A whole number that is not divisible by 2.

Operation: Refers to addition, subtraction, multiplication, or division.

Ordinal: Ordinal numbers give relative position in a set: first, second, third, etc.

Order of Operations: A set of rules used to solve mathematical problems in the correct order. This is often remembered with acronyms BEDMAS and PEMDAS.

Outcome: Used in probability to refer to the result of an event.

Parallelogram: A quadrilateral with two sets of opposite sides that are parallel.

Parabola: An open curve whose points are equidistant from a fixed point called the focus and a fixed straight line called the directrix.

Pentagon: A five-sided polygon. Regular pentagons have five equal sides and five equal angles.

Percent: A ratio or fraction with the denominator 100.

Perimeter: The total distance around the outside of a polygon. This distance is obtained by adding together the units of measure from each side.

Perpendicular: Two lines or line segments intersecting to form a right angle.

Pi: Pi is used to represent the ratio of a circumference of a circle to its diameter, denoted with the Greek symbol π.

Plane: When a set of points join together to form a flat surface that extends in all directions, this is called a plane.

Polynomial: The sum of two or more monomials.

Polygon: Line segments joined together to form a closed figure. Rectangles, squares, and pentagons are just a few examples of polygons.

Prime Numbers: Prime numbers are integers greater than 1 that are only divisible by themselves and 1.

Probability: The likelihood of an event happening.

Product: The sum obtained through multiplication of two or more numbers.

Proper Fraction: A fraction whose denominator is greater than its numerator.

Protractor: A semi-circle device used for measuring angles. The edge of a protractor is subdivided into degrees.

Quadrant: One quarter (qua) of the plane on the Cartesian coordinate system. The plane is divided into 4 sections, each called a quadrant.

Quadratic Equation: An equation that can be written with one side equal to 0. Quadratic equations ask you to find the quadratic polynomial that is equal to zero.

Quadrilateral: A four-sided polygon.

Quadruple: To multiply or to be multiplied by 4.

Qualitative: Properties that must be described using qualities rather than numbers.

Quartic: A polynomial having a degree of 4.

Quintic: A polynomial having a degree of 5.

Quotient: The solution to a division problem.

Radius: A distance found by measuring a line segment extending from the center of a circle to any point on the circle; the line extending from the center of a sphere to any point on the outside edge of the sphere.

Ratio: The relationship between two quantities. Ratios can be expressed in words, fractions, decimals, or percentages. Example: the ratio given when a team wins 4 out of 6 games is 4/6, 4:6, four out of six, or ~67%.

Ray: A straight line with only one endpoint that extends infinitely.

Range: The difference between the maximum and minimum in a set of data.

Rectangle: A parallelogram with four right angles.

Repeating Decimal: A decimal with endlessly repeating digits. Example: 88 divided by 33 equals 2.6666666666666…(«2.6 repeating»).

Reflection: The mirror image of a shape or object, obtained from flipping the shape on an axis.

Remainder: The number left over when a quantity cannot be divided evenly. A remainder can be expressed as an integer, fraction, or decimal.

Right Angle: An angle equal to 90°.

Right Triangle: A triangle with one right angle.

Rhombus: A parallelogram with four sides of equal length and no right angles.

Scalene Triangle: A triangle with three unequal sides.

Sector: The area between an arc and two radii of a circle, sometimes referred to as a wedge.

Slope: Slope shows the steepness or incline of a line and is determined by comparing the positions of two points on the line (usually on a graph).

Square Root: A number squared is multiplied by itself; the square root of a number is whatever integer gives the original number when multiplied by itself. For instance, 12 x 12 or 12 squared is 144, so the square root of 144 is 12.

Stem and Leaf: A graphic organizer used to organize and compare data. Similar to a histogram, stem and leaf graphs organize intervals or groups of data.

Subtraction: The operation of finding the difference between two numbers or quantities by «taking away» one from the other.

Supplementary Angles: Two angles are supplementary if their sum is equal to 180°.

Symmetry: Two halves that match perfectly and are identical across an axis.

Tangent: A straight line touching a curve from only one point.

Term: Piece of an algebraic equation; a number in a sequence or series; a product of real numbers and/or variables.

Tessellation: Congruent plane figures/shapes that cover a plane completely without overlapping.

Translation: A translation, also called a slide, is a geometrical movement in which a figure or shape is moved from each of its points the same distance and in the same direction.

Transversal: A line that crosses/intersects two or more lines.

Trapezoid: A quadrilateral with exactly two parallel sides.

Tree Diagram: Used in probability to show all possible outcomes or combinations of an event.

Triangle: A three-sided polygon.

Trinomial: A polynomial with three terms.

Unit: A standard quantity used in measurement. Inches and centimeters are units of length, pounds and kilograms are units of weight, and square meters and acres are units of area.

Uniform: Term meaning «all the same». Uniform can be used to describe size, texture, color, design, and more.

Variable: A letter used to represent a numerical value in equations and expressions. Example: in the expression 3x + y, both y and x are the variables.

Venn Diagram: A Venn diagram is usually shown as two overlapping circles and is used to compare two sets. The overlapping section contains information that is true of both sides or sets and the non-overlapping portions each represent a set and contain information that is only true of their set.

Volume: A unit of measure describing how much space a substance occupies or the capacity of a container, provided in cubic units.

Vertex: The point of intersection between two or more rays, often called a corner. A vertex is where two-dimensional sides or three-dimensional edges meet.

Weight: The measure of how heavy something is.

Whole Number: A whole number is a positive integer.

X-Axis: The horizontal axis in a coordinate plane.

X-Intercept: The value of x where a line or curve intersects the x-axis.

X: The Roman numeral for 10.

x: A symbol used to represent an unknown quantity in an equation or expression.

Y-Axis: The vertical axis in a coordinate plane.

Y-Intercept: The value of y where a line or curve intersects the y-axis.

Yard: A unit of measure that is equal to approximately 91.5 centimeters or 3 feet.

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The mathematics of physics should be explored, but new mathematical, statistical, and qualitative methodologies should also be sought.

The book was in two parts, the first part philosophical, the second dealing with mathematics and the natural sciences.

But they had — at the same time — to be war y that mathematics was not «so close» to applications as to produce unwanted philosophical conflict.

He thereby wished to purify mathematics of any reliance on the external world.

It is based on physical principles, but the mathematics is limited so that interested undergraduate students will be able to follow.

As such, they are the outcome both of a historical process of reciprocal interaction with the body of mathematics and of external influences.

Clear physical explanations are given throughout, making it possible to follow easily in sections where the mathematics are difficult.

According to the traditional hierarchy of the sciences, psychology is far separated from physics and mathematics by the intervening sciences of chemistry, biology, and neurophysiology.

Therefore, however complicated this simplicity might be for the human mind, mathematics is the master tool with which to seize the truth of the world.

The notation for the potential formulation is also simpler, making it easier to understand the underlying physics and mathematics.

Those artifacts include language, logic, mathematics, graphics, symbols, and various tools and devices, whether mechanical or electronic.

And a group with no need for higher mathematics may package complex arithmetic in its numbering system.

Such changes also produced a revolution in the images of geometrical knowledge and simultaneously triggered interesting and fruitful processes of reflexive thinking in mathematics.

Not only did such canons apply to natural science and to mathematics, but they applied to «mixed sciences» such as optics and mechanics as well.

This fact helps explain the small number of women staff in disciplines such as mathematics, physics, and mechanics.

These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.

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This shows grade level based on the word’s complexity.

noun

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CAN YOU ANSWER THESE COMMON GRAMMAR DEBATES?

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Origin of math

First recorded in 1845–50; by shortening

Words nearby math

Mater Turrita, mateship, mate’s rates, matey, mat grass, math, mathematical, mathematical expectation, mathematical induction, mathematical logic, mathematical probability

Other definitions for math (2 of 4)

noun British Dialect.

a mowing; a leveling or cutting down of grass, grain, etc., with a mowing machine or scythe.

the crop mowed.

Origin of math

First recorded before 900; Middle English (bede)-mad, a kind of manorial duty to mow for one’s lord, Old English mǣth “mowing, hay harvest”; cognate with German Mahd,Old Frisian mēth, Old Saxon mād(dag) “mowing (day)”; cf. aftermath, mow¹, mow²

Other definitions for math (3 of 4)

Also ma·tha [muhth—uh]. /ˈmʌθ ə/.

Origin of math

First recorded in 1825–35; from Hindi maṭh, from Sanskrit maṭha “hut, cottage, cell, monastery”

Other definitions for math (4 of 4)

abbreviation

mathematical.

mathematician.

mathematics.

Dictionary.com Unabridged
Based on the Random House Unabridged Dictionary, © Random House, Inc. 2023

Words related to math

analytical, numerical, scientific, algebra, calculation, calculus, geometry, addition, division, figures, multiplication, numbers, subtraction, trigonometry, algebraic, algorithmic, arithmetical, computative, geometrical, measurable

How to use math in a sentence

What’s telling about Risch’s remarks, however, is less the math than the defeatism behind it.
They blame funding but they didn’t let the math teacher go when they canceled the class.
Such calculations require strong math skills, but that’s just the beginning.
For Ivan, though, the idea of pure math or pure language is exciting.

Now we don’t need complicated math to tell us that 190,000 Americans have died from the novel coronavirus, with no end in the sight.
After the curtain calls, Christopher comes back to explain a complicated math problem.
Supporters pointed to math and literacy gains, while critics noted that those improvements disappeared in elementary school.
Jackson was an exceptional math and science student; the dreaded Bartlett was one of his favorite professors.
And last year, 4th and 8th grade students showed the biggest math and reading gains in the country.
The risk-benefit math becomes less favorable for the older patient.
Aftermath, aft′ėr-math, n. a second mowing of grass in the same season.
Darryl and I once tried to write our own better spam filter and when you filter spam, you need Bayesian math.
Im only an enlisted man and they dont give enlisted men enough math to answer questions like that.
Maybe Miss Graham can find me a book on math problems that a man can do in his head.
She could send several cheeses to table,—new milk cheese, nettle-cheese, floaten milk cheese and eddish or after-math cheese.

British Dictionary definitions for math (1 of 2)

noun

US and Canadian informal short for mathematics Brit equivalent: maths

British Dictionary definitions for math (2 of 2)

Collins English Dictionary — Complete & Unabridged 2012 Digital Edition
© William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins
Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009, 2012

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From Wikipedia, the free encyclopedia

A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

The most basic symbols are the decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the Latin alphabet. The decimal digits are used for representing numbers through the Hindu–Arabic numeral system. Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. Letters are used for representing many other sorts of mathematical objects. As the number of these sorts has remarkably increased in modern mathematics, the Greek alphabet and some Hebrew letters are also used. In mathematical formulas, the standard typeface is italic type for Latin letters and lower-case Greek letters, and upright type for upper case Greek letters. For having more symbols, other typefaces are also used, mainly boldface , script typeface (the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur , and blackboard bold ${displaystyle mathbb {N,Z,Q,R,C,H,F} _{q}}$ (the other letters are rarely used in this face, or their use is unconventional).

The use of Latin and Greek letters as symbols for denoting mathematical objects is not described in this article. For such uses, see Variable (mathematics) and List of mathematical constants. However, some symbols that are described here have the same shape as the letter from which they are derived, such as and .

These letters alone are not sufficient for the needs of mathematicians, and many other symbols are used. Some take their origin in punctuation marks and diacritics traditionally used in typography; others by deforming letter forms, as in the cases of and forall . Others, such as + and =, were specially designed for mathematics.

Layout of this article[edit]

Normally, entries of a glossary are structured by topics and sorted alphabetically. This is not possible here, as there is no natural order on symbols, and many symbols are used in different parts of mathematics with different meanings, often completely unrelated. Therefore, some arbitrary choices had to be made, which are summarized below.

The article is split into sections that are sorted by an increasing level of technicality. That is, the first sections contain the symbols that are encountered in most mathematical texts, and that are supposed to be known even by beginners. On the other hand, the last sections contain symbols that are specific to some area of mathematics and are ignored outside these areas. However, the long section on brackets has been placed near to the end, although most of its entries are elementary: this makes it easier to search for a symbol entry by scrolling.

Most symbols have multiple meanings that are generally distinguished either by the area of mathematics where they are used or by their syntax, that is, by their position inside a formula and the nature of the other parts of the formula that are close to them.

As readers may not be aware of the area of mathematics to which is related the symbol that they are looking for, the different meanings of a symbol are grouped in the section corresponding to their most common meaning.

When the meaning depends on the syntax, a symbol may have different entries depending on the syntax. For summarizing the syntax in the entry name, the symbol Box is used for representing the neighboring parts of a formula that contains the symbol. See § Brackets for examples of use.

Most symbols have two printed versions. They can be displayed as Unicode characters, or in LaTeX format. With the Unicode version, using search engines and copy-pasting are easier. On the other hand, the LaTeX rendering is often much better (more aesthetic), and is generally considered a standard in mathematics. Therefore, in this article, the Unicode version of the symbols is used (when possible) for labelling their entry, and the LaTeX version is used in their description. So, for finding how to type a symbol in LaTeX, it suffices to look at the source of the article.

For most symbols, the entry name is the corresponding Unicode symbol. So, for searching the entry of a symbol, it suffices to type or copy the Unicode symbol into the search textbox. Similarly, when possible, the entry name of a symbol is also an anchor, which allows linking easily from another Wikipedia article. When an entry name contains special characters such as [, ], and |, there is also an anchor, but one has to look at the article source to know it.

Finally, when there is an article on the symbol itself (not its mathematical meaning), it is linked to in the entry name.

Arithmetic operators[edit]

+: 1. Denotes addition and is read as plus; for example, 3 + 2.; 2. Denotes that a number is positive and is read as plus. Redundant, but sometimes used for emphasizing that a number is positive, specially when other numbers in the context are or may be negative; for example, +2.; 3. Sometimes used instead of for a disjoint union of sets.
–: 1. Denotes subtraction and is read as minus; for example, 3 – 2.; 2. Denotes the additive inverse and is read as negative or the opposite of; for example, –2.; 3. Also used in place of for denoting the set-theoretic complement; see in § Set theory.
×: 1. In elementary arithmetic, denotes multiplication, and is read as times; for example, 3 × 2.; 2. In geometry and linear algebra, denotes the cross product.; 3. In set theory and category theory, denotes the Cartesian product and the direct product. See also × in § Set theory.
·: 1. Denotes multiplication and is read as times; for example, 3 ⋅ 2.; 2. In geometry and linear algebra, denotes the dot product.; 3. Placeholder used for replacing an indeterminate element. For example, «the absolute value is denoted | · |» is clearer than saying that it is denoted as | |.
±: 1. Denotes either a plus sign or a minus sign.; 2. Denotes the range of values that a measured quantity may have; for example, 10 ± 2 denotes an unknown value that lies between 8 and 12.
∓: Used paired with ±, denotes the opposite sign; that is, + if ± is –, and – if ± is +.
÷: Widely used for denoting division in anglophone countries, it is no longer in common use in mathematics and its use is «not recommended».^[1] In some countries, it can indicate subtraction.
:: 1. Denotes the ratio of two quantities.; 2. In some countries, may denote division.; 3. In set-builder notation, it is used as a separator meaning «such that»; see {□ : □}.
/: 1. Denotes division and is read as divided by or over. Often replaced by a horizontal bar. For example, 3 / 2 or ${displaystyle {frac {3}{2}}}$ .; 2. Denotes a quotient structure. For example, quotient set, quotient group, quotient category, etc.; 3. In number theory and field theory, denotes a field extension, where F is an extension field of the field E.; 4. In probability theory, denotes a conditional probability. For example, denotes the probability of A, given that B occurs. Also denoted : see «|«.
√: Denotes square root and is read as the square root of. Rarely used in modern mathematics without a horizontal bar delimiting the width of its argument (see the next item). For example, √2.
√: 1. Denotes square root and is read as the square root of. For example, .; 2. With an integer greater than 2 as a left superscript, denotes an nth root. For example, denotes the 7th root of 3.
^: 1. Exponentiation is normally denoted with a superscript. However, is often denoted x^y when superscripts are not easily available, such as in programming languages (including LaTeX) or plain text emails.; 2. Not to be confused with ∧.

Equality, equivalence and similarity[edit]

=: 1. Denotes equality.; 2. Used for naming a mathematical object in a sentence like «let «, where E is an expression. On a blackboard and in some mathematical texts, this may be abbreviated as or This is related to the concept of assignment in computer science, which is variously denoted (depending on the programming language used)
≠: Denotes inequality and means «not equal».
≈: Means «is approximately equal to». For example, ${displaystyle pi approx {frac {22}{7}}}$ (for a more accurate approximation, see pi).
~: 1. Between two numbers, either it is used instead of ≈ to mean «approximatively equal», or it means «has the same order of magnitude as».; 2. Denotes the asymptotic equivalence of two functions or sequences.; 3. Often used for denoting other types of similarity, for example, matrix similarity or similarity of geometric shapes.; 4. Standard notation for an equivalence relation.; 5. In probability and statistics, may specify the probability distribution of a random variable. For example, means that the distribution of the random variable X is standard normal.^[2]; 6. Notation for showing proportionality. See also ∝ for a less ambiguous symbol.
≡: 1. Denotes an identity, that is, an equality that is true whichever values are given to the variables occurring in it.; 2. In number theory, and more specifically in modular arithmetic, denotes the congruence modulo an integer.
: 1. May denote an isomorphism between two mathematical structures, and is read as «is isomorphic to».; 2. In geometry, may denote the congruence of two geometric shapes (that is the equality up to a displacement), and is read «is congruent to».

Comparison[edit]

<: 1. Strict inequality between two numbers; means and is read as «less than».; 2. Commonly used for denoting any strict order.; 3. Between two groups, may mean that the first one is a proper subgroup of the second one.
>: 1. Strict inequality between two numbers; means and is read as «greater than».; 2. Commonly used for denoting any strict order.; 3. Between two groups, may mean that the second one is a proper subgroup of the first one.
≤: 1. Means «less than or equal to». That is, whatever A and B are, A ≤ B is equivalent to A < B or A = B.; 2. Between two groups, may mean that the first one is a subgroup of the second one.
≥: 1. Means «greater than or equal to». That is, whatever A and B are, A ≥ B is equivalent to A > B or A = B.; 2. Between two groups, may mean that the second one is a subgroup of the first one.
≪ , ≫: 1. Means «much less than» and «much greater than». Generally, much is not formally defined, but means that the lesser quantity can be neglected with respect to the other. This is generally the case when the lesser quantity is smaller than the other by one or several orders of magnitude.; 2. In measure theory, means that the measure is absolutely continuous with respect to the measure .
≦: 1. A rarely used symbol, generally used as a synonym of ≤.
≺ , ≻: Often used for denoting an order or, more generally, a preorder, when it would be confusing or not convenient to use < and >.

Set theory[edit]

∅: Denotes the empty set, and is more often written . Using set-builder notation, it may also be denoted .
#: 1. Number of elements: may denote the cardinality of the set S. An alternative notation is ; see .; 2. Primorial: denotes the product of the prime numbers that are not greater than n.; 3. In topology, denotes the connected sum of two manifolds or two knots.
∈: Denotes set membership, and is read «in» or «belongs to». That is, means that x is an element of the set S.
∉: Means «not in». That is, means .
⊂: Denotes set inclusion. However two slightly different definitions are common.; 1. may mean that A is a subset of B, and is possibly equal to B; that is, every element of A belongs to B; in formula, .; 2. may mean that A is a proper subset of B, that is the two sets are different, and every element of A belongs to B; in formula, .
⊆: means that A is a subset of B. Used for emphasizing that equality is possible, or when the second definition of is used.
⊊: means that A is a proper subset of B. Used for emphasizing that , or when the first definition of is used.
⊃, ⊇, ⊋: Denote the converse relation of , , and respectively. For example, is equivalent to .
∪: Denotes set-theoretic union, that is, is the set formed by the elements of A and B together. That is, .
∩: Denotes set-theoretic intersection, that is, is the set formed by the elements of both A and B. That is, .
∖: Set difference; that is, is the set formed by the elements of A that are not in B. Sometimes, is used instead; see – in § Arithmetic operators.
⊖ or: Symmetric difference: that is, or is the set formed by the elements that belong to exactly one of the two sets A and B.
∁: 1. With a subscript, denotes a set complement: that is, if , then ${displaystyle complement _{A}B=Asetminus B}$ .; 2. Without a subscript, denotes the absolute complement; that is, ${displaystyle complement A=complement _{U}A}$ , where U is a set implicitly defined by the context, which contains all sets under consideration. This set U is sometimes called the universe of discourse.
×: See also × in § Arithmetic operators.; 1. Denotes the Cartesian product of two sets. That is, is the set formed by all pairs of an element of A and an element of B.; 2. Denotes the direct product of two mathematical structures of the same type, which is the Cartesian product of the underlying sets, equipped with a structure of the same type. For example, direct product of rings, direct product of topological spaces.; 3. In category theory, denotes the direct product (often called simply product) of two objects, which is a generalization of the preceding concepts of product.
⊔: Denotes the disjoint union. That is, if A and B are sets then ${displaystyle Asqcup B=left(Atimes {i_{A}}right)cup left(Btimes {i_{B}}right)}$ is a set of pairs where i_A and i_B are distinct indices discriminating the members of A and B in .
∐: 1. An alternative to .; 2. Denotes the coproduct of mathematical structures or of objects in a category.

Basic logic[edit]

Several logical symbols are widely used in all mathematics, and are listed here. For symbols that are used only in mathematical logic, or are rarely used, see List of logic symbols.

¬: Denotes logical negation, and is read as «not». If E is a logical predicate, is the predicate that evaluates to true if and only if E evaluates to false. For clarity, it is often replaced by the word «not». In programming languages and some mathematical texts, it is sometimes replaced by «~» or «!«, which are easier to type on some keyboards.
∨: 1. Denotes the logical or, and is read as «or». If E and F are logical predicates, is true if either E, F, or both are true. It is often replaced by the word «or».; 2. In lattice theory, denotes the join or least upper bound operation.; 3. In topology, denotes the wedge sum of two pointed spaces.
∧: 1. Denotes the logical and, and is read as «and». If E and F are logical predicates, is true if E and F are both true. It is often replaced by the word «and» or the symbol «&«.; 2. In lattice theory, denotes the meet or greatest lower bound operation.; 3. In multilinear algebra, geometry, and multivariable calculus, denotes the wedge product or the exterior product.
⊻: Exclusive or: if E and F are two Boolean variables or predicates, denotes the exclusive or. Notations E XOR F and are also commonly used; see ⊕.
∀: 1. Denotes universal quantification and is read as «for all». If E is a logical predicate, means that E is true for all possible values of the variable x.; 2. Often used improperly^[3] in plain text as an abbreviation of «for all» or «for every».
∃: 1. Denotes existential quantification and is read «there exists … such that». If E is a logical predicate, means that there exists at least one value of x for which E is true.; 2. Often used improperly^[3] in plain text as an abbreviation of «there exists».
∃!: Denotes uniqueness quantification, that is, means «there exists exactly one x such that P (is true)». In other words,
is an abbreviation of .
⇒: 1. Denotes material conditional, and is read as «implies». If P and Q are logical predicates, means that if P is true, then Q is also true. Thus, is logically equivalent with .; 2. Often used improperly^[3] in plain text as an abbreviation of «implies».
⇔: 1. Denotes logical equivalence, and is read «is equivalent to» or «if and only if». If P and Q are logical predicates, is thus an abbreviation of , or of .; 2. Often used improperly^[3] in plain text as an abbreviation of «if and only if».
⊤: 1. denotes the logical predicate always true.; 2. Denotes also the truth value true.; 3. Sometimes denotes the top element of a bounded lattice (previous meanings are specific examples).; 4. For the use as a superscript, see □^⊤.
⊥: 1. denotes the logical predicate always false.; 2. Denotes also the truth value false.; 3. Sometimes denotes the bottom element of a bounded lattice (previous meanings are specific examples).; 4. In Cryptography often denotes an error in place of a regular value.; 5. For the use as a superscript, see □^⊥.; 6. For the similar symbol, see .

Blackboard bold[edit]

The blackboard bold typeface is widely used for denoting the basic number systems. These systems are often also denoted by the corresponding uppercase bold letter. A clear advantage of blackboard bold is that these symbols cannot be confused with anything else. This allows using them in any area of mathematics, without having to recall their definition. For example, if one encounters in combinatorics, one should immediately know that this denotes the real numbers, although combinatorics does not study the real numbers (but it uses them for many proofs).

: Denotes the set of natural numbers , or sometimes . It is often denoted also by . When the distinction is important and readers might assume either definition, $mathbb {N} _{1}$ and $mathbb {N} _{0}$ are used, respectively, to denote one of them unambiguously.
: Denotes the set of integers . It is often denoted also by .
$mathbb {Z} _{p}$: 1. Denotes the set of p-adic integers, where p is a prime number.; 2. Sometimes, ${displaystyle mathbb {Z} _{n}}$ denotes the integers modulo n, where n is an integer greater than 0. The notation is also used, and is less ambiguous.
: Denotes the set of rational numbers (fractions of two integers). It is often denoted also by .
$mathbb {Q} _{p}$: Denotes the set of p-adic numbers, where p is a prime number.
: Denotes the set of real numbers. It is often denoted also by .
: Denotes the set of complex numbers. It is often denoted also by .
: Denotes the set of quaternions. It is often denoted also by .
$mathbb {F} _{q}$: Denotes the finite field with q elements, where q is a prime power (including prime numbers). It is denoted also by GF(q).
: Used on rare occasions to denote the set of octonions. It is often denoted also by .

Calculus[edit]

□‘: Lagrange’s notation for the derivative: If f is a function of a single variable, , read as «f prime», is the derivative of f with respect to this variable. The second derivative is the derivative of , and is denoted .
: Newton’s notation, most commonly used for the derivative with respect to time: If x is a variable depending on time, then is its derivative with respect to time. In particular, if x represents a moving point, then is its velocity.
: Newton’s notation, for the second derivative: If x is a variable that represents a moving point, then is its acceleration.
d □/d □: Leibniz’s notation for the derivative, which is used in several slightly different ways.; 1. If y is a variable that depends on x, then ${displaystyle textstyle {frac {mathrm {d} y}{mathrm {d} x}}}$ , read as «d y over d x», is the derivative of y with respect to x.; 2. If f is a function of a single variable x, then ${displaystyle textstyle {frac {mathrm {d} f}{mathrm {d} x}}}$ is the derivative of f, and
${displaystyle textstyle {frac {mathrm {d} f}{mathrm {d} x}}(a)}$ is the value of the derivative at a.; 3. Total derivative: If ${displaystyle f(x_{1},ldots ,x_{n})}$ is a function of several variables that depend on x, then ${displaystyle textstyle {frac {mathrm {d} f}{mathrm {d} x}}}$ is the derivative of f considered as a function of x. That is, ${displaystyle textstyle {frac {mathrm {d} f}{dx}}=sum _{i=1}^{n}{frac {partial f}{partial x_{i}}},{frac {mathrm {d} x_{i}}{mathrm {d} x}}}$ .
∂ □/∂ □: Partial derivative: If ${displaystyle f(x_{1},ldots ,x_{n})}$ is a function of several variables, ${displaystyle textstyle {frac {partial f}{partial x_{i}}}}$ is the derivative with respect to the ith variable considered as an independent variable, the other variables being considered as constants.
𝛿 □/𝛿 □: Functional derivative: If ${displaystyle f(y_{1},ldots ,y_{n})}$ is a functional of several functions, ${displaystyle textstyle {frac {delta f}{delta y_{i}}}}$ is the functional derivative with respect to the nth function considered as an independent variable, the other functions being considered constant.
: 1. Complex conjugate: If z is a complex number, then is its complex conjugate. For example, .; 2. Topological closure: If S is a subset of a topological space T, then is its topological closure, that is, the smallest closed subset of T that contains S.; 3. Algebraic closure: If F is a field, then is its algebraic closure, that is, the smallest algebraically closed field that contains F. For example, is the field of all algebraic numbers.; 4. Mean value: If x is a variable that takes its values in some sequence of numbers S, then may denote the mean of the elements of S.
→: 1. denotes a function with domain A and codomain B. For naming such a function, one writes , which is read as «f from A to B«.; 2. More generally, denotes a homomorphism or a morphism from A to B.; 3. May denote a logical implication. For the material implication that is widely used in mathematics reasoning, it is nowadays generally replaced by ⇒. In mathematical logic, it remains used for denoting implication, but its exact meaning depends on the specific theory that is studied.; 4. Over a variable name, means that the variable represents a vector, in a context where ordinary variables represent scalars; for example, . Boldface () or a circumflex () are often used for the same purpose.; 5. In Euclidean geometry and more generally in affine geometry, denotes the vector defined by the two points P and Q, which can be identified with the translation that maps P to Q. The same vector can be denoted also ; see Affine space.
↦: Used for defining a function without having to name it. For example, is the square function.
○^[4]: 1. Function composition: If f and g are two functions, then is the function such that for every value of x.; 2. Hadamard product of matrices: If A and B are two matrices of the same size, then is the matrix such that ${displaystyle (Acirc B)_{i,j}=(A)_{i,j}(B)_{i,j}}$ . Possibly, is also used instead of ⊙ for the Hadamard product of power series.^{[citation needed]}
∂: 1. Boundary of a topological subspace: If S is a subspace of a topological space, then its boundary, denoted , is the set difference between the closure and the interior of S.; 2. Partial derivative: see ∂□/∂□.
∫: 1. Without a subscript, denotes an antiderivative. For example, ${displaystyle textstyle int x^{2}dx={frac {x^{3}}{3}}+C}$ .; 2. With a subscript and a superscript, or expressions placed below and above it, denotes a definite integral. For example, ${displaystyle textstyle int _{a}^{b}x^{2}dx={frac {b^{3}-a^{3}}{3}}}$ .; 3. With a subscript that denotes a curve, denotes a line integral. For example, ${displaystyle textstyle int _{C}f=int _{a}^{b}f(r(t))r'(t)operatorname {d} t}$ , if r is a parametrization of the curve C, from a to b.
∮: Often used, typically in physics, instead of for line integrals over a closed curve.
∬, ∯: Similar to and for surface integrals.
or: Nabla, the gradient or vector derivative operator ${displaystyle textstyle left({frac {partial }{partial x}},{frac {partial }{partial y}},{frac {partial }{partial z}}right)}$ , also called del or grad.
∇² or ∇⋅∇: Laplace operator or Laplacian: ${displaystyle textstyle {frac {partial ^{2}}{partial x^{2}}}+{frac {partial ^{2}}{partial y^{2}}}+{frac {partial ^{2}}{partial z^{2}}}}$ . The forms $nabla ^{2}$ and represent the dot product of the gradient ( or ) with itself. Also notated Δ (next item).
Δ: 1. Another notation for the Laplacian (see above).; 2. Operator of finite difference.
or $partial _{mu }$: Quad, the 4-vector gradient operator or four-gradient, ${displaystyle textstyle left({frac {partial }{partial t}},{frac {partial }{partial x}},{frac {partial }{partial y}},{frac {partial }{partial z}}right)}$ .
or ${displaystyle {Box }^{2}}$: Denotes the d’Alembertian or squared four-gradient, which is a generalization of the Laplacian to four-dimensional spacetime. In flat spacetime with Euclidean coordinates, this may mean either ${displaystyle ~textstyle -{frac {partial ^{2}}{partial t^{2}}}+{frac {partial ^{2}}{partial x^{2}}}+{frac {partial ^{2}}{partial y^{2}}}+{frac {partial ^{2}}{partial z^{2}}}~;}$ or ${displaystyle ;~textstyle +{frac {partial ^{2}}{partial t^{2}}}-{frac {partial ^{2}}{partial x^{2}}}-{frac {partial ^{2}}{partial y^{2}}}-{frac {partial ^{2}}{partial z^{2}}}~;}$ ; the sign convention must be specified. In curved spacetime (or flat spacetime with non-Euclidean coordinates), the definition is more complicated. Also called box or quabla.

Linear and multilinear algebra[edit]

∑ (Sigma notation): 1. Denotes the sum of a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in ${displaystyle textstyle sum _{i=1}^{n}i^{2}}$ or ${displaystyle textstyle sum _{0<i<j<n}j-i}$ .; 2. Denotes a series and, if the series is convergent, the sum of the series. For example, ${displaystyle textstyle sum _{i=0}^{infty }{frac {x^{i}}{i!}}=e^{x}}$ .
∏ (Capital-pi notation): 1. Denotes the product of a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in ${displaystyle textstyle prod _{i=1}^{n}i^{2}}$ or ${displaystyle textstyle prod _{0<i<j<n}j-i}$ .; 2. Denotes an infinite product. For example, the Euler product formula for the Riemann zeta function is ${displaystyle textstyle zeta (z)=prod _{n=1}^{infty }{frac {1}{1-p_{n}^{-z}}}}$ .; 3. Also used for the Cartesian product of any number of sets and the direct product of any number of mathematical structures.
⊕: 1. Internal direct sum: if E and F are abelian subgroups of an abelian group V, notation means that V is the direct sum of E and F; that is, every element of V can be written in a unique way as the sum of an element of E and an element of F. This applies also when E and F are linear subspaces or submodules of the vector space or module V.; 2. Direct sum: if E and F are two abelian groups, vector spaces, or modules, then their direct sum, denoted is an abelian group, vector space, or module (respectively) equipped with two monomorphisms and such that is the internal direct sum of and . This definition makes sense because this direct sum is unique up to a unique isomorphism.; 3. Exclusive or: if E and F are two Boolean variables or predicates, may denote the exclusive or. Notations E XOR F and are also commonly used; see ⊻.
⊗: Denotes the tensor product. If E and F are abelian groups, vector spaces, or modules over a commutative ring, then the tensor product of E and F, denoted is an abelian group, a vector space or a module (respectively), equipped with a bilinear map from to , such that the bilinear maps from to any abelian group, vector space or module G can be identified with the linear maps from to G. If E and F are vector spaces over a field R, or modules over a ring R, the tensor product is often denoted ${displaystyle Eotimes _{R}F}$ to avoid ambiguity.
□^⊤: 1. Transpose: if A is a matrix, $A^{top }$ denotes the transpose of A, that is, the matrix obtained by exchanging rows and columns of A. Notation ${displaystyle ^{top }!!A}$ is also used. The symbol is often replaced by the letter T or t.; 2. For inline uses of the symbol, see ⊤.
□^⊥: 1. Orthogonal complement: If W is a linear subspace of an inner product space V, then $W^{bot }$ denotes its orthogonal complement, that is, the linear space of the elements of V whose inner products with the elements of W are all zero.; 2. Orthogonal subspace in the dual space: If W is a linear subspace (or a submodule) of a vector space (or of a module) V, then $W^{bot }$ may denote the orthogonal subspace of W, that is, the set of all linear forms that map W to zero.; 3. For inline uses of the symbol, see ⊥.

Advanced group theory[edit]

⋉ ⋊: 1. Inner semidirect product: if N and H are subgroups of a group G, such that N is a normal subgroup of G, then and mean that G is the semidirect product of N and H, that is, that every element of G can be uniquely decomposed as the product of an element of N and an element of H. (Unlike for the direct product of groups, the element of H may change if the order of the factors is changed.); 2. Outer semidirect product: if N and H are two groups, and is a group homomorphism from N to the automorphism group of H, then ${displaystyle Nrtimes _{varphi }H=Hltimes _{varphi }N}$ denotes a group G, unique up to a group isomorphism, which is a semidirect product of N and H, with the commutation of elements of N and H defined by .
≀: In group theory, denotes the wreath product of the groups G and H. It is also denoted as or ; see Wreath product § Notation and conventions for several notation variants.

Infinite numbers[edit]

∞: 1. The symbol is read as infinity. As an upper bound of a summation, an infinite product, an integral, etc., means that the computation is unlimited. Similarly, in a lower bound means that the computation is not limited toward negative values.; 2. and are the generalized numbers that are added to the real line to form the extended real line.; 3. is the generalized number that is added to the real line to form the projectively extended real line.
𝔠: denotes the cardinality of the continuum, which is the cardinality of the set of real numbers.
ℵ: With an ordinal i as a subscript, denotes the ith aleph number, that is the ith infinite cardinal. For example, $aleph _{0}$ is the smallest infinite cardinal, that is, the cardinal of the natural numbers.
ℶ: With an ordinal i as a subscript, denotes the ith beth number. For example, $beth _{0}$ is the cardinal of the natural numbers, and $beth _{1}$ is the cardinal of the continuum.
ω: 1. Denotes the first limit ordinal. It is also denoted $omega _{0}$ and can be identified with the ordered set of the natural numbers.; 2. With an ordinal i as a subscript, denotes the ith limit ordinal that has a cardinality greater than that of all preceding ordinals.; 3. In computer science, denotes the (unknown) greatest lower bound for the exponent of the computational complexity of matrix multiplication.; 4. Written as a function of another function, it is used for comparing the asymptotic growth of two functions. See Big O notation § Related asymptotic notations.; 5. In number theory, may denote the prime omega function. That is, is the number of distinct prime factors of the integer n.

Brackets[edit]

Many sorts of brackets are used in mathematics. Their meanings depend not only on their shapes, but also on the nature and the arrangement of what is delimited by them, and sometimes what appears between or before them. For this reason, in the entry titles, the symbol □ is used as a placeholder for schematizing the syntax that underlies the meaning.

Parentheses[edit]

(□): Used in an expression for specifying that the sub-expression between the parentheses has to be considered as a single entity; typically used for specifying the order of operations.
□(□) □(□, □) □(□, …, □): 1. Functional notation: if the first is the name (symbol) of a function, denotes the value of the function applied to the expression between the parentheses; for example, , . In the case of a multivariate function, the parentheses contain several expressions separated by commas, such as .; 2. May also denote a product, such as in . When the confusion is possible, the context must distinguish which symbols denote functions, and which ones denote variables.
(□, □): 1. Denotes an ordered pair of mathematical objects, for example, .; 2. If a and b are real numbers, , or , and a < b, then denotes the open interval delimited by a and b. See ]□, □[ for an alternative notation.; 3. If a and b are integers, may denote the greatest common divisor of a and b. Notation is often used instead.
(□, □, □): If x, y, z are vectors in $mathbb {R} ^{3}$ , then may denote the scalar triple product.^{[citation needed]} See also [□,□,□] in § Square brackets.
(□, …, □): Denotes a tuple. If there are n objects separated by commas, it is an n-tuple.
(□, □, …) (□, …, □, …): Denotes an infinite sequence.
${displaystyle {begin{pmatrix}Box &cdots &Box \vdots &ddots &vdots \Box &cdots &Box end{pmatrix}}}$: Denotes a matrix. Often denoted with square brackets.
: Denotes a binomial coefficient: Given two nonnegative integers, is read as «n choose k«, and is defined as the integer ${displaystyle {frac {n(n-1)cdots (n-k+1)}{1cdot 2cdots k}}={frac {n!}{k!,(n-k)!}}}$ (if k = 0, its value is conventionally 1). Using the left-hand-side expression, it denotes a polynomial in n, and is thus defined and used for any real or complex value of n.
(□/□): Legendre symbol: If p is an odd prime number and a is an integer, the value of $left({frac {a}{p}}right)$ is 1 if a is a quadratic residue modulo p; it is –1 if a is a quadratic non-residue modulo p; it is 0 if p divides a. The same notation is used for the Jacobi symbol and Kronecker symbol, which are generalizations where p is respectively any odd positive integer, or any integer.

Square brackets[edit]

[□]: 1. Sometimes used as a synonym of (□) for avoiding nested parentheses.; 2. Equivalence class: given an equivalence relation, often denotes the equivalence class of the element x.; 3. Integral part: if x is a real number, often denotes the integral part or truncation of x, that is, the integer obtained by removing all digits after the decimal mark. This notation has also been used for other variants of floor and ceiling functions.; 4. Iverson bracket: if P is a predicate, may denote the Iverson bracket, that is the function that takes the value 1 for the values of the free variables in P for which P is true, and takes the value 0 otherwise. For example, is the Kronecker delta function, which equals one if , and zero otherwise.
□[□]: Image of a subset: if S is a subset of the domain of the function f, then is sometimes used for denoting the image of S. When no confusion is possible, notation f(S) is commonly used.
[□, □]: 1. Closed interval: if a and b are real numbers such that , then denotes the closed interval defined by them.; 2. Commutator (group theory): if a and b belong to a group, then ${displaystyle [a,b]=a^{-1}b^{-1}ab}$ .; 3. Commutator (ring theory): if a and b belong to a ring, then .; 4. Denotes the Lie bracket, the operation of a Lie algebra.
[□ : □]: 1. Degree of a field extension: if F is an extension of a field E, then denotes the degree of the field extension . For example, .; 2. Index of a subgroup: if H is a subgroup of a group E, then denotes the index of H in G. The notation |G:H| is also used
[□, □, □]: If x, y, z are vectors in $mathbb {R} ^{3}$ , then may denote the scalar triple product.^[5] See also (□,□,□) in § Parentheses.
${displaystyle {begin{bmatrix}Box &cdots &Box \vdots &ddots &vdots \Box &cdots &Box end{bmatrix}}}$: Denotes a matrix. Often denoted with parentheses.

Braces[edit]

{ }: Set-builder notation for the empty set, also denoted or ∅.
{□}: 1. Sometimes used as a synonym of (□) and [□] for avoiding nested parentheses.; 2. Set-builder notation for a singleton set: denotes the set that has x as a single element.
{□, …, □}: Set-builder notation: denotes the set whose elements are listed between the braces, separated by commas.
{□ : □} {□ | □}: Set-builder notation: if is a predicate depending on a variable x, then both and denote the set formed by the values of x for which is true.
Single brace: 1. Used for emphasizing that several equations have to be considered as simultaneous equations; for example, ${displaystyle textstyle {begin{cases}2x+y=1\3x-y=1end{cases}}}$ .; 2. Piecewise definition; for example, ${displaystyle textstyle |x|={begin{cases}x&{text{if }}xgeq 0\-x&{text{if }}x<0end{cases}}}$ .; 3. Used for grouped annotation of elements in a formula; for example, ${displaystyle textstyle underbrace {(a,b,ldots ,z)} _{26}}$ , ${displaystyle textstyle overbrace {1+2+cdots +100} ^{=5050}}$ , ${displaystyle textstyle left.{begin{bmatrix}A\Bend{bmatrix}}right}m+n{text{ rows}}}$

Other brackets[edit]

|□|

1. Absolute value: if x is a real or complex number, |x|

denotes its absolute value.

2. Number of elements: If S is a set, |x|

may denote its cardinality, that is, its number of elements.

is also often used, see #.

3. Length of a line segment: If P and Q are two points in a Euclidean space, then

often denotes the length of the line segment that they define, which is the distance from P to Q, and is often denoted

4. For a similar-looking operator, see |.

|□:□|

Index of a subgroup: if H is a subgroup of a group G, then

denotes the index of H in G. The notation [G:H] is also used

${displaystyle textstyle {begin{vmatrix}Box &cdots &Box \vdots &ddots &vdots \Box &cdots &Box end{vmatrix}}}$

${displaystyle {begin{vmatrix}x_{1,1}&cdots &x_{1,n}\vdots &ddots &vdots \x_{n,1}&cdots &x_{n,n}end{vmatrix}}}$

denotes the determinant of the square matrix ${displaystyle {begin{bmatrix}x_{1,1}&cdots &x_{1,n}\vdots &ddots &vdots \x_{n,1}&cdots &x_{n,n}end{bmatrix}}}$

||□||

1. Denotes the norm of an element of a normed vector space.

2. For the similar-looking operator named parallel, see ∥.

⌊□⌋

Floor function: if x is a real number,

is the greatest integer that is not greater than x.

⌈□⌉

Ceiling function: if x is a real number,

is the lowest integer that is not lesser than x.

⌊□⌉

Nearest integer function: if x is a real number,

is the integer that is the closest to x.

]□, □[

Open interval: If a and b are real numbers, -infty

, or

, and

, then

denotes the open interval delimited by a and b. See (□, □) for an alternative notation.

(□, □] ]□, □]

Both notations are used for a left-open interval.

[□, □) [□, □[

Both notations are used for a right-open interval.

⟨□⟩

1. Generated object: if S is a set of elements in an algebraic structure,

denotes often the object generated by S. If ${displaystyle S={s_{1},ldots ,s_{n}}}$

, one writes ${displaystyle langle s_{1},ldots ,s_{n}rangle }$

(that is, braces are omitted). In particular, this may denote

the linear span in a vector space (also often denoted Span(S)),
the generated subgroup in a group,
the generated ideal in a ring,
the generated submodule in a module.

2. Often used, mainly in physics, for denoting an expected value. In probability theory, E(X)

is generally used instead of

⟨□, □⟩ ⟨□ | □⟩

Both

and

are commonly used for denoting the inner product in an inner product space.

⟨□| and |□⟩

Bra–ket notation or Dirac notation: if x and y are elements of an inner product space,

is the vector defined by x, and

is the covector defined by y; their inner product is

Symbols that do not belong to formulas[edit]

In this section, the symbols that are listed are used as some sorts of punctuation marks in mathematical reasoning, or as abbreviations of English phrases. They are generally not used inside a formula. Some were used in classical logic for indicating the logical dependence between sentences written in plain English. Except for the first two, they are normally not used in printed mathematical texts since, for readability, it is generally recommended to have at least one word between two formulas. However, they are still used on a black board for indicating relationships between formulas.

■ , □: Used for marking the end of a proof and separating it from the current text. The initialism Q.E.D. or QED (Latin: quod erat demonstrandum, «as was to be shown») is often used for the same purpose, either in its upper-case form or in lower case.
☡: Bourbaki dangerous bend symbol: Sometimes used in the margin to forewarn readers against serious errors, where they risk falling, or to mark a passage that is tricky on a first reading because of an especially subtle argument.
∴: Abbreviation of «therefore». Placed between two assertions, it means that the first one implies the second one. For example: «All humans are mortal, and Socrates is a human. ∴ Socrates is mortal.»
∵: Abbreviation of «because» or «since». Placed between two assertions, it means that the first one is implied by the second one. For example: «11 is prime ∵ it has no positive integer factors other than itself and one.»
∋: 1. Abbreviation of «such that». For example, is normally printed «x such that «.; 2. Sometimes used for reversing the operands of ; that is, has the same meaning as . See ∈ in § Set theory.
∝: Abbreviation of «is proportional to».

Miscellaneous[edit]

!: 1. Factorial: if n is a positive integer, n! is the product of the first n positive integers, and is read as «n factorial».; 2. Subfactorial: if n is a positive integer, !n is the number of derangements of a set of n elements, and is read as «the subfactorial of n».
*: Many different uses in mathematics; see Asterisk § Mathematics.
|: 1. Divisibility: if m and n are two integers, means that m divides n evenly.; 2. In set-builder notation, it is used as a separator meaning «such that»; see {□ | □}.; 3. Restriction of a function: if f is a function, and S is a subset of its domain, then ${displaystyle f|_{S}}$ is the function with S as a domain that equals f on S.; 4. Conditional probability: denotes the probability of X given that the event E occurs. Also denoted ; see «/».; 5. For several uses as brackets (in pairs or with ⟨ and ⟩) see § Other brackets.
∤: Non-divisibility: means that n is not a divisor of m.
∥: 1. Denotes parallelism in elementary geometry: if PQ and RS are two lines, means that they are parallel.; 2. Parallel, an arithmetical operation used in electrical engineering for modeling parallel resistors: ${displaystyle xparallel y={frac {1}{{frac {1}{x}}+{frac {1}{y}}}}}$ .; 3. Used in pairs as brackets, denotes a norm; see ||□||.
∦: Sometimes used for denoting that two lines are not parallel; for example, .
: 1. Denotes perpendicularity and orthogonality. For example, if A, B, C are three points in a Euclidean space, then means that the line segments AB and AC are perpendicular, and form a right angle.; 2. For the similar symbol, see .
⊙: Hadamard product of power series: if ${displaystyle textstyle S=sum _{i=0}^{infty }s_{i}x^{i}}$ and ${displaystyle textstyle T=sum _{i=0}^{infty }t_{i}x^{i}}$ , then ${displaystyle textstyle Sodot T=sum _{i=0}^{infty }s_{i}t_{i}x^{i}}$ . Possibly, is also used instead of ○ for the Hadamard product of matrices.^{[citation needed]}

References[edit]

^ ISO 80000-2, Section 9 «Operations», 2-9.6
^ «Statistics and Data Analysis: From Elementary to Intermediate».
^ ^a ^b ^c ^d Letourneau, Mary; Wright Sharp, Jennifer (2017). «AMS style guide» (PDF). American Mathematical Society. p. 99.
^ The LaTeX equivalent to both Unicode symbols ∘ and ○ is circ. The Unicode symbol that has the same size as circ depends on the browser and its implementation. In some cases ∘ is so small that it can be confused with an interpoint, and ○ looks similar as circ. In other cases, ○ is too large for denoting a binary operation, and it is ∘ that looks like circ. As LaTeX is commonly considered as the standard for mathematical typography, and it does not distinguish these two Unicode symbols, they are considered here as having the same mathematical meaning.
^ Rutherford, D. E. (1965). Vector Methods. University Mathematical Texts. Oliver and Boyd Ltd., Edinburgh.

External links[edit]

Jeff Miller: Earliest Uses of Various Mathematical Symbols
Numericana: Scientific Symbols and Icons
GIF and PNG Images for Math Symbols
Mathematical Symbols in Unicode
Detexify: LaTeX Handwriting Recognition Tool

Some Unicode charts of mathematical operators and symbols:

Index of Unicode symbols
Range 2100–214F: Unicode Letterlike Symbols
Range 2190–21FF: Unicode Arrows
Range 2200–22FF: Unicode Mathematical Operators
Range 27C0–27EF: Unicode Miscellaneous Mathematical Symbols–A
Range 2980–29FF: Unicode Miscellaneous Mathematical Symbols–B
Range 2A00–2AFF: Unicode Supplementary Mathematical Operators

Some Unicode cross-references:

Short list of commonly used LaTeX symbols and Comprehensive LaTeX Symbol List
MathML Characters — sorts out Unicode, HTML and MathML/TeX names on one page
Unicode values and MathML names
Unicode values and Postscript names from the source code for Ghostscript

Источник

It’s time to review some terms you hear in math lectures. If you are not doing well in math, it’s probably because of miscommunications and not for any other reason.

BRIEFLY: I’m running out of time, so I’ll just write and talk faster.

BRUTE FORCE: Four special cases, three counting arguments, two long inductions, and a partridge in a pair tree.

BY A PREVIOUS THEOREM: I don’t remember how it goes (come to think of it, I’m not really sure we did this at all), but if I stated it right, then the rest of this follows.

CANONICAL FORM: 4 out of 5 mathematicians surveyed recommended this as the final form for the answer.

CHECK FOR YOURSELF: This is the boring part of the proof, so you can do it on your own time.

CLEARLY: I don’t want to write down all the in-between steps.

ELEGANT PROOF: Requires no previous knowledge of the subject, and is less than ten lines long.

FINALLY: Only ten more steps to go…

HINT: The hardest of several possible ways to do a proof.

IT IS WELL KNOWN: See “Mathematische Zeitschrift”, vol XXXVI, 1892.

LET’S TALK THROUGH IT: I don’t want to write it on the board because I’ll make a mistake.

OBVIOUSLY: I hope you weren’t sleeping when we discussed this earlier, because I refuse to repeat it.

ONE MAY SHOW: One did, his name was Gauss.

PROCEED FORMALLY: Manipulate symbols by the rules without any hint of their true meaning.

PROOF OMITTED: Trust me, it’s true.

Q.E.D. : T.G.I.F.

QUANTIFY: I can’t find anything wrong with your proof except that it won’t work if x is 0.

RECALL: I shouldn’t have to tell you this, but for those of you who erase your memory after every test, here it is again.

SIMILARLY: At least one line of the proof of this case is the same as before.

SKETCH OF A PROOF: I couldn’t verify the details, so I’ll break it down into parts I couldn’t prove.

SOFT PROOF: One third less filling (of the page) than your regular proof, but it requires two extra years of course work just to understand the terms.

THE FOLLOWING ARE EQUIVALENT: If I say this it means that, and if I say that it means the other thing, and if I say the other thing…

TRIVIAL: If I have to show you how to do this, you’re in the wrong class.

TWO LINE PROOF: I’ll leave out everything but the conclusion.

WITHOUT LOSS OF GENERALITY (WLOG): I’m not about to do all the possible cases, so I’ll do one and let you figure out the rest.

Please use the comment form to share your own commonly used words in your math lectures.

Source:

Источник

Origin of math

Words nearby math

Other definitions for math (2 of 4)

Origin of math

Other definitions for math (3 of 4)

Origin of math

Other definitions for math (4 of 4)

Words related to math

How to use math in a sentence

British Dictionary definitions for math (1 of 2)

British Dictionary definitions for math (2 of 2)

Layout of this article[edit]

Arithmetic operators[edit]

Equality, equivalence and similarity[edit]

Comparison[edit]

Set theory[edit]

Basic logic[edit]

Blackboard bold[edit]

Calculus[edit]

Linear and multilinear algebra[edit]

Advanced group theory[edit]

Infinite numbers[edit]

Brackets[edit]

Parentheses[edit]

Square brackets[edit]

Braces[edit]

Other brackets[edit]

Symbols that do not belong to formulas[edit]

Miscellaneous[edit]

See also[edit]

Related articles[edit]

[edit]

Unicode symbols[edit]

References[edit]

External links[edit]