Roman letters used in mathematics

Monday, 24 December 2012

Many Roman letters, both capital and small, are used in mathematics, science and engineering to denote by convention specific or abstracted constants, variables of a certain type, units, multipliers, physical entities. Certain letters, when combined with special formatting, take on special meaning.

Below is an alphabetical list of the letters of the alphabet with some of their uses. The field in which the convention applies is mathematics unless otherwise noted.

Aa

• A represents:
  • the first corner of a triangle
  • the digit "10" in hexadecimal and other positional numeral systems with a radix of 11 or greater
  • the unit ampere for electrical current
  • area
• represents the algebraic numbers or affine space in Algebraic Geometry
• a represents:
  • the first side of a triangle (opposite corner A)
  • the acceleration in mechanics equations
  • the x-intercept of a line
  • the unit are for area (100 m²)
  • the unit prefix atto (10⁻¹⁸)
  • the first term in a sequence or series (eg. S_n = n(a+l)/2)

Bb

• B represents:
  • the digit "11" in hexadecimal and other positional numeral systems with a radix of 12 or greater
  • the second corner of a triangle
  • a ball (also denoted by or )
  • a basis of a vector space or of a filter (both also denoted by )
• B with various subscripts represents several variations of Brun's constant and Betti numbers
• b represents:
  • the second side of a triangle (opposite corner B)
  • the y-intercept of a line
  • (usually with an index, sometimes with an arrow over it) a basis vector

 Cc

• C represents:
  • the third corner of a triangle
  • the digit "12" in hexadecimal and other positional numeral systems with a radix of 13 or greater
  • the unit coulomb of electrical charge
• C with indices denotes the number of combinations, a binomial coefficient
• represents the set of complex numbers
• A vertically elongated C with an integer subscript n sometimes denotes the n-th coefficient of a formal power series.
• c represents:
  • the third side of a triangle (opposite corner C)
  • the unit prefix centi (10⁻²)
• c represents:
  • the speed of light in vacuum
• Small bold C denotes the cardinality of the set of real numbers (the "continuum"), or,

equivalently, of the power set of natural numbers

 Dd

• D represents the digit "13" in hexadecimal and other positional numeral systems with a radix of 14 or greater
• d represents
  • the differential operator
  • the unit day of time (86 400 s)
  • the difference in an arithmetic sequence (eg. S_n = n(2a+(n-1)d)/2)

 Ee

• E represents:
  • the digit "14" in hexadecimal and other positional numeral systems with a radix of 15 or greater
  • an exponent in decimal numbers 1.2E3 is 1.2×10³ or 1200
  • the set of edges in a graph or matroid
  • the unit prefix exa (10¹⁸)
  • Energy in physics
• e represents:
  • Euler's number, a transcendental number equal to 2.71828182845… which is used as the base for natural logarithms
  • a vector of unit length, especially in the direction of one of the coordinates axes
  • the elementary charge in physics

Ff

• F represents
  • the digit "15" in hexadecimal and other positional numeral systems with a radix of 16 or greater
  • force in mechanics equations
  • the probability distribution function in statistics
  • the unit farad of electrical capacity
• f represents:
  • the generic designation of a function
  • the unit prefix femto (10⁻¹⁵)

 Gg

• G represents
  • an arbitrary graph, as in: G(V,E)
  • an arbitrary group
  • the unit prefix giga (10⁹)
  • Newton's gravitational constant
  • the Einstein tensor
• g represents:
  • the generic designation of a second function
  • the acceleration due to gravity on Earth

 Hh

• H represents:
  • a Hilbert space
  • the unit henry of magnetic inductance
  • the homology and cohomology functors
• h represents:
  • a small increment in the argument of a function
  • the unit hour for time (3600 s)
  • the unit prefix hecto (10²)
• represents the quaternions (after William Rowan Hamilton, representing the rationals)
• represents the Hamiltonian in Hamiltonian mechanics

Ii

• I represents:
  • the closed unit interval, which contains all real numbers from 0 to 1, inclusive
  • the identity matrix
• i represents:
  • the imaginary unit, a complex number that is the square root of −1
  • a subscript to denote the ith term (that is, a general term or index) in a sequence or list
  • the index to the elements of a vector, written as a subscript after the vector name
  • the index to the rows of a matrix, written as the first subscript after the matrix name
  • an index of summation using the sigma notation

Jj

• J represents the unit joule of energy
• j represents:
  • the index to the columns of a matrix, written as the second subscript after the matrix name
  • in electrical engineering, the square root of −1, instead of i
  • in electrical engineering, the principal cube root of 1:

 Kk

• K represents:
  • the unit kelvin of temperature
  • the functors of K-theory
  • an unspecified (real) constant
• k represents
  • the unit prefix kilo- (10³)
  • the Boltzmann constant
  • an integer, e.g. a dummy variable in summations, or an index of a matrix.
  • an unspecified (real) constant

 Ll

• L represents:
  • the unit litre of volume
  • the space of all integrable real (or complex) functions
  • the space of linear maps, as in L(E,F) or L(E) = End(E)
  • the Likelihood function
• l represents:
  • the length of a side of a rectangle or a rectangular prism (eg. V = lwh; A = lw)
  • the last term of a sequence or series (eg. S_n = n(a+l)/2)
• (or sometimes just L) represents the Lagrangian

 Mm

• M represents:
  • a manifold
  • a metric space
  • a matroid
  • the unit prefix mega (10⁶)
• m represents:
  • the number of rows in a matrix
  • the slope in a linear regression or in any line
  • the mass in mechanics equations
  • the unit metre of length
  • the unit prefix milli (10⁻³)

Nn

• N represents
  • the unit newton of force
• N_A represents the Avogadro constant which is the number of entities in one mole (used mainly in the counting of molecules and atoms)
• represents the natural numbers
• n represents
  • the number of columns in a matrix
  • the "number of" in algebraic equations.
  • the unit prefix nano (10⁻⁹)
  • the nth term of a sequence or series (eg. t_n = a+(n-1)d)

 Oo

• O represents the order of asymptotic behavior of a function; see Big O notation
• O represents — the origin of the coordinate system in Cartesian coordinates

 Pp

• P represents:
  • the pressure in physics equations
  • the unit prefix peta (10¹⁵)
• represents
  • the prime numbers
  • projective space
• p represents the unit prefix pico (10⁻¹²)

 Qq

• represents the rational numbers

 Rr

• R represents:
  • the Ricci tensor
• represents the set of real numbers and various algebraic structures built upon the set of real numbers, such as
• r represents:
  • the radius of a circle or sphere
  • the ratio of a geometric series (eg. ar^n-1)

 Ss

• S represents
  • a sum
  • the unit siemens of electric conductance
  • the unit sphere (with superscript denoting dimension)
  • the scattering matrix
• s represents:
  • an arclength
  • the distance traveled in mechanics equations
  • the unit second of time
• represents a system's action in physics

Tt

• T represents:
  • the top element of a lattice
  • a tree (a special kind of graph)
  • temperature in physics equations
  • the unit tesla of magnetic flux density
  • the unit prefix tera (10¹²)
  • the stress-energy tensor
• t represents:
  • time in graphs, functions or equations
  • a term in a sequence or series (eg. t_n = t_n-1+5)

 Uu

• U represents:
  • a U-set which is a set of uniqueness
  • a unitary operator
• U(n) represents the unitary group of degree n
• ∪ represents the union operator

 Vv

• V represents:
  • volume
  • the unit volt of voltage
  • the set of vertices in a graph
• v represents the velocity in mechanics equations

 Ww

• W represents the unit watt of power

 Xx

• x represents
  • an unknown variable, most often (but not always) from the set of real numbers, while a complex unknown would rather be called z, and an integer by a letter like m from the middle of the alphabet.
  • the coordinate on the first or horizontal axis in a cartesian coordinate system, the viewport in a graph or window in computer graphics

Yy

• Y represents:
  • the unit prefix yotta (10²⁴)
• y represents:
  • the unit prefix yocto (10⁻²⁴)
• y represents:
  • a second unknown variable
  • the coordinate on the second or vertical axis (backward axis in three dimensions) in a linear coordinate system, or in the viewport of a graph or window in computer graphics

 Zz

• Z represents:
  • the unit prefix zetta (10²¹)
  • a standarized normal random variable in Probability Theory and Statistics
• represents the integers
• z represents:
  • the unit prefix zepto (10⁻²¹)
  • the coordinate on the third or vertical axis in three dimensional space
  • the view depth in computer graphics, see also "z-buffering"
  • the argument of a complex function, or any other variable used to represent a complex value .

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Logic symbols

In logic, a set of symbols is commonly used to express logical representation. As logicians are familiar with these symbols, they are not explained each time they are used. So, for students of logic, the following table lists many common symbols together with their name, pronunciation and related field of mathematics. Additionally, the third column contains an informal definition, and the fourth column gives a short example.

Be aware that, outside of logic, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings.

 Basic logic symbols

Symbol	Name	Explanation	Examples	Unicode Value	HTML Entity	LaTeX symbol
	Should be read as
	Category
⇒ → ⊃	material implication	A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B. → may mean the same as ⇒ (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols). ⊃ may mean the same as ⇒ (the symbol may also mean superset).	x = 2  ⇒  x2 = 4 is true, but x2 = 4   ⇒  x = 2 is in general false (since x could be −2).	8658 8594 8835	⇒ → ⊃	\Rightarrow \rightarrow \supset
	implies; if .. then
	propositional logic, Heyting algebra
⇔ ≡ ↔	material equivalence	A ⇔ B means A is true if B is true and A is false if B is false.	x + 5 = y +2  ⇔  x + 3 = y	8660 8801 8596	⇔ ≡ ↔	\Leftrightarrow \equiv \leftrightarrow
	if and only if; iff
	propositional logic
¬ ˜	logical negation	The statement ¬A is true if and only if A is false. A slash placed through another operator is the same as "¬" placed in front.	¬(¬A) ⇔ A x ≠ y  ⇔  ¬(x =  y)	172 732	¬ ˜ ~	\lnot \tilde{}
	not
	propositional logic
∧ &	logical conjunction	The statement A ∧ B is true if A and B are both true; else it is false.	n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number.	8743 38	&and; &	\land \&[1]
	and
	propositional logic
∨	logical disjunction	The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.	n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number.	8744	&or;	\lor
	or
	propositional logic
⊕ ⊻	exclusive or	The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same.	(¬A) ⊕ A is always true, A ⊕ A is always false.	8853 8891	&oplus;	\oplus
	xor
	propositional logic, Boolean algebra
⊤ T 1	logical truth	The statement ⊤ is unconditionally true.	A ⇒ ⊤ is always true.	8868	T	\top
	top
	propositional logic, Boolean algebra
⊥ F 0	logical falsity	The statement ⊥ is unconditionally false.	⊥ ⇒ A is always true.	8869	&perp; F	\bot
	bottom
	propositional logic, Boolean algebra
∀	universal quantification	∀ x: P(x) means P(x) is true for all x.	∀ n ∈ N: n2 ≥ n.	8704	&forall;	\forall
	for all; for any; for each
	predicate logic
∃	existential quantification	∃ x: P(x) means there is at least one x such that P(x) is true.	∃ n ∈ N: n is even.	8707	&exist;	\exists
	there exists
	first-order logic
∃!	uniqueness quantification	∃! x: P(x) means there is exactly one x such that P(x) is true.	∃! n ∈ N: n + 5 = 2n.	8707 33	&exist; !	\exists !
	there exists exactly one
	first-order logic
:= ≡ :⇔	definition	x := y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence). P :⇔ Q means P is defined to be logically equivalent to Q.	cosh x := (1/2)(exp x + exp (−x)) A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B)	58 61 8801 58 8660	:= : &equiv; &hArr;	: = := \equiv \Leftrightarrow
	is defined as
	everywhere
( )	precedence grouping	Perform the operations inside the parentheses first.	(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.	40 41	( )	( )
	everywhere
⊢	inference	x ⊢ y means y is derived from x.	A → B ⊢ ¬B → ¬A	8866		\vdash
	infers or is derived from
	propositional logic, first-order logic

 See also

·      Table of mathematical symbols

·      Polish notation

 Special characters

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