Sides of an equation
Encyclopedia
In mathematics
, LHS is informal shorthand for the left-hand side of an equation
. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric
. This abbreviation is seldom if ever used in print; it is very informal.
More generally, these terms may apply to an inequation
or inequality. In the inequality case, there is no symmetry. The right-hand side is everything on the right side of a test operator in an expression
. Conversely, the left-hand side is everything on the left side.
Take
where "x + 5" would be the left-hand side and "y + 8" would be the right-hand side
s and integral equation
s, the terminology homogeneous is often used for equations with the RHS set equal to zero. The corresponding inhomogeneous or nonhomogeneous equation then has the RHS with some given data, but of a general character.
The typical case is of some operator L, with the difference being that between the equation
to be solved for a function f, and the equation
with g a fixed function, to solve again for f. The point of the terminology appears for L a linear operator. Then any solution of the inhomogeneous equation may have a solution of the homogeneous equation added to it, and still remain a solution.
For example in mathematical physics
, the homogeneous equation may correspond to a physical theory formulated in empty space
, while the inhomogeneous equation asks for more 'realistic' solutions with some matter, or charged particles.
the term T stands as the left-hand side and U as the right-hand side of the operator
*. This usage is less common, though.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, LHS is informal shorthand for the left-hand side of an equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
. This abbreviation is seldom if ever used in print; it is very informal.
More generally, these terms may apply to an inequation
Inequation
In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. This relation is written with a crossed-out equal sign as inx \neq y....
or inequality. In the inequality case, there is no symmetry. The right-hand side is everything on the right side of a test operator in an expression
Expression (mathematics)
In mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers , variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic...
. Conversely, the left-hand side is everything on the left side.
Some examples
The equation on the right side/right part of the "=" sign is the right side of the equation and the left of the "=" is the left side/left part of equation.Take
- x + 5 = y + 8
where "x + 5" would be the left-hand side and "y + 8" would be the right-hand side
Homogeneous and inhomogeneous equations
In solving mathematical equations, particularly linear simultaneous equations, differential equationDifferential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s and integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
s, the terminology homogeneous is often used for equations with the RHS set equal to zero. The corresponding inhomogeneous or nonhomogeneous equation then has the RHS with some given data, but of a general character.
The typical case is of some operator L, with the difference being that between the equation
- Lf = 0,
to be solved for a function f, and the equation
- Lf = g,
with g a fixed function, to solve again for f. The point of the terminology appears for L a linear operator. Then any solution of the inhomogeneous equation may have a solution of the homogeneous equation added to it, and still remain a solution.
For example in mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, the homogeneous equation may correspond to a physical theory formulated in empty space
Empty space
Empty space may refer to:* Outer space, especially the relatively empty regions of the universe outside the atmospheres of celestial bodies* Vacuum, a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure* Free space, a perfect...
, while the inhomogeneous equation asks for more 'realistic' solutions with some matter, or charged particles.
Syntax
More abstractly, when using infix notationInfix notation
Infix notation is the common arithmetic and logical formula notation, in which operators are written infix-style between the operands they act on . It is not as simple to parse by computers as prefix notation or postfix notation Infix notation is the common arithmetic and logical formula notation,...
- T*U
the term T stands as the left-hand side and U as the right-hand side of the operator
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
*. This usage is less common, though.