Sides of an equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Conversely, 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.
Some examples
In
- 2a + 5 = a/3,
the term
- a/3
is the RHS.
In
- x ≤ 10,
just
- 10
is the RHS.
Homogeneous and inhomogeneous equations
In solving mathematical equations, particularly linear simultaneous equations, differential equations and integral equations, the terminology homogeneous is often used for equations with the RHS set equal to zero. The corresponding inhomogeneous 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, 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.
Syntax
More abstractly, when using infix notation
- T*U
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.