An equation concerns usually elements of a certain set , where one can say if two elements are equal. In the simplest case, has one binary operation “” producing as result some elements of , and these can be compared. Then, an equation in is a proposition of the form
where one has equated two expressions and formed with “” of the elements or indeterminates of . We call the expressions and respectively the left hand side and the right hand side of the equation (1).
(which is always true).
Of course, may be equipped with more operations or be a module with some ring of multipliers — then an equation (1) may them.
But one need not assume any algebraic structure for the set where the expressions and are values or where they elements. Such a situation would occur e.g. if one has a continuous mapping from a topological space to another ; then one can consider an equation
A somewhat case is the equation
Root of equation
If an equation (1) in one indeterminate, say , then a value of which satisfies (1), i.e. makes it true, is called a root or a solution of the equation.
Especially, if we have a polynomial equation , we may speak of the or the ; it is the multiplicity of the zero of the polynomial . A multiple root has multiplicity greater than 1.
Example. The equation
in the system of the complex numbers has as its roots the numbers
which, by the way, are the primitive sixth roots of unity. Their multiplicities are 1.
|Date of creation||2013-03-22 15:28:33|
|Last modified on||2013-03-22 15:28:33|
|Last modified by||pahio (2872)|
|Defines||root of an equation|
|Defines||left hand side|
|Defines||right hand side|
|Defines||multiplicity of the root|
|Defines||order of the root|