equation
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equation
Equation
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
| (1) |
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).
Example. Let be a set and the set of its subsets. In the groupoid , where “” is the set difference, we can write the equation
(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 contain them.
But one need not assume any algebraic structure for the set where the expressions and are values or where they represent generic 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 comparable case is the equation
where is a certain or a generic vector space; both sides represent elements of the extended real number system.
Root of equation
If an equation (1) in contains 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 multiplicity or the order of a root ; 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.
Mathematics Subject Classification
20N02 no label found
