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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
equation
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(Definition)
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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
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(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.
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"equation" is owned by pahio.
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(view preamble)
See Also: algebraic equation, Diophantine equation, trigonometric equations, difference equation, differential equation, functional equation, homogeneous equation, proportion equation, finite difference, recurrence relation, characteristic polynomial, expression, equality
| Also defines: |
equate, side, root, solution, root of an equation, left hand side, right hand side, multiplicity of the root, order of the root, multiple root |
| Keywords: |
equation, root, solution |
This object's parent.
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Cross-references: roots of unity, primitive, complex numbers, multiplicity, polynomial, extended real number, vector space, topological space, continuous mapping, algebraic structure, ring, module, operations, set difference, groupoid, subsets, indeterminates, expressions, proposition, binary operation
There are 958 references to this entry.
This is version 26 of equation, born on 2005-08-17, modified 2008-04-24.
Object id is 7330, canonical name is Equation.
Accessed 33027 times total.
Classification:
| AMS MSC: | 20N02 (Group theory and generalizations :: Other generalizations of groups :: Sets with a single binary operation ) |
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Pending Errata and Addenda
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