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# equation

Equation

An equation concerns usually elements of a certain set $M$, where one can say if two elements are equal. In the simplest case, $M$ has one binary operation “$*$” producing as result some elements of $M$, and these can be compared. Then, an equation in $(M,\,*)$ is a proposition of the form

$\displaystyle E_{1}=E_{2},$ | (1) |

where one has equated two expressions $E_{1}$ and $E_{2}$ formed with “$*$” of the elements or indeterminates of $M$. We call the expressions $E_{1}$ and $E_{2}$ respectively the left hand side and the right hand side of the equation (1).

Example. Let $S$ be a set and $2^{S}$ the set of its subsets. In the groupoid $(2^{S},\,\smallsetminus)$, where “$\smallsetminus$” is the set difference, we can write the equation

$(A\!\smallsetminus\!B)\!\smallsetminus\!B=A\!\smallsetminus\!B$ |

(which is always true).

Of course, $M$ 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 $M$ where the expressions $E_{1}$ and $E_{2}$ are values or where they represent generic elements. Such a situation would occur e.g. if one has a continuous mapping $f$ from a topological space $L$ to another $M$; then one can consider an equation

$f(x)=y.$ |

A somewhat comparable case is the equation

$\dim{V}=2$ |

where $V$ 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 $M$ contains one indeterminate, say $x$, then a value of $x$ 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 $f(x)=0$, we may speak of the multiplicity or the order of a root $x_{0}$; it is the multiplicity of the zero $x_{0}$ of the polynomial $f(x)$. A multiple root has multiplicity greater than 1.

Example. The equation

$x^{2}\!+\!1=x$ |

in the system $\mathbb{C}$ of the complex numbers has as its roots the numbers

$x:=\frac{1\!\pm\!i\sqrt{3}}{2},$ |

which, by the way, are the primitive sixth roots of unity. Their multiplicities are 1.

## Mathematics Subject Classification

20N02*no label found*

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