# 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

${E}_{1}={E}_{2},$ | (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 $({2}^{S},\setminus )$, where “$\setminus $” is the set difference^{}, we can write the equation

$$(A\setminus B)\setminus B=A\setminus B$$ |

(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 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 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 case is the equation

$$dimV=2$$ |

where $V$ is a certain or a vector space; both elements of the extended real number system.

Root of equation

If an equation (1) in $M$ 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 or the ${x}_{0}$; 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 $\u2102$ 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.

Title | equation |

Canonical name | Equation |

Date of creation | 2013-03-22 15:28:33 |

Last modified on | 2013-03-22 15:28:33 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 30 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 20N02 |

Related topic | Equality2 |

Related topic | AlgebraicEquation |

Related topic | DiophantineEquation |

Related topic | TrigonometricEquation |

Related topic | DifferenceEquation |

Related topic | DifferentialEquation |

Related topic | IntegralEquation |

Related topic | FunctionalEquation |

Related topic | HomogeneousEquation |

Related topic | ProportionEquation |

Related topic | FiniteDifference |

Related topic | RecurrenceRelation |

Related topic | CharacteristicEquation |

Defines | equate |

Defines | side |

Defines | root |

Defines | solution |

Defines | root of an equation |

Defines | left hand side |

Defines | right hand side |

Defines | multiplicity of the root |

Defines | order of the root |

Defines | multiple root |