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

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.