What is a canonical form?

Almost all mathematical objects can be expressed in several ways. For example, a fraction of 2/6 is equivalent to 5/15 and -4/-12. A canonical form is a specific scheme that mathematicians use to describe objects from a given class in a unique way. Each object in the classroom has a single canonical representation corresponding to the template of a canonical form.

For rational numbers, canonical form is and / b , where and and b is positive. Such a fraction is usually described as "lowest". When inserted into a canonical form, 2/6 becomes 1/3. If there are two fractions of the same value, their canonical representations are identical. The two -dimensional linear equations have a canonical form of ax + from + c = 0, where c is 1 or. The form of hitting the slope is not canonical; Cannot be used to describe the x = 4 line.

Mathematics consider canonical forms especially useful in analyzing abstract systems in which two objects might seem significantly different, but are mathematically equivalent. The set of all closed donut paths has the same mathematical structure as a set of all arranged pairs ( and , b ) cent numbers. A mathematician can easily see this connection if it uses canonical forms to describe both sets. Both sets have the same canonical representation, so they are equivalent. In order to answer the topological question of donut curves, it may be easier for a mathematician to answer the equivalent algebraic question about the organized pairs of integers.

Many study fields employs matrices to describe systems. The matrix is ​​defined by its individual items, but those items often do not pass the character of the matrix. Canonical forms help mathematicians know when two matrix are somehow connected, whichIt may not be obvious otherwise.

Boolean algebers, the structure used by logici in description of the designs have two canonical forms: a disjunctive normal form and a conjunctive normal form. These are algebraically equivalent factoring or expansion of polynomials. A short example illustrates this connection.

High school director could say: "The football team must win one of their first two games and defeat our opponents, Hornets, in its third game, otherwise the coach will be released." This statement can be logically written as ( w 1 + w _{2 h} + f where " +" is a logical "or" operation and " *" is a logical "and" operation. Disjunctive normal formaexpression is w _{1 * h + w 2 * + f . Its conjunctive normal form for ( w 1 + w 2 + f ) * ( h + f ). All three of these expressions are PRand they are logically equivalent to exactly the same conditions.}

engineers and physics also use canonical forms to consider physical systems. Sometimes one system will be mathematically similar to another, although nothing like this appears. Differential matrix equations that are used to model one could be identical to equal to the modeling of the other. These similarities are manifested when systems are occupied in canonical form, such as an observable canonical form or controllable canonical form.