Canonical forms are unique schemas used by mathematicians to describe objects in a specific class. Rational numbers are in the form a/b, with no common factors and b positive. Canonical forms are useful in analyzing abstract systems and matrices, and in Boolean algebras. They help identify mathematical equivalences and similarities between physical systems.
Almost all mathematical objects can be expressed in multiple ways. For example, the fraction 2/6 is equivalent to 5/15 and -4/-12. A canonical form is a specific schema that mathematicians use to describe objects of a given class in a codified and unique way. Each object in the class has a single canonical representation that matches the model of the canonical form.
For rational numbers, the canonical form is a/b, where a and b have no common factors and b is positive. Such a fraction is typically described as “in minimal terms”. When put into canonical form, 2/6 becomes 1/3. If two fractions have the same value, their canonical representations are identical.
Canonical forms are not always the most common way of denoting a mathematical object. Two-dimensional linear equations have the canonical form Ax + By + C = 0, where C is 1 or 0. Yet mathematicians often use the form of the slope intercept – y = mx + b – when performing basic calculations. The slope-intercept form is not canonical; cannot be used to describe the line x = 4.
Mathematicians find canonical forms especially useful when analyzing abstract systems, where two objects might look vastly different but are mathematically equivalent. The set of all closed paths on a donut has the same mathematical structure as the set of all ordered pairs (a, b) of integers. A mathematician can easily see this connection if he uses canonical forms to describe both sets. The two sets have the same canonical representation, so they are equivalent. To answer a topological question about curves on a donut, a mathematician might find it easier to answer an equivalent algebraic question about ordered pairs of integers.
Many fields of study use matrices to describe systems. A matrix is defined by its individual entries, but these entries often do not convey the character of the matrix. Canonical forms help mathematicians know when two matrices are related in some way that would otherwise not be obvious.
Boolean algebras, the structure that logicians use when describing propositions, have two canonical forms: disjunctive normal form and conjunctive normal form. These are algebraically equivalent to factoring or expanding polynomials, respectively. A brief example illustrates this connection.
A high school principal might say, “The football team has to win one of its first two games and beat our rivals, the Hornets, in the third, or the coach will be fired.” This statement can be logically written as (w1 + w2) * H + F, where “+” is the logical operation “or” and “*” is the logical operation “and”. The disjunctive normal form for this expression is w1 *H + w2 *H + F. Its conjunctive normal form for is (w1 + w2 + F) * (H + F). All three of these expressions are true under exactly the same conditions, so they are logically equivalent.
Engineers and physicists also make use of canonical forms when considering physical systems. Sometimes one system will be mathematically similar to another even if it doesn’t look similar at all. The differential matrix equations used to model one may be identical to those used to model the other. These similarities become apparent when systems are expressed in a canonical form, such as the observable canonical form or the controllable canonical form.
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