Canonical Forms

In the field of mathematics, scientists and students alike might need to represent numbers in a variety of ways for making accurate calculations and also for recording data in a specific manner. One of those representations, called the canonical form of presentation, is very popular and often used by both academicians and scientists. In this article, we would be talking about the canonical form and how to work with them.  Read on to find how the canonical form of presentation is helpful for us and makes work simpler. Canonical forms are one of the easiest and simplest forms of representing a number or value. Consequently, they are also easy to read and understand.

Definition of Canonical Forms

A canonical form refers to the standard way of representing an element or entity as a mathematical expression. In most cases, the canonical form representation is unique and helps in recognizing an element or object.

Some popular examples of canonical forms include the row echelon form and Jordan form which are canonical forms for matrices. Similarly, the canonical representation of a positive integer in decimal representation is a finite sequence of digits not beginning with zero. To understand the application of canonical forms better, some examples are given below. Using canonical forms in large number notation and Number Theory really shows how it makes work easy for researchers and academicians who have to tirelessly work with such large numbers and perform complex calculations. 

Large Number Notation using Canonical Forms

Some numbers have a value, which is significantly higher than what is used in our daily lives. In some cases, the number might also be so small that it is not used on a regular basis. For example, the number of cells in a human body is a large number, while the height of paper might be a pretty small value. These kinds of numbers appear often in the fields of mathematics, cosmology, cryptography, and statistical mechanics, but large number notation in its native form makes no sense. Though there are many ways to represent large numbers, using the canonical form is still one of the simplest and easiest ways to do so. In this method of representation, powers of 10 and decimals are used. The examples given below would help you understand the notation better. 

444 can be written as shown below

444=43=25121.3410154=(10)22.2

This is an example of using canonical forms for large number notation.

Using Canonical Forms in Number Theory

Number Theory is a branch of pure Mathematics that works with and is devoted to the study of integers and integer-valued functions. Canonical forms can be used in Number Theory for representing a positive integer and also for showing a continued fraction. 

The canonical representation of a positive integer is under the premise that every positive integer can be represented in exactly one way as a product of prime powers. So, mathematically, n = p1n1p22…..pknk = i=0kpini

Hence, 126 = 2*32*7

930 = 2*3*5*31

Every rational number can be written as a unique simple continued fraction. So, each rational number has exactly two ways to be represented – either in its native rational form or as a unique simple continued fraction. The shorter one is usually chosen as the canonical representation of the rational number and in most cases, it is the continued fraction. 

Conclusion

After going through this article, you must have a brief understanding of what a canonical form is and how it helps scientists and academicians. Not only can this form be used for simplifying calculations, but can also be used to record data in a specified format. Additionally, many fields of Mathematics seem incomplete without the necessary canonical forms. You check out other uses of canonical forms in fields other than Mathematics. It would greatly broaden your horizon and make your understanding deeper.