Irrational numbers

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An irrational number is a number that cannot be expressed in the form m/n where n is not equal to zero and m and n are integers. In layman terms, no irrational number can be represented as a fraction with integers as the numerator and denominator. Applying Pythagorean mathematics, if we were to assume m and n are line segments, the ratio of the length of the sides, is in-commensurable. Moreover, if irrational numbers cannot be expressed as a simple fraction then it is impossible to express an irrational number as a decimal with a terminating decimal point or with periodic (repeated) decimals. All decimal equivalents of irrational numbers are non-periodic.A perfect example of this is the irrational number pi. There is an infinite number of values following the decimal place, hence many times pi is only approximated to the first couple significant figures for convenience. Irrational numbers can be represented as surds to be used in calculations.



HIstory

Greek

He postulated that the ratio of the hypotenuse to the side of a triangle of unit length is incommensurable (i.e. irrational). This implied that some lengths in geometry could not be expressed as a ratio of two lengths. The value he achieved none other than the square root of two. Pythagoreans however, rejected the idea as it challenged the foundations of their theories: it was thought that all numbers were rational

Indian

From about 800 BC or earlier, concepts involving numbers inexpressible as ratios began to emerge in Vedic Mathematics.Evidence of this study is present in the Sulbha sutras’ and the Brahmanas. Towards the 4th and 5th Century AD, Indian mathematicians coined the term āsanna, which meant approximately, to represent numbers that could not be represented as ratios of lengths (4, p.215). Incommensurability became a point or critical study from the 10th century AD. The study of in-commensurable lengths bred research on surd arithmetic, as well as rationalization, as a means to calculate and resolve irrational numbers. A surd is an archaic term for an irrational number. Rationalization is the simplification of an expression or equation by eliminating surds without changing the value of the expression.

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Conceptualizing and Representing Irrational Numbers

The concept of the number line makes it possible to represent all real numbers on the line (a real number is any number that is rational or irrational). Since both rational and irrational numbers are real, we can place both rational and irrational numbers on the number line. To identify irrational numbers we much apply Richard Dedekind’s Dedekind Cuts. In any separation of all rational numbers into 2 groups, those of the first group less than those of the second group, there is a number that occupies the point of separation. This point of separation is an irrational number.” Additionally, if the set of irrational numbers is a dense set, then between every rational number there is an irrational numbers. The set of irrational numbers is uncountable and there is an infinity of irrational numbers.

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Irrational numbers expressed as a decimal have an infinite number of values after the decimal point. The infinite number of values illustrate that there is no terminating decimal and their decimal representation is not period.


Transcendental Numbers

We can use irrational numbers to explore this concept of transcendental numbers. First we music define algebraic numbers: these are numbers that are the roots of non-zero integral polynomials. All rational numbers are algebraic. Transcendental numbers are numbers that are not algebraic. It is clear then that transcendental numbers are the opposite of algebraic numbers.All transcendental numbers are irrational. Let us note however, that though all transcendental numbers are irrational, not all irrational numbers are transcendental numbers. Examples of transcendental numbers include e, pi, and the values of trigonometric functions. Transcendental numbers are appear in various fields and disciplines. Pi is used in geometry as well as trigonometry and e is used in natural logarithmic function and in everyday calculations of growth over a specific period of time.


Resources

1. Flannery, David. "The Square Root of 2." Springer.com. Praxis Publishing, 2006. Web. 17 Nov. 2012. http://www.springer.com/mathematics/book/978-0-387-20220-4>. 2. Manning, Henry Parker. Irrational Numbers and Their Representation by Sequences and Series,. New York: J. Wiley & Sons; [etc., 1906. 3. Niven, Ivan. Irrational Numbers. [Buffalo]: Mathematical Association of America; Distributed by J. Wiley [New York, 1956. Print. 4. Selin, Helaine. Mathematics across Cultures: The History of Non-western Mathematics. Dordrecht: Kluwer Academic, 2000. Print.

--Khjamieson 21:34, 17 December 2012 (EST)