Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 5 - Number Theory and the Real Number System - 5.4 The Irrational Numbers - Exercise Set 5.4 - Page 299: 109


Irrational numbers are those numbers, which cannot be expressed as fraction \[\frac{p}{q}\] for any integer p and q, they have decimal expansion that neither shows periodicity nor terminates.

Work Step by Step

For example: \[\sqrt{2}\]. It is known that the first person to recognize the irrational numbers is a Greek philosopher, who is working on Pythagorean Theorem for calculating the sides of pentagram. The Greek mathematician made his discovery while out on sea. His discovery posed a serious problem to Pythagorean mathematics, because it shattered their all the assumptions that number and geometry were inseparable. Now, after 300 years it’s Euclid who proofed irrationality of irrational number such as\[\sqrt{2}\]. The irrationality of pi was proven by Lambert in 1760. The irrationality of Riemann zeta function was proven by Apery’s in 1979. Recently, the irrationality of many integers n such as Riemann zeta \[\left( 2n+1 \right)\]was proven by T. Reveal in 2000.
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