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Transcendental Numbers

Transcendental Numbers. Pi and e Zach Geiser, Rick Hanlon, Martha Smith. Transcendental Numbers. The investigation of transcendental numbers provided the first proof that squaring the circle is impossible with only a compass and straightedge.

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Transcendental Numbers

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  1. Transcendental Numbers Pi and e Zach Geiser, Rick Hanlon, Martha Smith

  2. Transcendental Numbers The investigation of transcendental numbers provided the first proof that squaring the circle is impossible with only a compass and straightedge. Transcendental numbers are numbers that are not roots of algebraic equations with integral coefficients. They do not terminate. Their decimals continue infinitely and never repeat.

  3. Transcendental Numbers • There are many different transcendental numbers. • Pi and e are the most common and the most used of all the transcendental numbers.

  4. Importance of Pi • Pi is the ratio of the circumference of a circle to its diameter. • It is used in equations for circumference, area, and volume. • It is constant and does not change from circle to circle. • It is used in probability as well as geometry.

  5. Importance of e • e can be defined in terms of infinite series, limits, logarithmic functions, and many other ways. • It is used in applications from logarithmic curves to studying interest. • It has many diverse and unique uses.

  6. Background of Pi • Ratio used as far back as Bible times, even though the approximation was not very good • The Babylonians and Egyptians used slightly better approximations. • Archimedes gave the first rigorous mathematical calculation of pi by circumscribing and inscribing polygons with circles. • The Islamic world worked the most with pi in medieval times.

  7. Pi Continued • Many well-known mathematicians such as Newton, Leibniz, Gregory, and Euler came up with formulas for pi, but they were not very efficient. • In India, two mathematicians named Aryabhata and Brahmagupta worked on calculating pi. • William Jones, an English mathematician, was the first to use the symbol “π” to represent pi.

  8. Pi Continued • Johann Heinrich Lambert proved that pi is irrational, and his proof was published in Elements de Geometrie. • German mathematician, F. Lindemann proved pi was transcendental in 1882. • Yasumasa Kanada calculated the digits of pi to 6.4 billion places, but there is no practical purpose for more than a few hundred digits.

  9. Pi Continued • Countless articles and books written on it • March 14 is dedicated to pi because it begins with 3.14. • There is a song about pi and a movie about Maximillian Cohen entitled Pi. • Some consider pi the world’s most mysterious number.

  10. Background of e • John Napier made the first reference to e in 1618. • Henry Briggs began working with Napier on e, and he gave an approximation of e but never mentioned it by name. • In 1647, Gre’goire de Saint-Vincent found that the area under a rectangular hyperbola from 1 to e is equal to 1. • This property makes e the base of natural logarithms, but it was not understood at that time.

  11. e Continued • Christian Huygens defined the exponential curve to be y=kak which is the logarithm base ten of e. • He also evaluated it to seventeen decimal places, and that was the most accurate calculation of the time. • Nicolas Mercator coined the term “natural logarithm” in his book.

  12. e Continued • In 1683, Jacob Bernoulli made the first approximation of e while working with his equation for compound interest. • Euler proved this value is exactly e. • In 1684, James Gregory realized that the log function is the inverse of the exponential function. • Euler is given credit for the notation “e”. • Some believe he gave it that symbol because of his last name, but others believe it is because it is the second vowel after “a”, which he was already using in his work.

  13. e Continued • Euler estimated e to eighteen decimal places and defined it as a continued fraction. • In 1873, Charles Hermite proved e was not algebraic and thus transcendental, but most mathematicians accept Euler as the first to prove e was irrational. • Since Euler, people have been trying to estimate e to more decimal places. • The latest example is from April 27, 2007 when Shigeru Kondo and Steve Paglaiarulo estimated it to 100 billion decimal places.

  14. Which is the better transcendental number Pi or e? You Decide

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