Tag Archives: pi

Transcendental Numbers: Beyond Algebra

The history of mathematics has been fraught with disappointments for mathematicians. This is particularly true in regard to the expected and continued failure of numbers, and math in general, to be pure and graceful. The Pythagoreans, in 6th century BCE Greece, venerated the whole numbers with an almost religious devotion because of their purity, and believed that the universe could be described by using only whole numbers. Unfortunately, math is not as pure as the Pythagoreans thought, which was revealed first by Hippasus of Metapontum when he discovered an undeniable proof for the existence of irrational numbers. Incidentally, Pythagoras had poor Hippasus drowned because of this. (The tale of the drowning of Hippasus may be merely a legend, like much of what is “known” about the Pythagoreans, due to a lack of reliable sources from the period.) Another impurity of numbers was wrestled with for millennia in the form of the square roots of negative numbers, a problem that was only put to rest with the advent of the imaginary number i. However, numbers turned out to be even weirder than previously imagined, because transcendental numbers were discovered by Gottfried Wilhelm Leibniz, in 1682.

In order to understand transcendental numbers, we need to understand algebraic numbers, or numbers that are not transcendental. An algebraic number is any number that is the solution to a polynomial with rational coefficients. Rational numbers are numbers that can be written as the ratio of two integers. All rational numbers are algebraic numbers, for instance the number 2 is a rational number because it can be written as 2/1. It is also an algebraic number because it is the root of the polynomial X – 2 = 0, which is a polynomial with only rational coefficients. While all rational numbers are algebraic, not all are algebraic numbers are rational, for example, √ (2) is an irrational number, but it is also algebraic because it is the solution to X² – 2 = 0. Strangely, the imaginary number i, although it is not real, is an algebraic number since it is the root of the polynomial X² + 1 = 0.

Transcendental numbers are numbers that cannot be written as the root to a polynomial with rational coefficients. All transcendental numbers are irrational. Leibniz coined the term “transcendental” in his 1682 paper in which he proved that the sin function is not an algebraic function.  Leonhard Euler (1707- 1783) was the first to generally define transcendental numbers in the modern sense, although it was Joseph Liouville, in 1844, who definitively proved the existence of the first transcendental number. That number is now called the Liouville Constant, and it is .110001000… with a 1 in every n! place after the decimal. The Liouville Constant was specifically constructed by Liouville to be a transcendental number. However, Charles Hermite first identified a transcendental number that was not created for that purpose in 1873. That number was the constant e, or Euler’s number, and is the base of the natural logarithm.

A famous transcendental number, called “Champernowne’s Number,” was discovered in 1933 and named after David G. Champernowne. It is formed by concatenating all the natural numbers behind the decimal point 0.12345678910…. Although, easily the most famous transcendental number is pi, which was proved to be transcendental by Ferdinand von Lindemann in 1882.

Pi. Image: Travis Morgan, via Flickr.

Pi. Image: Travis Morgan, via Flickr.

Georg Cantor, in the1870’s, proved that there are as many transcendental numbers as real numbers, a concept that is mind-boggling since the real numbers are uncountable. However, only a few numbers have ever been definitively proven to be transcendental, because it is extremely difficult to prove that any given number is transcendental.

Along with irrational and imaginary numbers, transcendental numbers have challenged and frustrated mathematicians throughout the ages. Undoubtedly, Pythagoras would be horrified by transcendental numbers, or maybe he would just drown anyone who tried to tell him about them. Today, however, transcendental numbers are embraced by mathematicians as a deep and important part of math.

Sources:

https://www.flickr.com/photos/morgantj/5575500301/in/photolist

http://nrich.maths.org/2671

http://individual.utoronto.ca/brucejpetrie/dissertation.html

http://sprott.physics.wisc.edu/pickover/trans.html

http://education-portal.com/academy/lesson/algebraic-numbers-and-transcendental-numbers.html

http://transcendence.co/transcendental-numbers/

http://www.daviddarling.info/encyclopedia/C/Champernownes_Number.html

http://www.britannica.com/EBchecked/topic/485235/Pythagoreanism

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Zu Chongzhi and his mathematics

https://upload.wikimedia.org/wikipedia/commons/thumb/0/07/%E6%98%86%E5%B1%B1%E4%BA%AD%E6%9E%97%E5%85%AC%E5%9B%AD%E7%A5%96%E5%86%B2%E4%B9%8B%E5%83%8F.jpg/1600px-%E6%98%86%E5%B1%B1%E4%BA%AD%E6%9E%97%E5%85%AC%E5%9B%AD%E7%A5%96%E5%86%B2%E4%B9%8B%E5%83%8F.jpg

Image:Zu Chongzhi. Author: Gisling, via Wikimedia Commons.

Zu Chongzhi was a Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties. He did a lot of famous mathematics during his life. His three most important contributions were studying The Nine Chapters on the Mathematical Art, calculating pi, and calculating the volume of sphere.

As we know,  The Nine Chapters on the Mathematical Art is the most famous book in the history of Chinese mathematics. In ancient China, most people could not understand “The Nine Chapters on the Mathematical Art”. Zu Chongzi read the book and then he used his comprehension to explain the formulas of the book. Zu Chongzhi, and his father wrote the “Zhui Shu”(缀术) together. The book made The Nine Chapters on the Mathematical Art easier to read. And the book also added some important formula by Zu. For example, the calculation of pi and the calculation of sphere volume. “Zhui Shu” also become math textbook at the Tang Dynasty Imperial Academy. Unfortunately, the book was lost in the Northern Song Dynasty.

Zu’s ratio, also called milü is named after Zu Chongzhi. Zu’s ratio was an early accurate approximation of pi. It was recorded in the “Book Of Sui” and “Zhui Shu”. (Book Of Sui is the official history of the Sui dynasty). According to the “Book Of Sui”, Zu Chongzhi discovered that pi is between 3.14159276 and 3.14159277. Today, we know the actual number is in accord with Zu’s ratio. But “Book Of Sui” did not record the method used to get the number. Most historians and mathematicians think Zu Chongzhi used Liu Hui’s π algorithm to get the number. Liu Hui’s algorithm means approximating circle with a 24,576 sided polygon. Japanese mathematician Yoshio Mikami pointed out, “22/7 was nothing more than the π value obtained several hundred years earlier by the Greek mathematician Archimedes, however milü π = 355/113 could not be found in any Greek, Indian or Arabian manuscripts, not until 1585 Dutch mathematician Adriaan Anthoniszoom obtained this fraction; the Chinese possessed this most extraordinary fraction over a whole millennium earlier than Europe”. Hence Mikami strongly urged that the fraction 355/113 be named after Zu Chongzhi as Zu’s fraction.( Yoshio Mikami)

https://i2.wp.com/upload.wikimedia.org/wikipedia/commons/e/e9/Sphere_volume_derivation_using_bicylinder.jpg

Image:Zu Chongzhi’s method (similar to Cavalieri’s principle) for calculating a sphere’s volume includes calculating the volume of a bicylinder. Author: Chen Bai, via WIkimedia Commons.

Zu Chongzhi’s other important contribution was calculation volume of the sphere. Together with his son Zu Geng, Zu Chongzhi used an ingenious method to determine the volume of the sphere.(Arthur Mazer). In The Nine Chapters on the Mathematical Art, the author used Steinmetz solid to get the volume of the sphere. The solid common to two (or three) right circular cylinders of equal radii intersecting at right angles is called the Steinmetz solid.

https://i0.wp.com/upload.wikimedia.org/wikipedia/commons/2/20/Bicylinder_Steinmetz_solid.gif

Image:Steinmetz solid. Author: Van helsing, via Wikimedia Commons.

But the book did not give the formula of how to get the volume of the sphere. Zu Chongzhi used “Zu Geng principle” (another name: Cavalieri’s principle) to show the volume of the sphere formula is (π*d³)/6. In order to commemorate the fact that Zu Chongzhi found the significant contribution of the principle with his son, people called the principle “Zu Geng principle”. “Zu Geng principle” is the same as “Cavalieri’s principle”, but “Zu Geng principle” is earlier than “Cavalieri’s principle”. “Cavalieri’s principle” means two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal.(Kern and Bland 1948, p. 26).

Work cited:

Yoshio Mikami , (1947). Development of Mathematics in China and Japan. 2nd ed. : Chelsea Pub Co;.

Arthur Mazer , (2010). The Ellipse: A Historical and Mathematical Journey. 1st ed. : Wiley;

Kern, W. F. and Bland, J. R. “Cavalieri’s Theorem” and “Proof of Cavalieri’s Theorem.” §11 and 49 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 25-27 and 145-146, 1948.

http://en.wikipedia.org/wiki/Cavalieri%27s_principle

http://en.wikipedia.org/wiki/Zu_Chongzhi