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 6^th 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.

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

Turing, Leibniz and Hilbert’s Entscheidungsproblem

Alan Turing. This image is in the public domain in the US because its copyright has expired.

What’s the first thing you think of when you hear the name Gottfried Leibniz? Let me guess: calculus.  Now what do you think when you hear of Alan Turing?  You might think of codebreaking during World War II, or the new movie coming out about him (The Imitation Game), or maybe you haven’t heard of him.  So why would I mention these two together? Computers of course! Wait, what do these two have to do with computers? Well let’s take a look and see.

The Entscheidungsproblem origins start with Gottfried Leibniz in the seventeenth century.  Leibniz had successfully created a mechanical calculating machine, one of the first of its kind.  This calculating machine led him to question if a machine could be made that could determine the truth values of mathematical statements.  In his research, he found that one would have to find a formal language to create this machine.  In 1900, David Hilbert, a German mathematician, included the following in his 23 unsolved (at the time) problems designed to further the disciplines in mathematics:

“10. Determination of the solvability of a Diophantine equation.  Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. (Winton Newson’s translation of HIlbert’s original problem, as quoted in D. Joyce)”

By 1928, Hilbert had broadened his question about Diophantine equations to a much more general question about mathematical statements in general: is there an algorithm that is universally valid. This created a new idea; is there an algorithm that can tell us if any algorithm will terminate?  The last of these three ideas was the beginning of the Entscheidungsproblem. In May, 1936 Alan Turing wrote a paper called “On Computable Numbers, with an Application to Entscheidungsproblem.”  In this paper, Turning reformatted Kurt Godels results on the limits of proof and computation.  He made a hypothetical device known as the Turing machine and went on to prove there was no solution to the Entscheidungsproblem.  He did this by using his Turing machine to show that the halting problem is undecidable; that it is impossible to know whether a program will finish running or continue forever.

The Turing machine itself can represent a computing machine.  It can change symbols on a strip of tape based on a set of tools.  A Turing machine has 3 main components, the first being an infinite tape. This infinite tape would be divided into cells in which a symbol can be placed.  In the tape there would be a head.  The head accesses one cell at a time, while moving either right to left or left to right.  The third component would be a member were there would be a fixed finite number of states.  After having these three components you have three actions: 1) write a symbol, 2) move either left or right, and 3) update its current state.  The formal definition of a Turing machine is defined as a 7-tuple.  The seven elements of the tuple would be as follows: a set of states, an input alphabet, the tape alphabet, the start state, a unique accept state, a unique reject state, and a transition function.

A Turing Machine, without infinite tape. Image: Rocky Acosta, via Wikimedia Commons.

Turing’s work on the Entscheidungsproblem and the Turing machine can be thought of as the birth of computer science and digital computers.  During World War II the idea of the Turing machine was used and manipulated into a simpler form, as well as into an actual electronic computer.  This led to machines such as the counter machine, register machine, and random access machine.  All of these machines launched us even further into the computer era.

It is interesting to see that the birth of the modern computer came from the Entscheidungsproblem, an idea that Leibniz had first thought of.  Why would I think this is interesting?  Leibniz had also worked on binary numbers and arithmetic, which is similar to what is used today in modern computing. It seems that Leibniz was ahead of his time.  Alan Turing seems to have just taken his ideas and brought them to our times.  We can see that without Turing we wouldn’t have modern computers the way they are. This means we wouldn’t be able to do any math that requires a computer to help with computations.  Think, how many times have you used your computer to access the Internet to get the answer to a math problem you were unable to solve? Not only that, but studying math wouldn’t have been as easy.  Knowledge that is passed through the Internet wouldn’t be possible without computers, with no YouTube to help show how to solve math problems, with no Khan Academy or Wolframalpha, and no easy access to any knowledge of any past essays that were written.

Sadly, Turing’s end wasn’t a happy one.  Living in England in the early 20th century as a gay man led him to commit suicide.  Leibniz lived almost twice as long as Turing.  It makes you wonder if we could have had even more interesting computing machines or ways of thinking of computational mathematics if he had lived a full life past the age of 41.

Sources

History on Turings life – http://www.math.rutgers.edu/courses/436/Honors02/turing.html

Hilberts Problems – http://aleph0.clarku.edu/~djoyce/hilbert/problems.html http://mathworld.wolfram.com/HilbertsProblems.html

Turings paper “On computable Numbers, with an application to the Entscheidungsproblem – http://plms.oxfordjournals.org/content/s2-42/1/230.full.pdf+html

Gottfried Wilhelm Leibniz on wikipedia – http://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz

Wikipedia article on the Turing Machine – http://en.wikipedia.org/wiki/Turing_machine

Invented or Discovered?

A philosophical question about math that has been asked since the times of the ancient Greeks (and possibly even before then) is whether mathematics is discovered or is it invented by man. People seem to think it has to be one or the other, but what if it is actually both?

Gottfried Wilhelm von Leibniz. Image: Christoph Bernhard Francke, via Wikimedia Commons.

Math is just a language, and like any other language that uses words to describe something (strings of symbols), math also uses symbols. Written language was developed both independently and simultaneously in ancient times. One person got an idea to use a written symbol to represent a tangible object. Sometimes multiple people got this same idea independently of each other, and other times a person would see this writing, it would spark the idea in their heads, and they would go on to develop their own written language. The same language was not developed by different people, rather each person used different symbols to represent different words. (Guns, Germs and Steel- Jared Diamond Chapter 12) The same can be said for math. Calculus was developed simultaneously, but independently by Isaac Newton and Gottfried Leibniz. Both developed different ways of doing calculus and each way gave the same results. Other times mathematicians have relied on the work of others to further their results.

Isaac Newton. Image: Sir Godfrey Kneller, via Wikimedia Commons.

The fact that math has been developed independently and yet yielded the same results shows that math is discovered. Math is the language used to describe the natural world and as long as the world exists someone can, at any time, develop a language to describe it. It may not be the same math that we use today (the Babylonians used an arithmetic system very different from our modern one), but it would still yield the same results. Given enough time, one would think, they would eventually be able to build the same skyscrapers and the same rocket ships that we have.

On the other hand, math was invented. We invent the symbols and decide what they represent; we invent the axioms and the particular system that we use. Newton invented infinitesimals in the use of calculus while Leibniz invented his own notation for calculus. The ancient Egyptians invented a different way to calculate the area of a circle than the one we use today. (A History of Mathematics- Uta c. Merzbach and Carl B. Boyer) Math does not exist without someone to invent the symbols we use to describe it.

Many people ask, and for good reason, if this question is even important, and it just may be. What if the concept of zero or negative numbers were never invented? Without these simple concepts would we still be able to build the same skyscrapers and rocket ships? It is possible that someone could have invented a concept similar to these but using different concepts? It is even possible that someone may have invented a way around them so we could avoid them altogether and this new invention could have even lead to a much simplified math system.