Tag Archives: Georg Cantor

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.










To Infinity and Beyond

Image: NASA/Paul E. Alers.

Image: NASA/Paul E. Alers.

In everyday life, infinity has multiple meanings. To most people, infinity means being bigger than any number, the entirety of everything, or something that has no ending. In mathematics, infinity is strictly defined, but for the purposes of this post we will use a more intuitive definition: bigger than any finite number or an unnamed number that is arbitrarily large.

Much like the number zero, infinity is a concept that had been controversial amongst ancient mathematicians and barely accepted by modern mathematicians before the twentieth century. The ancient Greeks are the first recorded society that used the concept of infinity, although they did not readily accept it. They believed (as did mathematicians and philosophers for two thousand years afterward) that there was a potential for something to be infinite but that it could not actually be infinite. Much like the set of integers could potentially be infinite by adding one to a get another larger number, but the set of all infinite integers did not exist.

It was not until the late 1800’s that mathematician Georg Cantor formalized and proved the concept of infinity. Not only did Cantor prove the existence of infinity but, much to everyone’s surprise, he also proved that there are different sizes of infinity.

Infinity. Image: Public domain, via Wikimedia Commons.

Infinity. Image: Public domain, via Wikimedia Commons.

How could something that is arbitrarily large be bigger than something else that is arbitrarily large? Take for example the set of all the even numbers, (2,4,6,…) and the set of all the natural numbers (1,2,3,…). The most obvious answer would be that the set of all even numbers is half as large as the set of all natural numbers, even though both of them are infinite. In fact, the correct answer is that the two sets of numbers are actually the same size. How is this so? Cantor showed this by assigning each element in each of the sets a number. The number 1 from the real’s and the number 2 from the evens was assigned the number 1, 2 and 4 were assigned the number 2 and so on. By doing this he showed that no matter how large either set becomes he can still assign them the same numbers, therefore proving that they are the same size. This is what is known as countable infinity.

So then how do we get infinities that are different sizes? Think of the integers on a number line, all the way from negative infinity (Yes, this is a real thing. It just means a negative number that is arbitrarily large) to positive infinity. It is quite large but as we talked about in the previous paragraph it is still countable. Now, take only the segment of line between one and two. In this segment you have the numbers 1.5, 1.2, 1.24, 1.247,… In fact you have so many numbers in this small segment that it would be impossible to list them all. To create a number that is different than any number you have chosen all I have to do is take the first digit of the first number and increase it by one, then take the second digit of the second number and increase that by one and so on and so forth. In doing this I have created a new number that you have not listed. We therefore have a set that is so large that it is uncountable.

The concept of infinity is very important in many areas of math. The field of calculus depends entirely on it and many other fields in mathematics use the concept in important theorems. In fact, the infinite arises much more in mathematics than does the finite.