An odd question in mathematics that always seems to creep up on us is, “What is in between those numbers?” That is to say, we learn early on that the fractions are between the whole numbers. Later on, we learn that some numbers, the so-called irrational numbers, are wedged between the fractions. Of course, we learn later on that there are actually more of those irrational numbers than real ones, but who is counting?
The irrational break down even further. Some, called algebraic numbers, are the answers to complicated polynomial equations. Oddly, those are comparatively simple to find, and surprisingly, there are only countably infinite algebraic numbers. To be an algebraic number, all you need it to be the answer to a polynomial equation with rational coefficients.
Finally, between and around all the algebraic numbers is the largest class of real numbers, and surprisingly, the hardest to find: the transcendental numbers.
Transcendental Numbers
Transcendental numbers are, by definition, irrational. They are also hard to find. Even when you find one, it is often hard to prove that the number you found is actually transcendental.
Two numbers commonly worked with in high level mathematics, pi and e, are both transcendental. That means that there is no polynomial that has either of those numbers as an answer. They somehow fit between all of the answers that we learn to find in an algebra class. These amazing numbers are very hard to find, yet more numerous than any other class of number.
Making a Transcendental Number
If you want another example, the Gelfond-Schneider Theorem gives us an easy way to find them. Originally proposed as Hilbert’s Seventh Problem for the then-coming 20th century, it showed that any rational number, such as your birthday, raised to an irrational power, like “the square root of two” or pi, is transcendental.
The number that your calculator outputs when you put that in is undiscoverable through algebraic means as the solution to a polynomial with rational coefficients. It may not sound like much, but it is very important to number theory.
Using Transcendental Numbers
Most times, you will use a transcendental number without even realizing it. Pi and e are ubiquitous in our calculations and you can’t avoid them. Most other transcendental numbers are unlikely to appear in your daily life. The existence of them is important mostly for providing the “filler” on the number line. Despite the fact that you might never find them, they are the most numerous class of real number, and they fill in the gaps between all of the others.
The Importance of Transcendental Numbers
For mathematicians, these numbers provided answers to problems that were centuries old. The classic “square the circle” construction, where a person armed with the classic geometry tools of a compass and straight edge had to make a square with the same area as a given circle, was found to be a fool’s errand. It is simply impossible without taking an infinite number of steps. This was found in 1882, though the problem dated back to around 500 B.C. with Anaxagoras.
Unlikely connections like these are all over mathematics, and it is for these reasons that we continue to push into strange spaces, examining numbers that seem almost pointless to the young algebra student. Who knows where our next great discovery will be, or how transcendental numbers will play a role?
Sources:
Wolfram-Mathworld: Transcendental Number
Wolfram-Mathworld: Gelfond’s Theorem
Wolfram-Mathworld: Circle Squaring