[Math Lair] Transcendental Numbers

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Besides classifying the real numbers into rational and irrational numbers, we can also separate them into algebraic and transcendental. If a real number satisfies an algebraic equation, which is an equation of the form

cnx n + cn-1x n-1 + cn-2x n-2 + ... + c2x 2 + c1x + c0 = 0

where all of the c's are integers, we say that it is an algebraic number. Real numbers that satisfy no such equations are called transcendental numbers. It is fairly easy to see that every rational number a/b satisfies the equation bxa = 0 and so all rational numbers are algebraic. Therefore, every transcendental number must be irrational.

Some irrational numbers are algebraic though. For example, all square roots (for example, root 2) satisfy quadratic equations, and other roots satisfy other equations of higher degree. On the other hand, numbers such as log 2 and π are transcendental.

The existence of transcendental numbers was not proved until 1840, when Joseph Liouville proved that the number 0.1100010000000000000000010000..., where 1's appear in the n!th position for all natural numbers n (! is the factorial symbol) is transcendental. The proof is too complicated to present here, but suffice to say that it can be shown that algebraic numbers can approximate this number but can never equal it.

It was shown in 1882 by Ferdinand von Lindemann that π was transcendental. Showing that π was transcendental also proved that squaring the circle, one of the three famous construction problems of antiquity was impossible.

In 1934, Gelfond and Schneider independently proved that ab is transcendental if a is algebraic (but not equal to 0 or 1), and b is irrational and algebraic. This theorem answers the question of whether log 2 is transcendental. Since 10 log 2 is equal to 2, 2 would be transcendental if log 2 were algebraic. Therefore, log 2 is either rational or transcendental. It is pretty easy to show it isn't rational, so it must be transcendental. Taking the work of Gelfond and Schneider with work done by Alan Baker in the 1960s, it can be shown that trigonometric functions (sine, cosine, tangent), logarithmic functions (logbx, ln x), and ex almost always produce transcendental values except at very specific locations.

In a sense, it can be said that almost all numbers are transcendental. In the 1870s, Georg Cantor showed that the real algebraic numbers are countably infinite, while the real numbers (which are comprised of the real algebraic numbers and real transcendental numbers) are uncountably infinite; thus, transcendental numbers are, in a sense, more numerous than numbers that are not transcendental.

There are still some gaps in our knowledge of what numbers are transcendental. We know that π and e are both transcendental, and, while we know that eπ is transcendental, we don't yet know if either π + e or πe are transcendental, let alone irrational.

Something to think about: Can you find two transcendental numbers that add to an integer? It's not too difficult to find such numbers. π, for example is transcendental. Therefore, 10 − π must also be transcendental. However, if you add the two numbers together, the sum is 10.