Transcendental number

In mathematics, numbers are often categorized by how they relate to polynomial equations. A transcendental number is a real or complex number that is not algebraic. An algebraic number is any number that serves as a root for a non-zero polynomial with integer coefficients. This means an algebraic number can be the solution to an equation like $x^2 - 2 = 0$. Because the square root of 2 solves this equation, it is algebraic. A transcendental number, however, cannot be the solution to any such equation. This quality of being transcendental defines a massive and mysterious part of the number system.

To understand the mechanism of transcendence, one must look at the relationship between numbers and polynomials. An integer polynomial is an expression made of variables and integer coefficients. For example, $3x^2 + 5x - 2$ is an integer polynomial. If a number is a root, it makes that expression equal to zero. Transcendental numbers are defined by their total failure to satisfy any such equation. This makes them distinct from rational numbers. Every rational number is algebraic because it can be the root of a simple degree-one polynomial. Consequently, all real transcendental numbers must also be irrational numbers. However, the reverse is not true. Some irrational numbers, like the square root of 2, are still algebraic.

Mathematicians view the real numbers as being made of non-overlapping sets. These sets include rational numbers, algebraic irrational numbers, and transcendental real numbers. While algebraic numbers are easy to define, proving a number is transcendental is extremely difficult. This difficulty is why we only know a few specific classes of transcendental numbers. One such class is known as Liouville numbers. These numbers can be more closely approximated by rational numbers than any irrational algebraic number can. Joseph Liouville proved that all such numbers are transcendental.

History shows a long journey toward understanding these numbers. The term "transcendental" traces back to Gottfried Wilhelm Leibniz in 1682. In that paper, he proved that $e^x$ is not an algebraic function of $x$. During the eighteenth century, Leonhard Euler likely provided the first modern definition. In 1768, Johann Heinrich Lambert conjectured that $\pi$ and $e$ were both transcendental. He also offered a tentative sketch of a proof for the transcendence of $\pi$. It was not until 1844 that Joseph Liouville proved transcendental numbers actually exist. He provided the first decimal examples, such as the Liouville constant.

Significant breakthroughs followed in the late nineteenth century. In 1873, Charles Hermite proved that $e$ is transcendental. This was the first time a number was proven transcendental without being specifically constructed for that purpose. In 1882, Ferdinand von Lindemann published the first complete proof that $\pi$ is transcendental. He used the fact that $e^{i\pi} = -1$ to show that if $i\pi$ is algebraic, then $\pi$ must be transcendental. This work was later generalized by Karl Weierstrass into the Lindemann–Weierstrass theorem. These discoveries changed how mathematicians viewed the density of numbers.

Georg Cantor revolutionized the field in 1874 by studying the size of different number sets. He proved that algebraic numbers are countable, meaning they can be put into a list. However, he proved that real and complex numbers are uncountable. Because the real numbers are the union of algebraic and transcendental numbers, the transcendental numbers must be uncountable. This means transcendental numbers are not rare; they are actually the vast majority. In fact, almost all real and complex numbers are transcendental. Cantor's work established the ubiquity of these numbers across the mathematical landscape.

Transcendental numbers have profound implications for geometry and logic. The transcendence of $\pi$ proves that certain geometric constructions are impossible. For example, you cannot "square the circle" using only a compass and a straightedge. This was a famous problem that remained unsolved until transcendence was understood. Mathematics also continues to explore these numbers through complex questions. In 1900, David Hilbert posed his seventh problem regarding whether $a^b$ is transcendental under certain conditions. This was answered in 1934 by the Gelfond–Schneider theorem. Modern math continues to seek answers for combinations like $\pi + e$ or $\pi e$, which remain unproven today.