The term transcendence comes from the latin word transcendere which directly translates to to climb over or beyond, meaning anything that is described as transcendental is seen as not of normal existence. This definition of transcendence being bizarre or odd holds true when applied to mathematics. The first recorded use of the phrase transcendental number was used by German mathematician Gottfried Wilhelm Leibniz in his paper proving that the function sin(x) was not algebraic in nature. But it was Leonhard Euler of Switzerland who first defined transcendental numbers in the modern sense, as any real or complex number that cannot be defined rationally (as a fraction) or as a root. Over the centuries, the complex topic of transcendence was tackled by many notable mathematicians. With Johann Heinrich Lambert writing about the transcendence of ???? and Euler’s number in 1768 and Ferdinand von Lindemann proving his hypothesis in 1882, we finally come to the present-day idea of transcendence that will be explored in the following paper. In this expository essay, we will discuss the categorization, proof, and application of transcendence in mathematics.

While the concept of transcendental numbers is fairly abstract, most numbers, real or complex, are categorized as transcendental. In 1850, Joseph Liouville discovered the Liouville constant, which was the first example to prove the existence of transcendental numbers. Due to the fact that it cannot be represented as a fraction, nor is it the root of any polynomial equation, this number became very integral in the evolution of the definition of transcendence. The constant is expressed by the function L=k=1?€10-k!, and is defined as 0.110001000000 where there is the digit ‘1’ in each decimal place corresponding with k! and the digit ‘0’ in any other position. This shows that the constant has no end and is therefore transcendental. Another common example of a transcendental number is Euler’s number (see figure 1). Found using the equation e=n?€(1+1n)n, and written as 1+1+12(1+13(1+14(1+15(1+…)))) in expanded form, e is useful in that it is the only number whose natural logarithm is equal to one, (ln(e) = 1). In 1932, German mathematician Kurt Mahler separated transcendental numbers into three categories, S, T, and U. He established these groups at a polynomial value at the complex number x, with a maximum degree n, and a positive integer maximum height H, with m(x,n,H) being the minimum nonzero absolute value of the polynomial at x. Using the equations: (x,n,H)=-log m(x,n,H)n log(H, (x,n)=H?€sup (x,n,H), and (x)=n?€sup (x,n), Mahler defined U as an infinite complex number, S as a number with a bounded ????(x,n) and finite ????(x), and T as a number where ????(x,n) is finite but unbounded, which only occurs when ????(x) is 0 . This means that although Liouville numbers all belong under the U category, a vast majority of complex numbers belong to set S. Using this classification system, Mahler was able to prove that the exponential function e can be used to create an S number using all nonzero algebraic numbers, and in addition shows that ???? is transcendental but is not a U number. A similar classification method is Koksma’s equivalent classification, where Koksma chose to divide transcendental numbers into three groups, S*, T*, and U*, but also chose to create a class representing the algebraic numbers, A*. However, the equations used to categorize S*, T*, and U* use the variables x, n, and H similarly, but add in the algebraic number of a finite set a. The categories are defined by *(x,n)=H?€sup *(x,n,H) and |x-a|=H-n*(x,H,n)-1. If *(x,n) is bounded and does not converge at 0, x is an S* number, when *(x,n) is unbounded and finite, x is an T* number, when *(x,n) is infinite, x is an U* number, and when *(x,n) converges at 0, x is an A* number.

Although there are theoretically infinite transcendental numbers, it is difficult to prove that a number is truly unable to be represented algebraically. The current prevailing way to prove transcendence is using the Lindemann-Weierstrass Theorem; in fact, this is the very formula that proved the transcendence of pi, ???? = 3.14159, and . The theorem states that the if a1, …, an are linearly-independent algebraic numbers over all rational numbers, meaning that a1, …, an are uncorrelated, then ea1, …,ean are also algebraically independent over all rational numbers. This proves not only that eais transcendental, for all rational numbers a, but also shows that ???? cannot be represented algebraically. Using Euler’s Identity, that ei+1=0, we arrive at the assumption that ei=(-1), where i represents the imaginary unit that satisfies i2=(-1) (see figure 2). However, if we were to assume that ???? is algebraic, that would imply that i???? is also algebraic. So after applying the Lindemann-Weierstrass theorem, we come to the contradiction that (-1) is transcendental. Thus ???? must be transcendental in nature. Another method used to prove transcendence is Baker’s theorem. To understand this principle we must first introduce the set of logarithms of nonzero algebraic numbers, L={C:eQ}.This shows that while ?» belongs to the set of all complex number, e is not rational. Similar to the Lindemann-Weierstrass theorem, Baker’s theorem states if 1,…,n are elements of L that are linearly independent for all rational numbers and all algebraic numbers are represented by 0,…,n, where not all ‘sare zero. Then we arrive at the function |0+11+…+nn|>H-C, where H represents the maximum heights of the’sand C is some computable nonzero number that depends on n, the number of ?»’s and the total degrees of ????’s (see figure 3). In other words, taking the absolute value of the sum of the products of many algebraic numbers, ????, and complex logarithmic numbers, ?», results in a transcendental number that is greater than H-C.The third proven method of showing transcendence is through the Gelfond-Schneider theorem. This statement is used to prove transcendence over a large classification of numbers. Originally theorized by Aleksandr Gelfond and Theodor Schneider, this principle states that for the algebraic numbers a and b, if a0, a1,and b is irrational, then all resulting values of ab are transcendental. This theorem led to two corollaries: Gelfond’s constant, e=23.1406… and the Gelfond-Schneider constant, 22=2.6651…, as well its square root, 22=1.6325…. Similar to this, the four exponential conjecture states that for two pairs of complex number x1,x2 and y1,y2 that are linearly independent over all rational numbers, then at least on of the following as transcendental:ex1y1,ex1y2.ex2y1,ex2y2. Although it has yet to be proven, the four exponentials conjecture is considered one of the strongest ideas relating to exponential functions using arithmetic values. In 1966, American mathematician Stephen Schanuel created a rule to generalize transcendental numbers further, known as Schanuel’s conjecture. The purpose of this theory was to find the degree of transcendence over certain field extensions of rational numbers. The conjecture states that if given n complex number z1,…,zn that are linearly independent over all rational numbers, the extension Q(z1,…,znez1,…,ezn) has a degree of transcendence of at least n over all rational numbers. This means that this theory ecompasses the Lindemann-Weierstrass theorem, in the event that z1,…,zn are all algebraic, Baker’s theorem, when z1,…,zn take the form exp(z1),…,exp(zn), and is also able to cover the unproven four exponentials conjecture and the Gelfond-Schneider theorem.

Many common functions can be used to create transcendence, these are known as transcendental functions. For example, any equation used to find the length of a curve, such as arc length: s=r180, area or circumference of a circle, or volume of a sphere rely on the transcendental number ???? to convert linear distances to curved or circular. The sine function is another example of a transcendental function, although there are several numbers that can be input into the sine function that output algebraic numbers, the only integer that puts out an algebraic solution is zero. There are hundreds of proven transcendental functions, often including multiple variables. Using the already identified transcendental numbers, for all real numbers x, x, ex, and logex are all fairly easy to understand and common examples of transcendental functions. Since the exponent, the base of the exponential function, and the base of the logarithm are all transcendental numbers, the transcendence is transferred to the functions and all of their solutions. While algebraic function such as square root functions or polynomials are not transcendental in nature, the indefinite integral of many algebraic functions is often found to be transcendental. For example, the logarithmic function was found while searching for the area of the multiplicative inverse function, f(x)=1x, and is now one of the most easily recognizable transcendental function. An additional example of a transcendental equation is Euler’s Gamma function. This equation is used to represent a factorial with an argument shifted down by one, (n)=(n-1)!, and can be expanded to all complex number except non-positive integers using the form (z)=0?€xz-1e-xdx (see figure 4). There are currently many rational values of z for which the answer is known and proven to be transcendental, for example z={16,14,13,12,23,34,56} is a set of values for which the argument (n-1)! is shown to be transcendental, indicating that the integral would also transcendental. It is fairly easy to understand that if f is an algebraic function and a is an algebraic number, then f(a) is also an algebraic number. However, there are entire transcendental functions that when evaluated at an algebraic number a, will also have an algebraic f(a). This set of algebraic numbers is known as an exceptional set of the transcendental function. For many functions the exceptional set is fairly small, like how the exponential function ex has an exceptional set of x={0}, also written at ?›(ex)={0}. Exceptional sets are often used to explain certain aspects of transcendental number theory, for example an exponential function of base 2 has an exceptional set of ?›(2x)=Q, meaning that 2x is only transcendental over irrational numbers, meaning it satisfies the Gelfond-Schneider theorem, but is not a transcendental function in itself. However, there are also functions with empty exceptional sets, which are often found using Schanuel’s conjecture. For example, ?›(eex)={?€…}, while f(x)=e1+x also has an empty exceptional set, but does not follow Schanuel’s conjecture. Using exceptional sets, we are able to prove that for any set of algebraic numbers, A, there exists a transcendental function whose exceptional set is A. This shows that there are transcendental functions who only output transcendental answers when given transcendental numbers.

In conclusion, transcendental numbers are abundant, fairly difficult to quantify, and have many current and possible future uses in mathematics. With new developments in the algebraic independence of modular functions being researched by the modern mathematicians Federico Pellarin and Yuri Valentinovich Nesterenko, there is no telling where transcendence will take us in the future. Although there are currently several ways to prove and use transcendence, we have just scratched the surface.

Transcendental Numbers. (2019, Jul 08).
Retrieved February 1, 2023 , from

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