Transcendental Numbers

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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.

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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.

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