Jonathan Henderson Abstract: The proposed question, Is π + e an irrational number is a pressing point in modern mathematics. With the first definition of transcendental numbers coming in the 1700 s there has been an attempt to explore these numbers further. As time has passed we have discovered that π and e are both transcendental, however proving that their sum is as well has stumped mathematicians for nearly 130 years. This paper explores the unsolved equation π+е, its history, what it means and the attempts to solve it. π + e When humanity began exploring mathematics they likely had no idea of the complex problems and equations that would be created to explain the way our world behaves. Over time we have accepted many operations and functions that assist in further expand our understanding of the way things work. Among these operations the addition operation, denoted +, is among the most elementary of them. It is a binary operation that combines two quantities (usually numbers) to express the combination as a single quantity. This simple operation is at the base of our mathematics structure, but there are still unsolved problems that deal with the adding of two quantities. Among the most well know open questions is: to determine if the expression π + e is a rational or irrational number? To understand this problem there may be a few things that must be defined, among the most important things to be defined are π, e, and the difference between rational and irrational numbers. To create some foundation to help understand the problem I have defined a few terms below In both definitions of πand e involved the idea of transcendental numbers, so trancendentals are worth exploring. A transcendental number, first defined in the modern sense by Eular, is a number that can t be defined algebraically. That is to say that there a transcendental number has no roots of a non-constant polynomial equation with rational coefficients that. Although only a few classes of transcendental numbers are currently known, mostly because of the difficulty in proving a number is transcendental, transcendental numbers are not rare. Almost all real and complex numbers are transcendental. All transcendental numbers are irrational, since all rational numbers must be algebraic. The opposite is not true however, all irrational numbers are not transcendental, an example of this is the number 2 which is known to be irrational but it is not a
transcendental number because it is an algebraic number, that is we can write it as x 2 2 = 0. The discovery of transcendental numbers allowed the proof of the impossibility of several ancient geometric problems involving ruler and compass construction; the most famous one, squaring the circle, is impossible because π is a transcendental number. π is a constant that gets its value from the ratio of any Euclidean circle s circumference to its diameter. π can also be obtained by finding the area of a given circle and rationing it to the square of its radius. π is also an irrational number, which means it can t be expressed as a fraction of two integers, and as a consequence its decimal representation never ends. Lastly we know that π is a transcendental number (first proven by Carl Louis Ferdinand von Lindemann) which means, but is not limited to, that there is no finite algebraic expression of operations on the integers that can produce its value. We similarly define e as a constant that is an irrational number and transcendental number (proven by Charles Hermite). e can be produced by finding lim n 1 + 1 n A rational number is simply any number that can be written as the quotient of two integers, in other words it s a number that can be written as a fraction. Irrational numbers are defined as being a number in the set of Real number and not a rational number. Some examples of irrational numbers are 2 5 π e the list of both rational numbers and irrational numbers are infinite, however when evaluating between the numbers 0 and 1 we notice that there are many more irrational numbers than rational numbers. Now that there is a basic understanding of the terms and vocabulary in the problem, we can attempt to understand what it is that make this particular problem so interesting. But first we should modify the question now that we know what we are asking about. I think it may be better to ask if π + e is a transcendental number. I say that because if π + e is in the set of transcendental numbers then it is implied that it also must be an irrational number. So, it s obviously unknown whether π + e is transcendental, but we can prove that at least π + e or πe is must be a transcendental number. The proof uses a few basic algebraic concepts to show that at least one of the two must be. The general proof that at least one of π + e or πe must be transcendental goes as fallows; Let π and e be trancendental numbers and consider the case: x pi x e =x 2 + π + e x + eπ n.
If both π + e and πe were algebraic, this would be a polynomial with algebraic coefficients. If that were true then it implies the roots of the polynomial π and e must be algebraic. This produces a contradiction, and shows that at least one of the coefficients of the polynomial must be transcendental. Now we see that there is even more to the problem. Perhaps this proof will be of help in solving the problem because now we know that if πe is not transcendental then π + e must be. I feel that the problem now needs to be evaluated again. Perhaps it should be stated; Is either π + e and/or eπ a transcendental number. If we can determine that one of the two is not a transcendental number than we immediately know that the other must be which tells us exactly what would be needed to prove. Of course if we indeed find that one of the two is a transcendental number it may not be of any help to have included the other in the problem because we cannot make the statement that only one of the two numbers is a rational number. However if it does turn out that somehow [although its widely believed π and e are completely unrelated (besides Euler s identity e iπ = 1)] and through the proof above believed imposable) the formula supplies both coefficients to be rational, we could use the polynomial in several areas of calculus and trigonometry, possibly deriving more and new formulas. The proof of π + e seems to be important because it s a basic case of adding two different trancendentals. There is possibility of creating an axiomatic system for determining if the addition of two trancendentals numbers are transcendental as well. This may lead to discovering and creating the set of numbers that are trancendentals. Perhaps the reason this problem is most interesting is because it may allow us to better define some mathematical operations. The addition property is proven, but if it is proven that a rational number is obtained from the adding of π and e there may be some new properties to assist in adding irrational numbers. As of now it would be imposable to manually add up the digits of π and e, but the potential for finding a simple algorithm for adding all irrationals drives part of this problem. Of course the same is true for the multiplication that I added to the problem. If the multiplication operation has an ability to have more discovered about it then we are going to explore it. Although it may seem improbable that the discovery of π + e as a rational or irrational number will provide us with any applications, we must remember that the discovery of e and π themselves have provided thousands of
applications. I did not locate any specific applications but I can speculate that the sum of the two would pay great roles in trigonometry, other theoretical mathematics, engineering and physics because of the roles π and e all ready take in those fields. That is the beauty of math, people through history have found solutions to the most abstract problems only to see the solutions be used in a completely different field than originally intended. Maybe the solution to this particular problem will be used as a means to solve for similar unknown transcendentals. Other similar problems involving transcendental are: Sums, products, powers, etc. (except for Gelfond's constant) of the number π and the number e: π + e, π e, π e, π/e, π, e, π e The Euler-Mascheroni constant γ (which has not even been proven to be irrational) Catalina s constant also not known to be irrational Apery s constant ζ(3), and in fact, ζ(2n + 1) for any positive integer n There are many modern problems that take a stab at attempting to prove or disprove transcendentalism. 1 In Edward Burger and Robert Tubbs, Making Transcendence Transparent they go through several proofs of transcendental numbers. They use several different processes to prove that numbers are transcendental. The book discusses that π and e are both not Liouville numbers but how that does not exclude them from being trancendentals. Of the few that are proven, the method to prove π was transcendental solved by Carl Louis Ferdinand von Lindemann, obtained the name Lindemann-Weierstrass theorem, which has taken a major role in guiding modern attempts to prove other numbers to be transcendental. In 1900, David Hilbert posted an influential question about transcendental numbers, called Hilbert s seventh problem: If a is an algebraic number, that is not zero or one, and b is an irrational algebraic number, is a b necessarily transcendental? 2The affirmative answer was provided in 1934 by the Gelfound-Schneider theorem. 3This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms. These works have laid some of the ground work for designing proofs as to which numbers fit the set of transcendental. Perhaps this problem truly is unsolvable, but only time will provide us with an answer. I believe these problems are solvable but demand an uncommon knowledge of math. Hopefully there is a future where we have discovered a
formula for finding useful transcendental numbers and ways to use them to better our society.
Bibliography 1. Edward Burger and Robert Tubbs, Making Transcendence Transparent, Springer (2004). 2. A. O. Gelfond, Transcendental and Algebraic Numbers, Dover reprint (1960). 3. Alan Baker, Transcendental Number Theory, Cambridge University Press, 1975.