ST. CLAIRSVILLE — Dr. Zijian Diao, associate professor of mathematics at Ohio University's Eastern Campus, recently found a new proof to a 300-year old problem in mathematics. Diao's discovery focused on one of the most important numbers in mathematics, e=2.71828...
First discovered in the financial investigation of compound interest, this number plays important and recurring roles across the full spectrum of mathematics. It lies at the heart of solutions to many real world problems, such as voltages and currents in electrical circuits, population growth, radioactive decay, Newton's laws of cooling and heating and pricing formulas for stock options.
Diao devised a new way to establish a fundamental property of e: it is an irrational number, i.e., it cannot be written as the ratio of two integers. This property was established first by the prominent Swiss mathematician Leonhard Euler in 1737, using a technique called continued fraction. A few decades later, another prominent French mathematician, Joseph Fourier, proved the same fact using a different tool called series analysis.
To understand both Euler and Fourier's proofs, calculus-level mathematical knowledge is required at the minimum. In contrast, Diao proves the irrationality of e with an elementary method that is accessible to high school students. Diao said, ''It is never a trivial task to show that a number is irrational. More than 2,500 years ago, a group of early Greek mathematicians, the school of Pythagoras, firmly believed that all numbers are rational, in other words, all numbers are ratios of two integers.
When the first irrational number, square root of 2, was discovered by Hippasus of Metapontum around 520 BC, this idea was deemed so dangerous that he was murdered in the sea off the coast of Greece. Although the irrationality of e is well-known to mathematicians, an important mission of mathematical research is to find easier and better ways to explain mathematical facts and make them accessible to the broadest audience possible.
"We are able to design an elegant argument for the irrationality of e by rewriting e, originally defined as an infinite summation, into an infinite product. The new structure allows us to achieve our goal utilizing minimal rudimentary mathematical facts, such as a reduced fraction has the smallest denominator among all fractions with the same value and one cannot count down to zero without stopping after finite steps. It is both surprising and gratifying to come up with a simple proof like this for a centuries-old problem.''
Diao's result was accepted on January 16, 2018 for publication in The American Mathematical Monthly, the flagship journal of the Mathematical Association of America. The American Mathematical Monthly is an extremely selective journal intended for a wide audience of mathematicians, from undergraduate students to research professionals, and is the most widely read mathematics journal in the world according to records on the digital library JSTOR.
Dr. Diao earned a Ph.D. in mathematics from Texas A&M University, an M.S. in computer science from the University of Illinois, and a B.S. in automation from the University of Science and Technology of China. He has been a faculty member at Ohio University Eastern since 2005.