Klaus Friedrich Roth was born on October 29, 1925, into a Jewish family in Breslau, Prussia (now Wrocław, Poland). In 1933, his parents moved the family to London to escape Nazi persecution. His father, who had been exposed to poison gas when fighting for Germany during World War I, died while Klaus was young. From 1939 to 1943, Klaus attended St. Paul’s School in London. at a time when The Blitz forced evacuation from London to Easthampstead Park. At that school, Klaus demonstrated a special aptitude for mathematics and chess. On graduation, he attempted to join the Air Training Corps but was rejected on account of his German ancestry. However, when this was no longer an impediment, his subsequent attempts were thwarted when he failed the tests for physical coordination required of pilots.
Roth graduated from Cambridge in 1945, and the following year entered a master’s program at University College London. He completed his master’s degree in 1948 and his doctorate in 1950. In his doctoral thesis, he proved that every positive integer is the sum of a square, a positive cube and a fourth power.
On completion of his doctorate, Roth was appointed to the position of lecturer at University College London. During this time, he made a remarkable mathematical breakthrough, solving a major unsolved problem involving the approximation of algebraic numbers by rationals. Before 1955, it had been shown, using convergents of the continued fraction of an irrational number x, that there are infinitely many rational numbers p/q such that |x – p/q| < 1 /q2. However, mathematicians wondered whether, for some algebraic numbers, x, the inequality could be tightened. That is, could 1 /q2 be replaced by 1/qn where n > 2?
In 1844 Liouville showed that for an algebraic number of degree n, the exponent of q could not be greater than n. Then in 1908, Thue proved that the exponent of q could not be greater than n/2 + 1, and in 1931, Siegel improved this by proving that the exponent of q could not be greater than 2√n. Finally, in 1955, Roth solved the problem completely by showing that for any algebraic number of any degree, the exponent of q cannot be any greater than 2. In other words, for any algebraic number x and for any ε > 0, there are only finitely many rational numbers p/q (in lowest terms) that satisfy the inequality: |x – p/q| < 1 /q2+ε. For this result, now known as Roth’s theorem, Klaus Roth was awarded the Fields Medal in 1958.
Roth’s theorem provides an upper bound on how closely an algebraic number can be approximated by rational numbers. The smaller the ε, the closer the approximation, but the number of such approximations is finite. Though Roth’s theorem may seem esoteric, it has applications in various areas of mathematics, including diophantine equations, and transcendental number theory. When number theorist Harold Davenport presented Roth with the Fields Medal at the International Congress in Edinburgh he said:
This achievement is one that speaks for itself: it closes a chapter, and a new chapter is now opened. Roth’s theorem settles a question which is both of a fundamental nature and of extreme difficulty. It will stand as a landmark in mathematics for as long as mathematics is cultivated.
In spite of this outstanding achievement, Roth was not appointed to a full professorship until 1961. Then, in 1966 Roth accepted the chair in pure mathematics at Imperial College London and remained there until his official retirement in 1988. After retirement, Roth moved to Inverness, Scotland where he died on November 10, 2015 at 90 years of age.
Klaus Roth’s groundbreaking work significantly advanced the understanding of Diophantine approximation, and his contributions continue to be highly regarded in the mathematical community. In addition to the Fields Medal, he received numerous other awards and honors during his career, including the De Morgan Medal of the London Mathematical Society in 1983 and the Sylvester Medal from the Royal Society in 1991.
