Klaus Friedrich Roth was a mathematician who spent his life proving that some numbers are harder to approximate than others. He was born on October 29, 1925 in Breslau, Germany. The current city is Wrocław, Poland. He died on 10 November 2015 in Inverness, Scotland.
He was born in Germany but became a British mathematician. He won the Fields Medal in 1958. This is often called the Nobel Prize of mathematics. He won awards for his research on number theory. In particular, he solved the question of how close rational numbers are to algebraic numbers.
Educational Paths of Fields Medal Winners
Roth didn’t go straight to the top. He studied at Peterhouse College, University of Cambridge, England. He received his B.A. there in 1945. Then he went to the University of London. There he completed his master’s degree. He received his Ph.D. in 1948. 1950.
He started his career at University College London. He worked there from 1948 to 1966, after which he moved to Imperial College. He became a professor of pure mathematics. He served in this position until 1988.
The math behind the medals
Why did he win? This boils down to rational approximations to algebraic numbers.
This is the core question. Consider any irrational number. It’s called alpha. It doesn’t matter if alpha is algebraic or not. There is always an infinite number of rational numbers p over q. These fractions are very close to alpha. The rules are simple. The distance between p/q and alpha is less than 1 divided by q squared.
This can be proved using continued fractions. The convergents of the continued fraction for alpha plays this role.
Roth looked deeper. He asked for the exponent mu. The exponent indicates how accurately the number is approximated. Find an approximation to p/q that has an error less than q raised to the mu power.
If mu bar is the upper limit of such an exponent, the value changes according to the number.
Joseph Liouville addressed this problem in 1844. He showed that mu bar is less than n. where n is the degree of algebraic alpha.
Axel Thue improved on this in 1908. He showed that μ bar is less than n divided by 2 plus 1.
Carl Ludwig Siegel took another leap forward in 1921. He showed that μ bar is less than the square root of 2n.
Freeman J. Dyson further improved this in 1947. He made μ bar less than the square root of 2 times of n.
Then Roth appeared. In 1955 he showed that mu bar equals 2 for any algebraic alpha.
This is not a simple solution. This is quite difficult. Solve the algebraic number problems completely.
Beyond the Approximation Theorem
Roth is known for more than just this proof. He also studied integer sequences. He used a Selberg sieve. He did research in the field of analytic number theory.
He co-authored a book
