The ancient Greeks believed that the universe could be described using only whole numbers and fractions. However, they encountered a problem when trying to represent the diagonal of a square with sides of length 1 as a fraction, leading to the discovery of irrational numbers like $\sqrt{2}$.
For centuries, mathematicians struggled to define and understand irrational numbers until Richard Dedekind introduced a new way to approach them. Dedekind defined irrational numbers using sets of rational numbers, creating what he called a cut. This method allowed for a rigorous definition of real numbers, combining rationals and irrationals.
Around the same time, Dedekind’s friend Georg Cantor also explored irrational numbers, leading to a different but mathematically equivalent definition. Cantor’s work raised questions about the nature of infinity and the size of different sets of numbers, challenging traditional notions of mathematics.
Dedekind and Cantor’s contributions marked a significant shift in the understanding of mathematics, paving the way for new concepts and advancements in various fields. Dedekind’s work particularly influenced calculus, sequences, and functions, expanding the scope of mathematical exploration.
Emmy Noether, a prominent mathematician, acknowledged Dedekind’s impact on algebraic number theory, highlighting his lasting influence on the development of abstract algebra. Dedekind’s formal definition of $\sqrt{2}$ opened up new possibilities for mathematical exploration, inspiring mathematicians to invent new concepts and broaden their understanding of mathematics as a whole.
