A New Realm of Numbers
Overview
Great strides were made in the 1800s toward moving back to a rigorous, logical base for mathematics. Essential to this effort was progress in number theory. Joseph Liouville (1809-1882) expanded the understanding of real numbers when he proved the existence of transcendental numbers. Later, Charles Hermite (1822-1901) demonstrated that e, the natural logarithmic constant, was a transcendental number. In 1882, Ferdinand Lindemann (1852-1939) answered "no" to the classic challenge, "Can the circle be squared?" when he proved that pi (π), the ratio of the circumference of any circle to its diameter, was also a transcendental number. Julius Wilhelm Richard Dedekind (1831-1916) completed the view of real numbers by explaining them in terms of irrational and irrational numbers. The establishment and characterization of real numbers extended the rigor of mathematics, improving the quality of proofs. It affirmed the concept of limits and allowed the rigorous development of analysis, which is essential to solving many difficult problems of engineering and science.
Background
During the 1800s, mathematicians took on the challenge of making their discipline more rigorous. While the Greeks carefully defined terms and worked out proofs based on logic and consistency, their successors took a more practical approach.
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