Transcendental friends
The emergence, in the time of the Greeks, of the notion of irrationality, with the impossibility, for certain numbers to be written as a quotient of integers, was a highlight of mathematics. Each new number was then subjected to the question: fraction or irrational? The proof, in the 19th century, of the irrationality of numbers like π or e, added a degree to the categorization: unlike others, such as √2, they were not solutions of a polynomial equation with integer coefficients. They were named transcendental, as if to remind that they came from a higher world. In fact, almost all real numbers are transcendental but for some irrationals, the question remains open.
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On the road to transcendence
Among irrational numbers, transcendental numbers are not roots of any polynomial equation with integer coefficients. They seem to transcend common sense, hence their name. Cantor showed that they make up the overwhelming majority of the real numbers, without identifying a single one!
Liouville, Hermite and Lindemann: paving the way | Tangente
A tribute to the pioneers who blazed the trail for others to follow.
The story of e
At the beginning of the 17th century, Galileo's telescopes and Newton's theories sparked an explosion in observations of the heavens. New mathematical tools were devised to handle the associated calculations, which were astronomical in every sense. Euler's number emerged naturally from this work.
In search of the universal number: Champernowne and Borel | Tangente
Sometimes, a concept that is relatively simple to define leads us into a world we dare not imagine, where our common sense struggles to find its bearings. Such is the case with universal numbers, which remain today as elusive as they are fascinating.
Transcendental, you say? The history of a term | Tangente
In Latin, the verb transcendere combines trans, "beyond," with scandere, "to climb"; it therefore literally means "to climb beyond," "to cross" or "to surpass."
One of them—or not…
A number for which, apparently, no definition involves a polynomial with integer coefficients has no reason to be algebraic. But that does not prove it is transcendental! Despite the many theorems on the subject, a great many numbers continue to resist classification.
Almost everything is normal
Looking at the distribution of a number's decimal digits leads to the definition of "normal" numbers.