Definition: A number is transcendental if it is not algebraic.
A transcendental number is a (possibly complex) number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree. Every real transcendental number must also be irrational, since a rational number is, by definition, an algebraic number of degree one.
Transcendental numbers are important in the history of mathematics because their investigation provided the first proof that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was, in fact, insoluble. Specifically, in order for a number to be produced by a geometric construction using the ancient Greek rules, it must be either rational or a very special kind of algebraic number known as a Euclidean number. Because the number XXX is transcendental, the construction cannot be done according to the Greek rules. [1]
References
[1] “Transcendental Number — From Wolfram MathWorld”. 2023. mathworld.wolfram.com. https://mathworld.wolfram.com/TranscendentalNumber.html.
Additional Reading
Maths and Musings. “Proving The Existence Of Transcendental Numbers And How To Compare The Infinite”. 2020. Medium. https://www.cantorsparadise.com/proving-the-existence-of-transcendental-numbers-and-how-to-compare-the-infinite-3d9ac5c298f9.
Here’s a cute way to prove the existence of Transcendental numbers. It requires *only* a keenness for mathematics, and is combined with a quick guide to the infinite and Cantor’s Diagonalisation argument! We’ll also see that there are ‘more’ transcendental numbers than non-transcendental.
“Recounting The History of Math’s Transcendental Numbers | Quanta Magazine”. 2023. Quanta Magazine. https://www.quantamagazine.org/recounting-the-history-of-maths-transcendental-numbers-20230627/.
Transcendental numbers include famous examples like e and π, but it took mathematicians centuries to understand them.
“The 15 Most Famous Transcendental Numbers – Cliff Pickover “. 2023. sprott.physics.wisc.edu. https://sprott.physics.wisc.edu/pickover/trans.html.
Thomas, Peter James. “In layman’s term, what exactly does it mean to be transcendental? How would a transcendental number be different from an ordinary number, say 7?”. 2020. Quora. https://qr.ae/pyB4jG.
An obvious question is “are there any numbers that are not Algebraic?” The answer is yes, there are both Real and Complex Numbers that are not the solutions of any polynomial with Integer coefficients. Such numbers are called Transcendental Numbers.
“Top Ten Transcendental Numbers”. 2010. David Richeson: Division By Zero. https://divisbyzero.com/2010/11/04/top-ten-transcendental-numbers/.
“Transcendental Number”. 2023. Vedantu. https://www.vedantu.com/maths/transcendental-number.
“Transcendental Numbers”. 2023. mathsisfun.com. https://www.mathsisfun.com/numbers/transcendental-numbers.html.
“Transcendental Numbers | Brilliant Math & Science Wiki”. 2023. brilliant.org. https://brilliant.org/wiki/transcedental-number/.
A transcendental number is a number that is not a root of any polynomial with integer coefficients. They are the opposite of algebraic numbers, which are numbers that are roots of some integer polynomial. e and π are the most well-known transcendental numbers.
“Transcendental Number – Wikipedia”. 2023. en.wikipedia.org. https://en.wikipedia.org/wiki/Transcendental_number.
Tripathi, Amitabha. “What is the exact difference between an algebraic and a transcendental number?”. 2021. Quora. https://qr.ae/pyB4QR.
“Where Transcendental Numbers Hide In Everyday Math | Quanta Magazine”. 2021. Quanta Magazine. https://www.quantamagazine.org/where-transcendental-numbers-hide-in-everyday-math-20211027/.
The transcendental number π is as familiar as it is ubiquitous, but how does Euler’s number e transcend the ordinary?
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