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# Number Theory/existence of transcendentals

Question
I'm wondering how transcendental numbers work! I can understand the proofs of irrational numbers.. But I'm not able to understand for transcendental.. Can you explain a simple proof of a transcendental number(any number like pi or e or something)? And How people came to know transcendental numbers exist? Help me please!

Hello Shameem
I'm sorry to say that a simple proof does not exist.  Unless you are at least at doctorate level.
Liouville gave the first proof that a transcendental number exists.
If you are familiar with the proof that the rationals are countable, then a similar stacking proof can be used to show that the algebraic numbers are countable, by listing them by the coefficients of their polynomial equation. That leaves an infinite number of uncountable numbers.  These are the transcendentals.
So can I refer you to http://en.wikipedia.org/wiki/Transcendental_number.

Best wishes

vijilant

Number Theory

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#### Vijilant

##### Expertise

Most questions on number theory, divisibility, primes, Euclidean algorithm, Fermat`s theorem, Wilson`s theorem, factorisation, euclidean algorithm, diophantine equations, Chinese remainder theorem, group theory, congruences, continued fractions.

##### Experience

Teacher of math for 53 years

Organizations
AQA Doncaster Bridge Club Danum Strings Orchestra Doncaster Conservative Club Danum Strings Orchestra Simply Voices Choir Doncaster TNS mystery shopping St Paul's Music Group Cantley

Publications
Journal of mathematics and its applications M500 magazine

Education/Credentials
BSc (Hons) Liverpool (Science). BA (Hons) OU (Mathematics)

Awards and Honors
State Scholarship 1955 Highest Score in Yorkshire on OU course MST209 £50 prize First class honours in OU BA Mathematics

Past/Present Clients
I taught John Birt, former Director of the BBC in 1961. His homework book was the most perfect I have ever marked. And also the most neat. I could tell he was destined for great things. One of my classmates was the poet Roger McGough, and I have a mention in his autobiography.