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1MileCrash
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Can you come to a transcendental number through operations involving non-transcendental numbers?
Or is it impossible as I presume?
Or is it impossible as I presume?
1MileCrash said:Can you come to a transcendental number through operations involving non-transcendental numbers?
Or is it impossible as I presume?
1MileCrash said:Can you come to a transcendental number through operations involving non-transcendental numbers?
micromass said:And what about [itex]2^{\sqrt{2}}[/itex]?? This is transcendental... See the Gelfond-Schneider theorem.
A transcendental number may not be computed using any finite number of algebraic operations.
gb7nash said:The answer to this depends entirely on what the OP considers a valid operation. Perhaps give us a list of rules/things you can do?
1MileCrash said::)
So any integer raised to the power of it's root is transcendental?
Or would it be better to say, an integer raised to a non-transcendental irrational number is transcendental?
1MileCrash said:I meant mainly though algebraic means, addition, multiplication, powers, and all of their inverses.
Hmm. What about micromass's answer: [itex]2^{2^\frac{1}{2}}[/itex]?KingNothing said:No, not a finite number of them.
pmsrw3 said:Hmm. What about micromass's answer: [itex]2^{2^\frac{1}{2}}[/itex]?
That looks like a finite combination of the allowed operations.
KingNothing said:I remember reading somewhere that a transcendental number may not be computed using any finite number of algebraic operations.
A transcendental number is a real number that is not a root of any non-zero polynomial equation with integer coefficients. In other words, it is a number that cannot be expressed as a fraction and is not a solution to any algebraic equation.
Transcendental numbers are different from algebraic numbers in that they cannot be expressed as a solution to any algebraic equation, whereas algebraic numbers can be expressed as such. Transcendental numbers are also considered to be "more irrational" than algebraic numbers.
The concept of transcendental numbers was first introduced by German mathematician Johann Heinrich Lambert in the 18th century. However, the first proof of the existence of transcendental numbers was not provided until the 19th century by French mathematician Joseph Liouville.
Famous examples of transcendental numbers include pi (π), e, and the golden ratio (φ). However, there are infinitely many transcendental numbers and it is estimated that most real numbers are transcendental.
Transcendental numbers have been a source of fascination for mathematicians because of their "unpredictable" and "random" nature. They have also been used in various fields such as physics and cryptography. The study of transcendental numbers has also led to important developments in the field of number theory.