Is lnn\ln n transcendental for all rational n>1n>1?

I know that lnn is transcendental for all integer n>1. But does this still hold for non-integer rational values of n>1? For example, is ln1.5 transcendental? EDIT: Somehow managed to overlook the fact that cases like lne=1 are not transcendental 😛 The question has been revised to only include rational numbers n. Answer No, what … Read more

What functions do we need besides polynomials to describe any real as the root of some equation?

If f(x) is some polynomial with integer coefficient of degree >0, then any solution to f(x)=0 is an algebraic number. If g(x) and h(x) are also polynomials with integer coefficients, is it possible for the solution of f(x)+g(sinx)=0 to be any real number? What about f(x)+g(sinx)+h(logx)=0? Answer There is no way of constructing all the … Read more

Where is the sine function transcendental? [duplicate]

This question already has answers here: When is sinx an algebraic number and when is it non-algebraic? (2 answers) Closed 4 years ago. Most if the the values of the sine function that I am familiar with are irrational, like sin(π/3) or sin(π/6), or even rational like sin(π) or sin(0). Surely the sine function must … Read more

if the sum of two numbers α\alpha and β\beta is algebraic, and their product is transcendental, what do we know about these numbers?

These are elements of a field. My intuition says that α=a+b, β=a−b, where, a is algebraic and b is transcendental, but I can’t prove it. I don’t even know where to start. Thanks in advance! Answer If α+β is algebraic and αβ is transcendental, then (α−β)2=(α+β)2−4αβ is transcendental, so α−β is transcendental. Thus α=((α+β)+(α−β))/2 is … Read more

Is this proof that tana\tan a is transcendental correct?

First, a proof that sina is transcendental, where a is algebraic not zero. Given that, eia=cosa+isina if sina were algebraic, then cosa=±√1−sin2a is also algebraic, implying that eia=±√1−sin2a+isina is algebraic, but eia is transcendental (Lindemann theorem). Doing the same thing we can prove that cosa is transcendental. If tana=sinacosa were algebraic, then sina/cosa is algebraic, … Read more

Extend a rational number field Q\mathbb{Q} by using a transcendental number?

Here denoting a set of real transcendental numbers T, what can we then say about the structure Q(t)={+∞∑k=0aktk|ak∈Q},for some t∈T. Now it is not a ring, is it? What do we have to add to make it a ring (or a field)? Is Q∪T a ring (or a field)? Answer Sure, what you write as Q(t) … Read more

How to prove this: If tt is a transcendental number, then 5t4+8t+35t^{4}+8t+3 is also transcendental?

Can I prove as follows? If 5t4+8t+3 is not transcendental, then 5t4+8t+3 is a solution of a polynomial p. If you expand p(5t4+8t+3)=0 and write it in the form of another polynomial p′(t)=0, this indicates that t is the solution of this polynomial, which contradicts the fact t is transcendental. Answer Yes, it is as … Read more

Is it correct: natural-logarithm maps algebraic numbers to transcendentals and vice-verse, over the domain it is defined?

Is it correct that the natural logarithm function maps algebraic numbers to transcendental and transcendental numbers to algebraic, other than 1? Of course, over the domain natural log is defined i.e. (0,∞)? i.e. ln:A+→T;ln:T+→A, andln:A+↛ where; A^+ is the set of positive algebraic numbers except 1. T^+ is the set of positive transcendental numbers. T … Read more

Is the golden ratio a transcedental number?

I have been studying the concept of transcedental numbers. Till now, I had taken it for granted that all important numbers like pi and e were transcedental. I have no reason for assuming this or for clustering them together. It’s just my intuition had placed numbers like pi, e and the golden ratio together and … Read more

Complicated series converges to π\pi.

How do I get this result? 426880√10005∑∞k=0(6k)!(545140134k+13591409)(k!)3(3k)!(−640320)3k=π It seems formidable. Context: I came across this when reading a book (in Chinese) on sequences for high school students where the author started to introduce infinite sequences. He listed a few other famous results and then this, as an illustration that this kind of series can be … Read more