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 October 27th, 2018, 12:09 PM #1 Senior Member   Joined: Nov 2011 Posts: 247 Thanks: 3 e^n Is the next sentence true or false: Every e^(number) equations are transcendental equation.
 October 27th, 2018, 12:19 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,264 Thanks: 1198 I wouldn't say that $e^0 = 1$ is transcendental. Thanks from shaharhada
October 27th, 2018, 12:32 PM   #3
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Quote:
 Originally Posted by romsek I wouldn't say that $e^0 = 1$ is transcendental.
O.K.
thanks.

is this sentence true/false:
every e^(complex number) is transcendental?

October 27th, 2018, 12:33 PM   #4
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Quote:
 Originally Posted by shaharhada O.K. thanks. is this sentence true/false: every e^(complex number) is transcendental?
How dare you ask this question right after quoting and thanking his post that you obviously didn't read????

October 27th, 2018, 12:47 PM   #5
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Quote:
 Originally Posted by Micrm@ss How dare you ask this question right after quoting and thanking his post that you obviously didn't read????
Sorry, but I don't understand.
The first question in about number that I don't understand if there is counterexample to the sentence...
And in second question I ask about particular thing, the complex number.

The second question is about all the complex number and my question is about counterexample of it.
So why question is:
Why it can't be a counterexample in the second question?
Is There a reason to it?

 October 27th, 2018, 01:02 PM #6 Senior Member     Joined: Sep 2015 From: USA Posts: 2,264 Thanks: 1198 $e^{\ln(k)} = k,~\forall k \in \mathbb{Z},~k>0$ Do you consider this transcendental? Certainly $k$ isn't. $\ln(k)$ however probably is
October 27th, 2018, 01:43 PM   #7
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Quote:
 Originally Posted by romsek $e^{\ln(k)} = k,~\forall k \in \mathbb{Z},~k>0$ Do you consider this transcendental? Certainly $k$ isn't. $\ln(k)$ however probably is
Is there example of counterexample to algebraic solution to this equation?

October 27th, 2018, 01:45 PM   #8
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Quote:
 Originally Posted by shaharhada In the second question I ask about complex numbers. Is there example of counterexample to algebraic solution to this equation?
ln(k) is a complex number.

October 27th, 2018, 02:02 PM   #9
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Quote:
 Originally Posted by shaharhada In the second question I ask about complex numbers. Is there example of counterexample to algebraic solution to this equation?
I really have no idea what you are asking here.

October 27th, 2018, 05:12 PM   #10
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Quote:
 Originally Posted by shaharhada Is the next sentence true or false: Every e^(number) equations are transcendental equation.
There's a little confusion and ambiguity in your question.

First, if $n$ is a number -- integer, real, or complex -- then $e^n$ is another number. It's not an equation. So it might or might not be a transcendental number, but it's always one single, particular number. Not an equation.

We could ask when $e^n$ is transcendental. It's pretty clear that if $n$ is an integer, $e^n$ is transcendental if and only if $n$ is nonzero.

If $n$ is real, $e^n$ is transcendental unless we specifically select $n$ so it's not. For example if $n$ = $\ln 2$, then $e^n = 2$, which is not transcendental. Of course I haven't given or thought about the exact criteria by which we can know if a real exponent leads to a transcendental number. But you can always use a natural log to undo raising $e$ to a power. And vice versa. That's the most important thing to know about log and exp. They undo each other.

If $n$ is complex it's much the same. Some specifically cook-up values will give you non-transcendentals (aka algebraics) and "random" complex numbers will generally lead to transcendental complex numbers.

So that's one problem I have with your question. The way you phrased it, it refers to numbers and not equations.

Secondly, what is a transcendental equation? Do you mean it in the sense of https://en.wikipedia.org/wiki/Transcendental_equation? Or do you mean $e^z$ as a function?

So your question is ambiguous in a couple of ways and that's perhaps why you are not happy with the responses you're getting.

Last edited by Maschke; October 27th, 2018 at 05:14 PM.

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