Can complex numbers be produced using only algebraic operations?

[1MileCrash]

Can you come to a transcendental number through operations involving non-transcendental numbers?

Or is it impossible as I presume?

 

[micromass]

What do you consider "an operation"? I can take the arcsine of 1 and it would get me a transcendental number...

 

[1MileCrash]

I consider arcsin a legitimate operation, but not one that doesn't involve transcendental numbers in your case. The transcendental number in that case is hidden within the definition of the radian. If a circle is 2pi radians, we're basing everything off of a transcendental number.

arcsin(1) is a transcendental number in radians, and an integer (90) in degrees.

Radians is just as valid of a measurement system, but the fact is that radians is based off of transcendental numbers. A (non-zero) integer in degrees is always a transcendental number in radians because we involve pi, a transcendental number.

So arcsin(1) = 90, or 1.5700705 (whatever) radians, and they are equally valid, but the radian measurement, by definition, involves transcendental numbers. So the transcendental number in the operation arcsin(1) is hidden in the definition of the radian.

 

[pmsrw3]

Well, I would disagree with that. You don't HAVE to incorporate pi into the definition of arcsin. For instance, you can define sin with a power series with rational coefficients, then say arcsin is the inverse of that. But how about this:

$$\int_1^2 \frac{dx}{x}$$

Is integration a "legitimate operation"?

How about "find x such that 2^x = -1"? Is that a "legitimate operation"?

 

[SteveL27]

Can you come to a transcendental number through operations involving non-transcendental numbers?

Or is it impossible as I presume?

What counts as an "operation?" Certainly if you are allowed to form an infinite series, you can express any transcendental as a sum of rationals, by using the number's decimal expansion. For example pi = 3 + 1/10 + 4/100 + ...

 

[micromass]

And what about $2^{\sqrt{2}}$?? This is transcendental... See the Gelfond-Schneider theorem.

 

[KingNothing]

Can you come to a transcendental number through operations involving non-transcendental numbers?

I remember reading somewhere that a transcendental number may not be computed using any finite number of algebraic operations.

 

[1MileCrash]

And what about $2^{\sqrt{2}}$?? This is transcendental... See the Gelfond-Schneider theorem.

:)

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?

A transcendental number may not be computed using any finite number of algebraic operations.

What about above?

 

[gb7nash]

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]

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?

I meant mainly though algebraic means, addition, multiplication, powers, and all of their inverses.

 

[micromass]

:)

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?

Gelfond-Schneider says that any algebraic number raised to an irrational algebraic number is transcendental.

 

[KingNothing]

I meant mainly though algebraic means, addition, multiplication, powers, and all of their inverses.

Using powers, you can. But not with a finite number multiplication/division/addition/subtraction operations.

 

[pmsrw3]

No, not a finite number of them.

Hmm. What about micromass's answer: $2^{2^\frac{1}{2}}$?

That looks like a finite combination of the allowed operations.

 

[I like Serena]

Nice! I didn't know about Gelfond-Schneider yet.

So perhaps we will only allow powers to integers?

 

[KingNothing]

Hmm. What about micromass's answer: $2^{2^\frac{1}{2}}$?

That looks like a finite combination of the allowed operations.

EDIT: Sorry, I didn't realize he said powers! My bad!

Using only addition, subtraction, multiplication and division, it is not possible with a finite number of these operations.

 

[Hurkyl]

I remember reading somewhere that a transcendental number may not be computed using any finite number of algebraic operations.

• 1 is algebraic
• The sum of two algebraic numbers is algebraic
• The product of two algebraic numbers is algebraic
• The roots of a polynomial equation in one variable with algebraic coefficients are algebraic

Every algebraic number can be produced by the above observations.

(note that one can prove the difference of algebraic numbers is algebraic, and similarly quotients and powers with rational exponents)

(A complex number is transcendental if and only if it is not algebraic)