last change: 2022-08-07

# Transcendental Numbers

Just some notes.

## Relation of Number Sets

The illustration works for both, β or β. Depending on the choice, πΈ is either the real algebraic numbers or the complex algebraic numbers, and likewise the same is true for π and π. Not affected by this choice is β, which is always the set of just the quotients of integers β€, which are also not affected.

β€: integer numbers

β: rational numbers, i.e. quotients

β: real numbers

β: complex numbers

πΈ: algebraic numbers

π := β\β (or β\β): irrational numbers (not a ring)

π := β\πΈ (or β\πΈ): transcendental numbers (not a ring)

Note that:

β β πΈ β β

π β π β β

πΈ and π intersect, but neither is a subset of the other.

There are numbers which are both irrational and algebraic, but some algebraic numbers are not irrational and some irrational numbers are not algebraic.

Examples:

β(2) β πΈ and β(2) β π (β(2) β π)

1 β πΈ, but 1 β π

Ο β π, but Ο β πΈ (Ο β π)

**Lemma 0**:

log_{b}a is irrational for a,b β β with a β b, b β 1.

Proof.

Let a, b be integers with a β b, b β 1. Then βi,j β β we have aβ± β bΚ² by the fundamental theorem of arithmetic.

And then

aβ± β bΚ²

β (aβ±)^{1/i} β (bΚ²)^{1/i}

β a^{i/i} β b^{j/i}

β aΒΉ β b^{j/i}

β log_{b}a β log_{b}(b^{j/i})

β log_{b}a β j/i (βi,j β β)

β βi,j β β: log_{b}a = j/i

β log_{b}a β β . β

## Gelfond-Schneider Theorem

**Theorem 1** (Gelfond-Schneider theorem):

If b β πΈβ{0,1}, p β πβ©πΈ, then bα΅ is transcendental.

**Corollary 1.1a** (Contrapositive A of the Gelfond-Schneider theorem):

If tα΅ is algebraic, p β πβ©πΈ, and t β {0,1}, then t is transcendental.

Proof.

Let tα΅ algebraic, p β πβ©πΈ, and t β {0,1}. Then tα΅ is not transcendental. With the contrapositive of theorem 1 it follows that t β πΈβ{0,1} or p β πβ©πΈ. Because we know that by the hypothesis p β πβ©πΈ, it follows that t β πΈβ{0,1}. We also know that t β {0,1}, so t β πΈβ{0,1} βͺ {0,1} β t β πΈ β t β π. β

**Corollary 1.1b** (Contrapositive B of the Gelfond-Schneider theorem):

If bα΅ is algebraic and b β πΈ\{0,1}, then p β πβ©πΈ.

Proof.

Let bα΅ algebraic and b β πΈβ{0,1}. Then bα΅ is not transcendental. With the contrapositive of theorem 1 it follows that b β πΈβ{0,1} or p β πβ©πΈ. Because we know by the hypotheis that b β πΈβ{0,1}, it follows that p β πβ©πΈ. β

**Corollary 1.2**:

log_{b}a is transcendental for a,b β β with a β b, b > 1.

Proof.

Let a,b β β with a β b, b > 1. Then b^{logba} = a is algebraic. With b^{logba} algebraic and algebraic b β {0,1} it follows with corollary 1.1b that log_{b}a β πβ©πΈ. For a proof by contradiction, assume log_{b}a β πΈ. From lemma 0 we know that log_{b}a β π. With log_{b}a β π, β πΈ it follows that log_{b}a β πβ©πΈ, which is a contradiction, because we already know that log_{b}a β πβ©πΈ, so the assumption is false and it follows that log_{b}a β πΈ β log_{b}a β π. β

**Corollary 1.3**:

e^Ο is transcendental, because

e^Ο

= e^(-Ο Β· -1)

= e^(-Ο Β· iΒ²)

= e^(iΟ Β· -i)

= (e^(iΟ))β»β±

= (-1)β»β±, which is transcendental by the Gelfond-Schneider theorem, because -1 is algebraic and -i is irrational.

## Hermite-Lindemann Theorem (aka. Lindemann-WeierstraΓ Theorem)

**Theorem 2** (Hermite-Lindemann theorem, aka. Lindemann-WeierstraΓ theorem):

if Ξ±β, ..., Ξ±β are algebraic numbers that are linearly independent over the rational numbers β, then e^{Ξ±β}, ..., e^{Ξ±β} are algebraically independent over β.

**Corollary 2.1** (Corollary to the Hermite-Lindemann theorem):

If πΌ β πΈ\{0}, then e^{πΌ} is transcendental.

Proof.

Let πΌ β πΈ\{0}. Then {πΌ} is a linearly independent set over the rationals. By the Hermite-Lindemann theorem, {e^{πΌ}} is an algebraically independent set, or in other words e^{πΌ} is transcendental. β

**Corollary 2.2**:

e is transcendental, because 1 is algebraic, so by corollary 2.1 eΒΉ is transcendental.

**Corollary 2.3** (Contrapositive to corollary 2.1):

If eα΅ is algebraic and t β 0, then t is transcendental.

Proof:

Let t β 0 and for a proof by contradiction, assume that t is not transcendental, i.e. t is algebraic, then by corollary 2.1 eα΅ is transcendental. That's a contradiction, so the assumption cannot be true, i.e. t is transcendental. β

**Corollary 2.4**:

The natural logarithm of any algebraic number πΌ β {0, 1} is transcendental, because e^{log(πΌ)} = πΌ is algebraic, so by corollary 2.3, the exponent is transcendental.

That means that all of log(2), log(3), log(4), log(5/7), and log(-7/45) are transcendental.