Transcendental Numbers Explained: Multiplication by Algebraic and Rational Numbers

Transcendental numbers are fascinating entities in mathematics, defined by their non-algebraic nature. This article delves into the properties of transcendental numbers, particularly focusing on how they behave when multiplied by algebraic and rational numbers. Understanding these interactions is crucial for mathematicians and students alike, as it sheds light on the fundamental characteristics of these unique numbers.

1. Introduction to Transcendental Numbers

A transcendental number is a real or complex number that is not algebraic, meaning it is not the root of any non-zero polynomial equation with rational coefficients. Some well-known examples include e (Euler's number) and π (pi).

2. Multiplication with Nonzero Algebraic Numbers

When a transcendental number is multiplied by a nonzero algebraic number, the result remains transcendental. For instance:

e is transcendental. 1/e is also transcendental (since the reciprocal of a transcendental number is still transcendental). Multiplying e by 1/e yields 1, which is not transcendental. Multiplying e by 1 results in the transcendental number e again.

The key point here is that no general statement can be made across all cases involving transcendental and algebraic numbers due to the non-algebraic nature of transcendental numbers.

3. Multiplication with Another Transcendental Number

Multiplication of two transcendental numbers may or may not yield a transcendental result. For example:

e multiplied by 1/e equals 1, which is an integer. This demonstrates that the product of two transcendental numbers can be a rational number or algebraic number.

4. Property of Transcendental Numbers Multiplied by Non-Zero Rational Numbers

A fundamental property of transcendental numbers states that multiplying a transcendental number by a non-zero rational number still results in a transcendental number. To prove this, let us consider the following argument:

4.1 Proof by Contradiction

Assume that our claim is false, meaning there exists a transcendental number x and a non-zero rational number r such that rx is not transcendental. This implies that rx is algebraic. Therefore, there exists a non-zero polynomial p with integer (or equivalently rational) coefficients such that prx 0.

We can express prx 0 as pt 0, where t rx. Since prx 0, we know that pt 0 holds for t, which is a polynomial with rational coefficients.

Assume pt sum_{i0}^{n} a_it^i, where a_i are integers and n is a positive integer. Since prx 0, there must exist some a_i neq 0 for some i.

Let q(x) prx. We can rewrite q(x) sum_{i0}^{n} a_ir^ix^i. Since a_ir^i neq 0 and r neq 0, q(x) 0 implies that x is a root of the polynomial q, which contradicts the fact that x is transcendental.

Therefore, our initial assumption must be false, and we conclude that x multiplied by any non-zero rational number r remains a transcendental number.

5. Conclusion

This article has highlighted the unique properties of transcendental numbers, particularly focusing on their behavior when multiplied by algebraic and rational numbers. Understanding these properties not only enriches our knowledge of number theory but also provides a deeper insight into the nature of numbers in mathematics.

6. Keywords

transcendental number, algebraic number, rational number