Understanding the Roots of Cubic Equations and the Nature of e

The exploration of cubic equations often involves questioning whether certain irrational numbers, such as e, can serve as roots to such equations. However, these investigations often lead to fascinating insights into the nature of transcendental numbers and algebraic numbers.

The Myth of e as a Root

The first problem is that it is NOT the root of that equation. The given cubic equation can be rewritten as:

x^3 a / b

If e was the root of this cubic equation, then e would indeed not be transcendental. This is because the root of a polynomial is an algebraic number, and transcendental numbers are those which are not roots of any non-zero polynomial with rational coefficients.

Proof by Approximation

Let's see if e is the root of the given cubic equation. We have:

Therefore, the equation becomes:

e^3 b / a

Using a calculator, we evaluate both sides: e^3 20.0855369232 b/a 20.0855369171

As you can see, while these values are very close, they are not identical. Therefore, e is not the solution of this cubic equation.

b/a is a very good approximation of e^3, but only an approximation and not identical to e^3.

Transcendental Number Insights

The same principle applies to π. Even though π satisfies the equation 25x^2 - 936 0, it does not fulfill the equation exactly. This highlights the distinction between algebraic and transcendental numbers.

The Nature of e and Other Irrational Numbers

If e were the root of a cubic equation, it would not be transcendental. As it is known, e is an irrational and transcendental number. The exact solution of the given cubic equation is:

x 271801 / 99990

Here, x 2.71828182818281828… while e 2.7182818284590…. The difference between x and e is x - e 2.7 * 10^{-10}, which is a minute difference but significant for precise evaluations.

Further Exploration

For those interested in exploring more closely: find 4 integers a, b, c, d in mathbb{Z} such that their absolute value is less than 1000 and such that the cubic equation ax^3 bx^2 cx d 0 has a root x_0 with x_0 - e ≤ 10^{-12}. One example close to solving this is:

81x^3 - 188x^2 - 983x - 344 0

One of the roots of this equation is close to e; the error is 1.04 * 10^{-12}.

Conclusion

It is safe to say that e is not a root of any cubic equation. It is a good approximation but not a root of a rational number. Roots of rational numbers are algebraic numbers, whereas transcendental numbers, like e, are not roots of any non-zero polynomial with rational coefficients.