The Non-Zero Real Numbers in Algebraic Equations

Exploring algebraic equations and finding non-zero real numbers can often involve intricate mathematical operations. With modern computer algebra systems like Mathematica, solving such equations becomes more feasible and straightforward. In this article, we will delve into a specific system of equations and analyze the non-zero real numbers that the equations yield. We will also introduce a method for explaining the relationship between these numbers.

Using Mathematica to Solve the Equations

Consider the system of equations given by:

a b cab c b c abc a c a bca b

These equations can be solved using Mathematica, which employs symbolic computation to provide precise results. The inputs and outputs from Mathematica are as follows:

Mathematica Code

Input: Solve[{a*b  c, b*c  a, c*a  b}, {a, b, c}] // FullSimplify

Output

{{a - 0, b - 0, c - 0}, {a - Root[-3   3*x^2 - 3*x^3, 1], b - Root[-3   3*x^2 - 3*x^3, 2], c - Root[-3   3*x^2 - 3*x^3, 3]}, {a - Root[-3   3*x^2 - 3*x^3, 2], b - Root[-3   3*x^2 - 3*x^3, 3], c - Root[-3   3*x^2 - 3*x^3, 1]}, {a - Root[-3   3*x^2 - 3*x^3, 3], b - Root[-3   3*x^2 - 3*x^3, 1], c - Root[-3   3*x^2 - 3*x^3, 2]}}

The numerical output of the above code is:

{{a - 0.0, b - 0.0, c - 0.0}, {a - -2.53209, b - -1.3473, c - 0.879385}, {a - -1.3473, b - 0.879385, c - -2.53209}, {a - 0.879385, b - -2.53209, c - -1.3473}}

The results show that the non-zero solutions for (a), (b), and (c) are the roots of the cubic polynomial (x^3 - 3x^2 - 3 0). This polynomial has three roots:

(a approx -2.53209) (b approx -1.3473) (c approx 0.879385)

The roots can be rearranged in different cyclic permutations, such that each element moves to the position of the next element and the last element moves to the position of the first. This indicates that the splitting field of the polynomial is (mathbb{Q}(alpha)), where (alpha) is one of the roots of the polynomial.

Explaining the Relationship Between the Roots

Alon provided another method for explaining the relationship between the roots. For instance, we can define:

(a [0, 1, 7, 3, 2, 3, 1, 1, 6, 11, 1, 3, 2, 1, 9, 36, 1, 1]) (b -[2, 1, 1, 7, 3, 2, 3, 1, 1, 6, 11, 1, 3, 2, 1, 9, 36, 1, 1]) (c -[1, 2, 1, 7, 3, 2, 3, 1, 1, 6, 11, 1, 3, 2, 1, 9, 36, 1, 1])

These sequences demonstrate that (a, b,) and (c) differ only in the first or second terms. Through the use of continued fractions, we can express (b) and (c) in relation to (a):

(b -2a frac{3}{a} 1) (c -a frac{3}{a} 2)

These expressions verify that the sum, product, and sum of pairs of the roots (a, b,) and (c) are consistent with the original system of equations. This method provides a clear and concise way to understand the interrelation of the roots without involving complex computations.

Conclusion

By utilizing Mathematica and exploring the properties of roots in algebraic equations, we can efficiently determine non-zero real numbers that satisfy given conditions. The methodology illustrated in this article not only simplifies the process of solving such equations but also deepens our understanding of the underlying mathematical structures. The relationship between the roots and the continued fractions offer an interesting perspective on solving algebraic systems and provide a foundation for further educational exploration.