Question

Suppose that a line of length 1 is divided in extreme and mean ratio, that is, that the line is divided at \(x\) so that \(\frac{1}{x}=\) \(\frac{x}{x-1}\). Show by the method of the Euclidean algorithm that 1 and \(x\) are incommensurable. In fact, show that \(1: x\) can be expressed using Theaetetus's definition as \((1,1,1, \ldots)\).

Short answer

Question: Prove that the ratio 1:x, where x is divided in extreme and mean ratio, is incommensurable and can be expressed using Theaetetus's definition as (1,1,1, ...). Solution: By solving the given equation, we show that the roots of the quadratic equation have no rational solutions. With the aid of the Euclidean algorithm, we prove that 1 and x have no common measure, so they are incommensurable. We then express the ratio 1:x using Theaetetus's definition as (1,1,1, ...).

Step-by-step solution

Solve the given equation

We are given the equation \(\frac{1}{x} = \frac{x}{x-1}\). Cross multiply to remove the fractions: \(x^2 = x - 1\) Now, rearrange the equation: \(x^2 - x + 1 = 0\) We can notice that this is a quadratic equation.

Find the roots of the quadratic equation

Since it's a quadratic equation, we can try to find its roots (i.e., values of x for which the equation is satisfied). The quadratic formula is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) For our equation, \(a = 1\), \(b = -1\), and \(c = 1\). Plugging in these values, we have: \(x = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}\) Since the discriminant, \(b^2 - 4ac\), is negative, there are no rational roots for this equation.

Apply the Euclidean algorithm

Now, we'll prove that 1 and x are incommensurable using the Euclidean algorithm. Suppose that they do have a common measure m. Then: \(x = m \cdot p\) and \(1 = m \cdot q\) with \(p\) and \(q\) being integers. Now we have: \(\frac{1}{x} = \frac{m \cdot q}{m \cdot p}\) Using the given equation, we have: \(\frac{m \cdot q}{m \cdot p} = \frac{m \cdot p}{(m \cdot p) - m \cdot q}\) By canceling terms \((m \cdot q)\) from both sides, we get: \(\frac{1}{p} = \frac{p}{p - q}\) We get back to the same format of the given equation \(\frac{1}{x} = \frac{x}{x-1}\). Since the algorithm will never stop, we can conclude that 1 and x are incommensurable, meaning they have no common measure.

Express the ratio using Theaetetus's definition

Since 1 and x are incommensurable and their ratio is the same form as the equation \(\frac{1}{x}=\frac{x}{x-1}\), we can express the ratio 1:x using Theaetetus's definition as: (1,1,1, ...) Thus, we have successfully shown that the ratio 1:x is incommensurable and can be expressed using Theaetetus's definition as (1,1,1, ...).

Key concepts

Extreme and Mean Ratio

The extreme and mean ratio, also known as the golden ratio, is an irrational number often encountered in geometry, art, and architecture. It is symbolized by the Greek letter phi (\f\(\f\)). In the context of our exercise, a line of length 1 is divided at a point \f\(x\f\) such that the proportion of the entire line to the larger segment is the same as the proportion of the larger segment to the smaller one. In mathematical terms, this is represented as \f\(\frac{1}{x} = \frac{x}{1 - x}\f\). This ratio is found by solving a specific quadratic equation as seen in the solution steps.

It's vital for students to appreciate how the extreme and mean ratio ties into broader mathematical concepts, with its enduring presence throughout history providing noteworthy examples of its aesthetic and functional properties.

Euclidean Algorithm

The Euclidean algorithm is a time-honored procedure for finding the greatest common divisor (GCD) of two numbers. It relies on the principle that the GCD of two numbers also divides their difference. In the exercise, this algorithm helps prove that 1 and \f\(x\f\) are incommensurable, or have no common measure, by showing a continued process without reaching an integer solution. It's a cornerstone concept in number theory, predating even the quadratic equation, and understanding it provides students with a glimpse into the logical structure that underpins much of modern mathematics. Moreover, it emphasizes the recursive nature of mathematical algorithms, which have applications far beyond theoretical problems—extending to computing and various algorithmic designs.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \f\(ax^2 + bx + c = 0\f\). It can yield zero, one, or two real roots, depending on the discriminant \f\((b^2 - 4ac)\f\). In the given exercise, the quadratic equation \f\(x^2 - x + 1 = 0\f\) is solved using the quadratic formula. The solution displays that the absence of rational roots implies the two numbers in question, 1 and \f\(x\f\), are indeed incommensurable numbers. It's an example of how quadratic equations are not only pivotal in algebra but also serve as foundational tools for exploring ratios, proportions, and the nature of numbers.

Theaetetus's Definition

Theaetetus's definition of incommensurable lengths involves sequences of integers. If two lengths are incommensurable, we can keep finding integers that get us 'closer' to their ratio but never exactly there. In our textbook problem, the ratio of 1 to \f\(x\f\) can be expressed as an infinite sequence of 1's. This concept brings attention to the abstract nature of irrational numbers and the infinite processes that can define them. The interplay between clear, definable numbers and these more conceptual infinite sequences affords students insights into the depths of number theory and the foundations of real analysis.