Question

Show that the side and diagonal of a square are incommensurable by using the method of anthyphairesis. Show that the ratio \(d: s\) can be expressed using Theaetetus's definition. as \((1,2,2,2, \ldots)\). Hint: Draw the diagonal of the square; then cut off on it the side and draw a square on the remaining segment.

Short answer

Answer: By performing the steps mentioned in the provided solution, we can apply Theaetetus's definition to identify that the ratio of the diagonal to the side of a square can be expressed as (1,2,2,2,…) in an anthyphairesis progression. The self-similarity of the ratio between the diagonal and side of the newly formed squares as we continue this process indefinitely shows that the side and diagonal of a square are incommensurable.

Step-by-step solution

Draw a square and its diagonal

First, draw a square with a side length equal to s. Then, draw the diagonal of the square which has a length equal to d.

Cut off the side of the square

Draw a line from one vertex of the square, parallel to the opposite side, and cutting the diagonal at a point where the length from the vertex equals s (the side of the square). Let's label this new point A.

Form a new square

Now, draw a new square with side length equal to the remaining segment of the diagonal after cutting off s (the length from point A to the farthest vertex of the square). Label the length of this segment as x.

Apply the Pythagorean Theorem

Since the original square had side length s and the diagonal is d, we can use the Pythagorean theorem for a right triangle: \(d^2 = s^2 + s^2 = 2s^2\). Similarly, for the new square with side length x, its diagonal will also be a part of the original diagonal with the length equal to s. So, the Pythagorean theorem for this new square will be: \(s^2 = x^2 + x^2 = 2x^2\).

Calculate the ratio of diagonal to side using anthyphairesis

Divide \(d^2\) by \(s^2\): \(\frac{d^2}{s^2} = \frac{2s^2}{s^2}\) which simplifies to \(\frac{d^2}{s^2} = 2\). Divide both sides by 2: \(\frac{d}{s} = \sqrt{2}\).

Apply Theaetetus's definition

Using Theaetetus's definition, the ratio between consecutive segments in an anthyphairesis progression is \((1,2,2,2,\ldots)\). We can see that \(s\) and \(d\) are in a geometrical progression, since \(d = s\cdot\sqrt{2}\). Further, observe that the ratio of \(x\), the side of the new square, to \(s\), satisfies the same relationship: \(s = x\cdot\sqrt{2}\). This means that the problem reduces down to a self-similar problem, where the new square and the original square have the same ratio between their diagonal and side. Consequently, we can continue to apply Theaetetus's definition indefinitely, iterating through squares of progressively smaller side lengths, while maintaining the ratio between them as \((1,2,2,2,\ldots)\). In conclusion, we have shown that the side and diagonal of a square are incommensurable using the method of anthyphairesis, and that the ratio \(d:s\) can be expressed using Theaetetus's definition as \((1,2,2,2,\ldots)\).

Key concepts

Anthyphairesis

The concept of anthyphairesis, also known as the Euclidean algorithm for real numbers, is a process used to find the common measure between two incommensurable quantities by successively subtracting the smaller quantity from the larger one.

In the context of our exercise, anthyphairesis is applied to show the incommensurability between the side and diagonal of a square. By continuously subtracting the side length of the square from the diagonal, we engage in an infinite process that reveals there is no finite measure that is a common divisor for both lengths. This method reflects an ancient mathematical understanding of irrational numbers, even before they were formally defined as such.

The step-by-step solution of our exercise has us perform a geometric version of anthyphairesis by constructing a sequence of smaller and smaller squares whose sides and diagonals reflect this endless subtraction process. This visual and tactile approach helps students grasp the concept of incommensurability beyond an abstract numerical procedure.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, stating that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In our exercise, we use the Pythagorean Theorem to determine that the diagonal of a square, when squared, is double the side squared (\(d^2 = 2s^2\)). This relationship is essential to understanding the incommensurability of the side and diagonal of the square, as it sets the stage for the subsequent step of anthyphairesis. By applying the theorem, we've effectively begun to compare the side and diagonal, eventually to discover that their ratio cannot be expressed as a ratio of two integers, thus proving them incommensurable.

Theaetetus's Definition

Theaetetus's definition of incommensurability relates to his study of irrational numbers. He defined that two lengths are incommensurable if their ratio is not a ratio of whole numbers.

In our exercise, Theaetetus's definition plays a crucial part in expressing the relationship between the side and diagonal of a square as \((1,2,2,2,\textellipsis\)). This expression does not present a definitive pair of integers, but rather an infinite sequence, indicating the impossibility of finding a common unit that evenly measures both the side and diagonal. The definition provides a solid theoretical backing to the practical demonstration given through the geometric construction of the problem.

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

In the context of our exercise, we observe that the ratio between the side and diagonal of subsequent squares constructed by the method of anthyphairesis forms a geometric progression. Going through the steps of the solution, we note that each new side length and diagonal can be described as being multiplied by the square root of 2. This insight not only confirms the unending nature of the process—characteristic of the incommensurability of the side and diagonal—but also underscores the self-similar pattern that emerges from the geometric progression, thus linking Theaetetus's definition to the concrete geometric construction of the squares.