Problem 13 - A History of Mathematics: An Introduction

Question

Show that if a right triangle has one leg of length 1 and a hypotenuse of length 2, then the second leg is incommensurable with the first leg. (In modern terms, this is equivalent to showing that \(\sqrt{3}\) is irrational.) Use an argument similar to the proposed Pythagorean argument that the diagonal of a unit square is incommensurable with the side.

Step-by-Step Solution

Answer

Question: Prove that the length of the second leg in a right triangle with a hypotenuse of \(2\) and one leg of length \(1\) is incommensurable with the first leg. Answer: We derived that the length of the second leg is \(\sqrt{3}\) and proved that this length is irrational using a proof by contradiction. As a result, we can conclude that the lengths of the two legs are incommensurable.

Step 1: Set up the Pythagorean theorem equation

Begin by stating the Pythagorean theorem, \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the lengths of the two legs and \(c\) is the length of the hypotenuse. In this problem, we know that \(a = 1\) and \(c = 2\). Plug these values into the Pythagorean theorem to get the equation for the unknown side \(b\): \(1^2 + b^2 = 2^2\).

Step 2: Solve for the length of the second leg

Solve the equation: \(1^2 + b^2 = 2^2\). \(1 + b^2 = 4\) \(b^2 = 3\) \(b = \sqrt{3}\) Now we have found that the length of the second leg is \(\sqrt{3}\).

Step 3: Assume the second leg is a rational number

To prove that the second leg is incommensurable (or that \(\sqrt{3}\) is irrational), first assume that \(\sqrt{3}\) is a rational number. This means it can be expressed as a reduced fraction such that \(b = \frac{p}{q}\), where \(p\) and \(q\) are integers that share no common factors other than 1.

Step 4: Plug in the assumed rational number for \(b\) into our equation

Substitute our assumed rational representation of \(b\) into the equation we derived in step 2: \((\frac{p}{q})^2 = 3\) \(p^2 = 3q^2\) This equation states that \(p^2\) is equal to 3 times some square integer (\(3q^2\)). Therefore, \(p^2\) must be divisible by 3.

Step 5: Show contradiction by examining divisibility by 3

Since we assumed that \(p\) and \(q\) share no common factors other than 1, this means that \(p\) must be divisible by 3 (as its square, \(p^2\), is divisible by 3). Let's replace \(p\) with \(3k\) for some integer \(k\). The equation becomes: \((3k)^2 = 3q^2\) \(9k^2 = 3q^2\) Dividing both sides by 3, we get: \(3k^2 = q^2\) Now, we can see that \(q^2\) is also divisible by 3, which means that \(q\) must also be divisible by 3. This contradicts our original assumption that \(p\) and \(q\) share no common factors other than 1, since they both are divisible by 3.

Step 6: Conclude that the second leg is incommensurable

Since our assumption that the length of the second leg (\(\sqrt{3}\)) is rational led to a contradiction, we can now conclude that the second leg is not a rational number, and is thus incommensurable with the first leg. This shows that \(\sqrt{3}\) is irrational.

Key Concepts

Right Triangle Properties

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. Because of this distinct feature, the right triangle has some special properties that make it crucial in geometry and trigonometry.

In a right triangle, the side opposite the 90-degree angle is called the hypotenuse, which is the longest side. The other two sides are known as the legs of the triangle. A fundamental characteristic of right triangles is that the square of the hypotenuse equals the sum of the squares of the other two sides. This is known as the Pythagorean theorem.

• Hypotenuse: The longest side, opposite the right angle.
• Legs: The two sides forming the right angle.
• Pythagorean Theorem: An essential relation among the sides.

Understanding these basic properties helps solve problems involving right triangles, such as finding unknown side lengths or angles.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction of two integers. In other words, they cannot be written as a ratio (\(rac{p}{q}\), where both \(p\) and \(q\) are integers and \(q e 0\)).

These numbers often appear in contexts where exact division into equal parts is impossible. A classic example of an irrational number is \(\sqrt{3}\), which appears in the current exercise. The decimal representation of \(\sqrt{3}\) is non-terminating and non-repeating, much like \(\pi\) or \(\sqrt{2}\).

This means irrational numbers have infinite decimal places without a regular pattern. Importantly, if a triangle involves these irrational numbers, one property of the triangle might be that the sides are incommensurable, meaning no common measure exists between the two, just like in our problem.

Recognizing whether a number is irrational or not is essential, as it's a foundation upon which many logical deductions, such as proving the incommensurability of triangle sides, rely.

Pythagorean Theorem

The Pythagorean theorem is a crucial concept in geometry that describes the relationship between the sides of right triangles. Simply put, the theorem asserts that for any right triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)): \(a^2 + b^2 = c^2\).

This equation offers a powerful tool for calculating the length of one side of a right triangle if the lengths of the other two sides are known.

Let's break down the application as seen in the original problem:

• We know one leg has a length of 1 and the hypotenuse has a length of 2.
• Plugging these into the Pythagorean theorem, we get \(1^2 + b^2 = 2^2\). Simplifying, \(b^2\) becomes equal to 3.

This process revealed \(\sqrt{3}\) as the length of the second leg, leading to further exploration around its nature as an irrational number.

The Pythagorean theorem is more than just a rule for triangles; it's a cornerstone of mathematical reasoning and problem-solving, extending even beyond geometry into fields like physics, engineering, and computer science.