Question

Consider a square whose side-length is one unit. Select any five points from inside this square. Prove that at least two of these points are within \(\frac{\sqrt{2}}{2}\) units of each other.

Short answer

According to the Pigeonhole Principle, since there are 5 points and 4 smaller squares, at least one smaller square contains 2 points. The maximum distance within this smaller square is its diagonal length, which is \(\frac{\sqrt{2}}{2}\). Therefore, at least two of the chosen points are within \(\frac{\sqrt{2}}{2}\) units of each other.

Step-by-step solution

Divide the Square

Divide the given square with side-length one unit into four equal squares. Each of these smaller squares will also have side-length of 0.5 units.

Apply Pigeonhole Principle

Select any five points from inside the large square. According to the Pigeonhole Principle, since there are five points and only four smaller squares, at least one smaller square must contain at least two of these points.

Calculate Distance

The maximum distance between two points in the smaller square will be equal to the length of the diagonal of that square. Using the Pythagorean theorem, the length of the diagonal can be calculated as \(\sqrt{(0.5^2 + 0.5^2)} = \frac{\sqrt{2}}{2}\). Thus, the two points within the smaller square are within \(\frac{\sqrt{2}}{2}\) units of each other.

Key concepts

Understanding Mathematical Proof

A mathematical proof is a logical argument that establishes the truth of a proposition. It is a cornerstone of mathematics that transforms assumptions into conclusions based on agreed-upon rules and definitions. Proofs are imperative because they provide certainty to mathematical claims. In our exercise, the proposition to be proven is that at least two points out of five chosen within a unit square are no more than \(\frac{\sqrt{2}}{2}\) units apart.To prove this, we can't rely on trial-and-error or intuition; we need an irrefutable argument. This exercise uses a classic direct proof technique by applying the Pigeonhole Principle—a concept stating that if you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon. Here, our 'pigeons' are the five points, and the 'pigeonholes' are the four smaller squares. The application of this principle directly leads to the conclusion that two points must be within a certain distance of each other, which is the crux of our proof.

Pythagorean Theorem Explained

The Pythagorean theorem is fundamental in geometry, particularly when discussing the distance between two points in a plane. It states 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 lengths of the other two sides. Symbolically, if 'c' represents the length of the hypoteness and 'a' and 'b' the lengths of the other two sides, the theorem can be expressed as \(c^2 = a^2 + b^2\).

In the context of our square exercise, we utilize the theorem to calculate the maximum possible distance between any two points within a smaller square, which is always the length of the diagonal. Here, as each side of the smaller square measures 0.5 units, the diagonal's length can be deduced by substituting into the Pythagorean theorem, resulting in \(\sqrt{(0.5^2 + 0.5^2)} = \frac{\sqrt{2}}{2}\), which aligns with the proof's requirements.

Interpreting Geometric Distance

Geometric distance, also known as Euclidean distance, is the straight-line distance between two points in Euclidean space. It is the most common way to think about the distance and is easy to visualize. For instance, if you place two dots on a piece of paper, the geometric distance between them is the length of the straight line you would draw to connect the dots.

In our square exercise, we've used geometric distance to ascertain how close the points are to each other. When the exercise asks whether any two points are within \(\frac{\sqrt{2}}{2}\) units of each other, it's calling for a calculation of the straight-line distance between them. Once we find that two points lie in the same smaller square, this maximum distance is the diagonal of the smaller square, which we've calculated using the Pythagorean theorem. This concept is vital in various fields, including physics, engineering, and computer science, as it is the most natural way to quantify how far apart two entities are in space.