What is the number feared by Pythagoreans since it lies halfway between the only two integers that can be both the perimeter and the area of the same rectangle?
The number feared by Pythagoreans is known as the square root of 2 (√2). Pythagoreans were a group of ancient Greek mathematicians who believed in the power of numbers and their mystical properties.
To understand why √2 was feared by Pythagoreans, we need to recall some basic properties of a rectangle. The perimeter of a rectangle is the sum of all its sides, while the area is the product of its length and width.
Let's say the length and width of the rectangle are both integers. In that case, the perimeter would be the sum of these two integers multiplied by 2. Similarly, the area would be the product of these two integers.
Now, let's assume there exists a rectangle where both the perimeter and the area are integers. This means that the sum of the length and width, multiplied by 2, is an integer, and their product is also an integer.
If we consider the length and width to be x and y, respectively, we can write two equations based on the above conditions:
2(x + y) = integer equation 1
xy = integer equation 2
The Pythagoreans believed in the beauty and perfection of whole numbers and ratios. Therefore, they focused on finding rectangles where the length and width are both integers. In such cases, they could easily find both the perimeter and area as whole numbers.
However, there is no pair of integers that fulfills both equations simultaneously except for x = y = 1, which corresponds to a square. Any other combination of integers for x and y will lead to one of these two equations being violated. This is where the square root of 2 (√2) comes into play.
The Pythagoreans discovered that the ratio of the diagonal to the side length of a square is equal to √2. In a square with sides of length 1, the diagonal is √2. Since √2 is an irrational number, it cannot be expressed as a fraction or a ratio of two integers.
This finding was considered a threat to the Pythagoreans' belief in the perfection of whole numbers. They regarded √2 as an imperfect and "unholy" number because it lies between two integers and cannot be expressed exactly.
Hence, the square root of 2 (√2) was feared by Pythagoreans since it represents a departure from their ideal of perfect whole numbers and reveals the existence of irrational numbers.