the number feared by phythagoreans since it ties halfway between the only two integers that can be both the perimeter and the area fo the same triangle
Might that be 15?
The Pythagorean triangle with sides 5-12-13 has a perimeter of 30 and an area of 30.
The number feared by Pythagoreans is called the square root of 2, often denoted as √2. It is an irrational number, meaning it cannot be expressed as a simple fraction or ratio of two integers. The reason Pythagoreans feared this number is because it represents a fundamental contradiction to their belief that all numbers could be expressed as ratios of integers.
To understand why the square root of 2 is feared by Pythagoreans, let's explore the concept of the perimeter and area of a right triangle. In a right triangle, one side is the hypotenuse (the side opposite the right angle) and the other two sides are the legs. The Pythagorean theorem states that the square of the hypotenuse's length is equal to the sum of the squares of the lengths of the two legs.
Now, let's consider the case where the lengths of both the legs and the hypotenuse are integers. In such a triangle, we can calculate both the perimeter and the area using these lengths.
If we assume that one leg has a length of 1 unit, then the other leg can be any integer greater than 1. Let's say that the other leg has a length of n units. The hypotenuse can then be calculated using the Pythagorean theorem as √(1^2 + n^2), which simplifies to √(1 + n^2).
Now, let's look at the perimeter and area of this triangle. The perimeter is simply the sum of the lengths of the three sides: 1 + n + √(1 + n^2). The area of a right triangle is given by (1/2) * base * height, and in this case, the base and height are the two legs with lengths 1 and n. Therefore, the area is (1/2) * 1 * n = n/2.
To find the values where the perimeter and area are equal, we can set their expressions equal to each other:
1 + n + √(1 + n^2) = n/2
To solve this equation, we need to isolate the square root term. Here's how you can do it:
1. Subtract n from both sides of the equation:
1 + √(1 + n^2) = n/2 - n
2. Combine like terms on the right side:
1 + √(1 + n^2) = -n/2
3. Square both sides of the equation to eliminate the square root:
(1 + √(1 + n^2))^2 = (-n/2)^2
4. Expand and simplify:
1 + 2√(1 + n^2) + 1 + n^2 = n^2/4
5. Cancel out the n^2 terms:
2√(1 + n^2) + 2 = n^2/4
6. Move the 2 to the other side:
2√(1 + n^2) = n^2/4 - 2
7. Simplify the right side:
2√(1 + n^2) = (n^2 - 8)/4
8. Divide both sides by 2:
√(1 + n^2) = (n^2 - 8)/8
9. Square both sides again to remove the square root:
1 + n^2 = (n^2 - 8)^2/64
10. Expand and simplify:
1 + n^2 = (n^4 - 16n^2 + 64)/64
11. Multiply both sides by 64:
64 + 64n^2 = n^4 - 16n^2 + 64
12. Rearrange to form a quadratic equation:
n^4 - 80n^2 + 128 = 0
Now, you can solve this quartic equation to find the values of n that satisfy both the perimeter and area conditions of the triangle. However, solving a quartic equation can be quite complex and involve lengthy calculations. You can use numerical methods or software tools like WolframAlpha to find the approximate solutions.
In summary, the Pythagorean fear of the square root of 2 arises from the fact that it represents an irrational number, which cannot be expressed as a ratio of integers. This number is significant because it breaks the Pythagorean belief that all numbers can be expressed in such ratios. The fear stems from the contradiction it presented to the Pythagorean worldview.