Pythagorean Theorem: Unexpected Finding

We get 2n+1, which follows the pattern of odd natural numbers. This can be further generalized to any combination of numbers for the Pythagorean theorem, showing that the difference between successive equations will always be the next odd natural number. This can be expressed as "1^2+n^2=\sqrt{(1^2+(n-1)^2)}+\sqrt{(2n+1)}". Further exploration is needed to fully understand this pattern.
  • #1
Angel11
11
0
Hello, today i was playing around with the pythagorean theorem and found out something that i can't really explaing or atleast explain it with probably a false answer. So i was putting every possible combination with the max digit of 10. For example 1^2+1^2=\sqrt{2}, 1^2+2^2=\sqrt{5}... 10^2+1^2=\sqrt{101},10^2+2^2=\sqrt{104}. And then i found out that the square roots of the resolts are increasing by 3,5,7,9,11 and then i found out that those number are increasing by 2. so something like "1^2+2^2=\sqrt{(1^2+1^2)}+\sqrt{3} and then 1^2+3^2=\sqrt{(1^2+2^2)}+\sqrt{5}" so any idea how it works? my guess is WAY off i thought about it more and it is awful so i would appreciate anyones response.
 
Mathematics news on Phys.org
  • #2
If I understand correctly, you have observed for example:

\(\displaystyle 1^2+0^2=1\)

\(\displaystyle 1^2+1^2=2\)

\(\displaystyle 1^2+2^2=5\)

\(\displaystyle 1^2+3^2=10\)

And you've seen that the difference between successive equations are the sequence of odd natural numbers. Let's look at the difference between two successive equations in general:

\(\displaystyle 1^2+n^2=n^2+1\)

\(\displaystyle 1^2+(n+1)^2=n^2+2n+2\)

What do we get when we subtract the former from the latter?
 

1. What is the Pythagorean Theorem?

The Pythagorean Theorem is a mathematical formula that states the relationship between the three sides of a right triangle. It states that 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.

2. Who discovered the Pythagorean Theorem?

The Pythagorean Theorem is named after the ancient Greek mathematician, Pythagoras, who is credited with discovering it. However, there is evidence that this theorem was known and used by other ancient civilizations before Pythagoras.

3. How is the Pythagorean Theorem used in real life?

The Pythagorean Theorem has many practical applications, such as in construction, engineering, and navigation. It can be used to calculate distances, determine the height of a building, and create right angles.

4. What is the unexpected finding related to the Pythagorean Theorem?

The unexpected finding related to the Pythagorean Theorem is that it can also be applied to non-right triangles, as long as the triangle can be divided into two right triangles. This discovery was made by the ancient Greek mathematician, Hippasus, and is known as the "Hippasus Theorem".

5. Can the Pythagorean Theorem be proved?

Yes, the Pythagorean Theorem can be proved using different methods, such as geometric proofs, algebraic proofs, and even using advanced mathematical concepts like calculus. This theorem has been proven and used for centuries, and is considered one of the most fundamental principles in mathematics.

Similar threads

  • General Math
Replies
2
Views
766
  • General Math
Replies
8
Views
2K
  • General Math
Replies
2
Views
739
  • General Math
Replies
3
Views
228
  • Calculus and Beyond Homework Help
Replies
10
Views
286
  • Special and General Relativity
3
Replies
73
Views
2K
Replies
7
Views
798
Replies
7
Views
1K
  • General Math
Replies
8
Views
1K
Replies
12
Views
2K
Back
Top