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Formula of Pythagoras

The formula of Pythagoras is

 


Explanation

The Greeks called odd numbers gnomons. A gnomon is a hook of a sundial.

6
5
4
3
2
1
1 3 5 7 9 11

The hooks contain odd numbers, represented by dots. Herein are the relations

12 = 1
22 = 1 + 3
32 = 1 + 3 + 5
42 = 1 + 3 + 5 + 7
52 = 1 + 3 + 5 + 7 + 9
62 = 1 + 3 + 5 + 7 + 9 + 11

In general, this can be written as

Each odd number appears as the difference of two squares

1 = 12 - 02
3 = 22 - 12
5 = 32 - 22
7 = 42 - 32
9 = 52 - 42
11 = 62 - 52

The following applies in general

You can also write for this

In this case, the odd number (2n + 1) itself can be the square of another odd number g. Then generally applies (2n + 1) = g2, so also n = (g2 – 1) / 2, and that gives the formula of Pythagoras

Here you can enter any odd number greater than 1 for g.

52 =  42 +  32
132 = 122 +  52
352 = 242 +  72
412 = 402 +  92
612 = 602 + 112
852 = 842 + 132

All multiples of 2, 3, and 4 are omitted, because in developing the formula we assumed the development of two squares.

 


Pythagorean theorem

This is the Pythagoras theorem and the formulas describe rectangular triangles. The complete list of these under 100 also contains

102 =  62 +  82
152 =  92 + 122
202 = 122 + 162
262 = 242 + 102
392 = 362 + 152
522 = 482 + 202

However, these do not comply with the Pythagoras formula.

 


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