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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.