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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 holds in general
You can also write for this
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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.
HistoryThe old Greek mathematician Pythagoras (572 BC - 500 BC) is famous for his number theory. |