Pythagorean Theorem
If a right triangle has a hypotenuse of length r and other two sides of lengths a and b,
then
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Complex Number Proof
Let z = a + i b = (r, A) be a point in the first quadrant. The triangle with vertices 0, a, a+ ib is congruent or isometric to the given right triangle
.
Multiplying a vector a + i b with angle A and length r by its complex conjugate a - ib gives a complex number with angle 0 = A + (-A) and length r2 units according to the add the angles, multiply the lengths polar coordinate, multiplication rule. The product has value r2 > 0 as shown below.
This gives
r2 = (a+ib) (a-ib)
But the previous formulas for expressing products in terms of their real and imaginary parts,
r2 = a2 + b2 + i(-ba+ab)
as the polar and rectangular methods for computing a product give the same result.
This implies
r2 = a2 + b2
Remark: There are over 100 different proofs of the Pythagorean theorem.
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Complex Numbers |
The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.
Easy Consequences
Hint: See the (newest) Complex
Number. Starter Lesson.
for a simple geometric introduction, then continue with easy consequence below.
The clearest geometric proof of the distributive law appears in the
folder.
First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision .
D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
Intro assumes the field properties
of real numbers in place of
deriving them geometrically