The following applet shows how to multiply and add complex numbers (vectors
in the plane). For multiplication, add the angles and multiply the lengths
to get the product. Hit the button C=A*B twice to turn on and off the arc
illustration of angle addition. The illustration works best when both factors
are in the first quadrants. Hit the button C=A+B to again and again to alternate
between illustrations of the head-to-tail method and the component method for
vector addition. Now experiment or play.
Note: Help function button should but does not take you to
the local Complex No Intro. Need to
recompile.
Note: Help function button should but does not take you to
the local Complex No Intro. Need to
recompile.
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Complex Numbers
with easy consequences of two ways
to multiply complex numbers in and between vectors & trig, etc
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[ Back ] [ Area Intro ] [ Next ]
The fundamental theorem of algebra and partial
fraction decomposition in calculus depend on complex numbers.
Easy Consequences
Vec & Cmplx No Applet B2 C. Conjugates B3 Pythagoras B4 Distance B5 Rt Triangle Similarity B6 Trig., Functions B7 Dot & Cross Products B8 Cosine Law B9 Exponential & cis fns B10 Easy Trig Identities B11 Set Viewpoint Links: Interactive Maths
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 Euclidean-Geometry/Complex
No.s
folder.
First Earlier (Old) exposition of complex numbers follows
in Z1 to B1 below - read for review or revision .
First (Old) Complex No Intro Distributive Law A1 Add Poiints A2 Polar Coords A3 Polar Multiply A4 Complex No.s A5 Real Numbers A6 Law of Signs A7 Key Properties B1 2nd Mult Method C1 Unsigned Coords C2 Signed Coords C3 Set Codification C4 More On Real No.s D1 Arrow Navigation D2 Sum of Motions D3 Addition Method I D4 Addition Method II D5 Addition Method III D6 Coordinate Addition D7 1st Distributive Law D8 2nd Distributive Law D9 3rd Distributive Law
D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
This further
Complex Number
Intro assumes the field properties
of real numbers in place of
deriving them geometrically
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