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Move points a and b with the mouse to explore the geometric behavior of complex multiplication.

f   =   ab   =   (axbx - ayby)  +  i (axby + aybx)

When the complex numbers a and b are multiplied together, it produces another complex number, which is labeled f.

As you move a and b, observe that the two triangles {0,1,a} and {0,b,f} are always congruent.  This congruent relationship means that the two triangles always have the same three angles, and the lengths of their corresponding sides are always in proportion.

To understand how the vertices correspond, you can individually multiply each vertex in the triangle {0,1,a} with b.  This produces the corresponding vertices {0,b,f}.

Observe that the side {0,1} is congruent to the side {0,b}, and therefore the two triangles have a side-length ratio of |b|.

Observe the angle between the line segment {0,a} and positive x axis.  This angle is called the phase of a, and is represented here by the symbol _a.  As you move a around, observe the relationship between _a and _f.

You can analyze the two congruent triangles to show that:

f = a + b

and

|f| = |a||b|

Therefore, multiplying two complex numbers together has the effect of simultaneously "adding their phase angles" and "multiplying their magnitudes".

Part 1: The unit circle

Move point d with the mouse to explore the integer powers of f.

f =  d |d|      g = f 2      h = f 3      k = f 4      m = f 5

The complex number f is located on the unit circle.  If a number is located on the unit circle, then all of its real powers are also located on the unit circle.

What are the cube roots of 1?  The number 1 has three different cube roots, all located on the unit circle.  You can find these roots by concentrating on h as you spin f around the entire circle.  Whenever h is equal to 1, f is then located at one of the cube roots of 1.

As you spin f around the circle, also note its location whenever g, k or m is equal to 1.  In general, the number 1 has n complex nth-roots, all located on the unit circle.  These n roots define the vertices of a regular n-sided polygon, for all integer values of n > 2.

The integer powers of f are equally spaced along the circumference of the unit circle.  If the phase angle of f is multiplied by n, then the result (when normalized) is equal to the phase angle of f ⁿ:

nf  =  (fn)     when normalized to (- ... ]

This relationship is expressed trigonometrically by DeMoivre's theorem:

(cos f  +  i sin f)n   =   (cos nf  +  i sin nf)

Part 2: The power spiral

Move point a with the mouse to explore the real powers of a.

g = a2     h = a3     k = a4     m = a5     blue: an (n = 0 to 20)

The real powers of a complex number generate a logarithmic spiral.  This spiral has the polar equation:

r  =  ew     where     w  =   log |a| a

The constant w indicates how rapidly the spiral explodes outward (or implodes inward) as increases.  However, if a = 0 or a = 1, then w is undefined, and the spiral collapses onto a.  Otherwise, if a is on the unit circle, then w = 0, and the spiral collapses onto the unit circle.  Otherwise, if a is on the positive x axis, then w is undefined, and the spiral collapses onto the positive x axis.

The logarithmic spiral is notable because all rays emanating from the origin intersect the spiral at a constant angle (arccot w).

DeMoivre's theorem can be employed to describe this spiral trigonometrically:

an  = (ax + iay)n   = (|a| cos a  +  i |a| sin a)n   = (|a| (cos a  +  i sin a))n   = |a|n (cos a  +  i sin a)n   = |a|n (cos na  +  i sin na)   {from DeMoivre's theorem}

A corollary of this result is that:

|an|   =   |a|n

Therefore, raising a complex number to the power of n has the effect of simultaneously "multiplying its phase angle by n" and "raising its magnitude to the nth power", for all real values of n.

Part 1: Horizontal

Move point c up and down with the mouse to explore how each complex number along a horizontal line (red) is transformed by the complex exponential to generate a new line (purple).

f = ec      red: g = t + icy      purple: eg

Move c left and right to explore the exponential relationship between c_x and |f|.

Part 2: Vertical

Move point c left and right with the mouse to explore how each complex number along a vertical line (red) is transformed by the complex exponential to generate a circle (purple).

f = ec      red: g = cx + it      purple: eg

Observe the angle between f's radial line and the positive x axis.  This angle is called the phase of f, and is represented here by the symbol .

Move c up and down to understand how c_y = (as measured in radians).

Move c up and down the y axis to verify Euler's formula:

ei = cos + i sin

Move c left and right to verify the generalized form of Euler's formula:

e c      =    e (cx + icy)      =    e cx     e icy      =    e cx     (cos cy  +   i sin cy)

Move point c up and down with the mouse to understand the geometric relationship between e^m and e^-m for all pure imaginary values of m.

The two numbers e^m and e^-m are labeled f and g on the graph.  Learn how to swap the positions of f and g.

m = icy      f = em      g = e-m      h =  f + g 2

The numbers f and g always have the same real component, but their imaginary components are opposite in sign.  This relationship between f and g is called a conjugate relationship.

Because f and g are conjugates, the midpoint between them (labeled h) is always a real number. The number h is called the hyperbolic cosine of m.  The standard abbreviation for "hyperbolic cosine" is cosh.

cosh m   =   h   =    f + g 2    =    em + e-m 2

Observe the angle between f's radial line and the positive x axis.  This angle is called the phase of f, and is represented here by the symbol .

Move c up and down to verify the following relationships:

cy  =   m  =   i h  =   cos

Combining these together, you can now see the relationship between cosh and cos:

cos   =   h   =   cosh m   =   cosh i

and you can now derive the complex definition of cos:

cos   =   h   =    f + g 2    =    em + e-m 2    =    ei + e-i 2

Move point a around with the mouse to understand the relationship between a, h, and k.

a = h + k      g = h - k      n = -g

Observe the following:

h  =   ax   =   |a| cos   =   |a| cosh i k  =   i ay   =   i |a| sin   =   |a| sinh i a  =   h + k   =   |a| (cos + i sin )   =   |a| ei  a  =   h + k   =   |a| (cosh i + sinh i)   =   |a| ei  g  =   h - k   =   |a| (cos - i sin )   =   |a| e-i  g  =   h - k   =   |a| (cosh i - sinh i)   =   |a| e-i  n  =   -g h  =    a + g 2    =   |a|  ei + e-i 2   =   |a| cos k  =    a + n 2    =    a - g 2    =   |a|  ei - e-i 2   =   |a| i sin |a|2    =   h2 - k2   =   ax2 - (i ay)2   =   ax2 - i2ay2   =   ax2 + ay2

Verify that the above observations are consistent with the following identities, which are true for all complex z:

cosh z   =    ez + e-z 2 sinh z   =    ez - e-z 2 cos z   =   cosh iz  =    eiz + e-iz 2 sin z   =    sinh iz i   =    eiz - e-iz 2i ez   =   cosh z  +  sinh z e-z   =   cosh z  -  sinh z eiz   =   cos z  +  i sin z e-iz   =   cos z  -  i sin z 1   =   (cos z)2  +  (sin z)2 1   =   (cosh z)2  -  (sinh z)2

Move point c around with the mouse to explore how each complex number along a straight line (green) is transformed by the complex exponential to generate the following pair of curves:

1:	the real component of the complex exponential (blue),

 

2:	the imaginary component of the complex exponential (red).

blue: y = Re e(cx)      red: y = Im e(cx)

Move c left and right along the x axis to explore how the function e^(cx) transforms the real numbers (green) into an exponential curve (blue).

Move c up and down the y axis to explore how the function e^(cx) transforms the pure imaginary numbers (green) into the curves y = cos c_yx (blue) and y = sin c_yx (red).

This demonstrates the unification of the exponential curves with the cosine and sine curves.  All these curves can be generated by applying different "viewpoints" to a single underlying mathematical formula.

This unification is expressed by the generalized form of Euler's formula:

e c      =    e cx    cos cy  +   i  e cx    sin cy

If you let c_y = 0, then cos c_y = 1 and sin c_y = 0, which causes the entire formula to collapse into a trivial identity:

e cx     =    e cx     + 0

If you let c_x = 0, then e⁰ = 1, which causes the real and imaginary components of the formula to individually yield the cosine and sine (respectively):

e icy     =   cos cy  +   i sin cy