Complex Numbers and Programming

scheme@(guile-user)> (define pi 3.141592)

scheme@(guile-user)> pi

$1 = 3.141592

scheme@(guile-user)> (define tau (* 2 pi))

scheme@(guile-user)> tau

$2 = 6.283184

scheme@(guile-user)> 1-0i

$6 = 1

scheme@(guile-user)> 0-1i

$7 = 0.0-1.0i

They can be constructed from the real number components:

scheme@(guile-user)> (make-rectangular 0 1)

$8 = 0.0+1.0i

Or from the angle and magnitude:

scheme@(guile-user)> (make-polar 1 (/ pi 2))

$10 = 3.2679489653813835e-7+0.9999999999999466i

The components can be extracted back out:

scheme@(guile-user)> (real-part 2+1i)

$11 = 2.0

scheme@(guile-user)> (imag-part 2+1i)

$12 = 1.0

scheme@(guile-user)> (magnitude 0+1i)

$13 = 1.0

scheme@(guile-user)> (magnitude 0+2i)

$14 = 2.0

scheme@(guile-user)> (magnitude 1+1i)

$15 = 1.4142135623730951

scheme@(guile-user)> (angle 1+0i)

$16 = 0.0

scheme@(guile-user)> (angle 0+1i)

$17 = 1.5707963267948966

The interesting operation is exponentiation, which rotates and scales the vector:

Oh, and Guile allows us to represent a complex number in polar format (magnitude@angle):

scheme@(guile-user)> 1@6.283184

$25 = 0.9999999999991457-1.3071795861522044e-6i

scheme@(guile-user)> (make-polar 1 tau)

$26 = 0.9999999999991457-1.3071795861522044e-6i

scheme@(guile-user)> (expt $26 3)

$27 = 0.9999999999923108-3.921538758447679e-6i

Those familiar with complex numbers would be expecting 1+τi raise to the power of three to be 1-0i, but instead we get an ugly approximation with lots of digits. I find it we get much more simple answers if we round to three digits past the decimal. There is not a function in guile scheme that does exactly that, so far as I know, so we have to multiply, round, and then multiply back again:

scheme@(guile-user)> (define (signif x) (* 0.001 (round (* 1000 x))))

scheme@(guile-user)> (signif pi)

$36 = 3.142

scheme@(guile-user)> (signif tau)

$37 = 6.283

Then we integrate that into a complex number approximation function:

scheme@(guile-user)> (make-polar 1 (/ tau 8))

$32 = 0.7071068967259818+0.7071066656470943i

scheme@(guile-user)> (define (appr z) (make-rectangular (round (real-part z)) (round (imag-part z))))

scheme@(guile-user)> (appr-z $32)

$38 = 0.707+0.707i

scheme@(guile-user)> (appr-z (expt $32 2))

$39 = 0.0+1.0i

The author considers the programming codes in this blog post to be trivial, and therefore they can be reused without an explicit license.

Oh, by the way, ever heard of this very simple recursive complex number function?:

fc(z) = z^2+c

You might recognize one particular visual representation: