Subject: Complex arithmetic and quaternion arithmetic

Q8a: How does complex arithmetic work?

A8a: It works mostly like regular algebra with a couple additional formulas:
(note: a, b are reals, x, y are complex, i is the square root of -1)

Powers of i:

i^² = -1

Addition:

(a+i*b)+(c+i*d) = (a+c)+i*(b+d)

Multiplication:

(a+i*b)*(c+i*d) = a*c-b*d + i*(a*d+b*c)

Division:

(a+i*b) / (c+i*d) = (a+i*b)*(c-i*d) / (c^²+d^²)

Exponentiation:

exp(a+i*b) = exp(a)*(cos(b)+i*sin(b))

Sine:

sin(x) = (exp(i*x) - exp(-i*x)) / (2*i)

Cosine:

cos(x) = (exp(i*x) + exp(-i*x)) / 2

Magnitude:

|a+i*b| = sqrt(a^²+b^²)

Log:

log(a+i*b) = log(|a+i*b|)+i*arctan(b / a) (Note: log is multivalued.)

Log (polar coordinates):

log(r e^^(i*a)) = log(r)+i*a

Complex powers:

x^^y = exp(y*log(x))

de Moivre's theorem:

x^ⁿ = r^ⁿ [cos(n*a) + i*sin(n*a)] (where n is an integer)

More details can be found in any complex analysis book.

Q8b: How does quaternion arithmetic work?

A8b: quaternions have 4 components (a + ib + jc + kd) compared to the two of complex numbers. Operations such as addition and multiplication can be performed on quaternions, but multiplication is not commutative.

Quaternions satisfy the rules

• i^² = j^² = k^² = -1
• ij = -ji = k
• jk = -kj = i,
• ki = -ik = j

See:

Frode Gill's quaternions page

http://www.krs.hia.no/~fgill/quatern.html