| | | | | | | | | | | | | S | u | b | j | e | c | t | : | | C | o | m | p | l | e | x | | a | r | i | t | h | m | e | t | i | c | | a | n | d | | q | u | a | t | e | r | n | i | o | n | | a | r | i | t | h | m | e | t | i | c | | | | | | | | | | | | | |
| |
| | 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 |
| |