## Complex Numbers

Complex numbers are constructed as an ordered pair of two real numbers representing the real and the imaginary parts of the number.

The imaginary number (i) is defined as the pair (<,> (r.0) (r.1)) /edit/peano_new/arithmetic/complex/add_multiply_min.gh/df-i meaning it has no real part and one as its imaginary part.

### Powers of (i)

• (= (exp (i) (2)) (-n (1))) /edit/peano_new/arithmetic/complex/add_multiply_complex.gh/iSquared2
• (= (exp (i) (3)) (-n (i))) /edit/peano_new/arithmetic/complex/add_multiply_complex.gh/iCubed
• (= (exp (i) (4)) (1)) /edit/peano_new/arithmetic/complex/add_multiply_complex.gh/iPower4

### Complex Arithmetic

• (= (+ (+ a (* b (i))) (+ c (* d (i)))) (+ (+ a c) (* (+ b d) (i)))) /edit/peano_new/arithmetic/complex/add_multiply_complex.gh/comadd
• (= (* (+ a (* b (i))) (+ c (* d (i)))) (+ (- (* a c) (* b d)) (* (+ (* a d) (* b c)) (i)))) /edit/peano_new/arithmetic/complex/add_multiply_complex.gh/commul
• (= (/ (+ a (* b (i))) (+ c (* d (i)))) (/ (+ (+ (* a c) (* b d)) (* (- (* b c) (* a d)) (i))) (+ (exp c (2)) (exp d (2))))) /edit/peano_new/arithmetic/complex/add_multiply_complex.gh/comdiv