Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONSIm zz 2z 1 + z 2z 1Re zFigure 3.3The addition of two complex numbers.or in component notationz 1 + z 2 =(x 1 ,y 1 )+(x 2 ,y 2 )=(x 1 + x 2 ,y 1 + y 2 ).The Argand representation of the addition of two complex numbers is shown infigure 3.3.By straightforward application of the commutativity and associativity of thereal and imaginary parts separately, we can show that the addition of complexnumbers is itself commutative and associative, i.e.z 1 + z 2 = z 2 + z 1 ,z 1 +(z 2 + z 3 )=(z 1 + z 2 )+z 3 .Thus it is immaterial in what order complex numbers are added.◮Sum the complex numbers 1+2i, 3 − 4i, −2+i.Summing the real terms we obtain1+3− 2=2,and summing the imaginary terms we obtain2i − 4i + i = −i.Hence(1 + 2i)+(3− 4i)+(−2+i) =2− i. ◭The subtraction of complex numbers is very similar to their addition. As in thecase of real numbers, if two identical complex numbers are subtracted then theresult is zero.86