Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS3.2.3 MultiplicationComplex numbers may be multiplied together and in general give a complexnumber as the result. The product of two complex numbers z 1 and z 2 is foundby multiplying them out in full and remembering that i 2 = −1, i.e.z 1 z 2 =(x 1 + iy 1 )(x 2 + iy 2 )= x 1 x 2 + ix 1 y 2 + iy 1 x 2 + i 2 y 1 y 2=(x 1 x 2 − y 1 y 2 )+i(x 1 y 2 + y 1 x 2 ). (3.6)◮Multiply the complex numbers z 1 =3+2i and z 2 = −1 − 4i.By direct multiplication we findz 1 z 2 =(3+2i)(−1 − 4i)= −3 − 2i − 12i − 8i 2=5− 14i. ◭ (3.7)The multiplication of complex numbers is both commutative and associative,i.e.z 1 z 2 = z 2 z 1 , (3.8)(z 1 z 2 )z 3 = z 1 (z 2 z 3 ). (3.9)The product of two complex numbers also has the simple propertiesThese relations are derived in subsection 3.3.1.|z 1 z 2 | = |z 1 ||z 2 |, (3.10)arg(z 1 z 2 )=argz 1 +arg z 2 . (3.11)◮Verify that (3.10) holds for the product of z 1 =3+2i and z 2 = −1 − 4i.From (3.7)We also find|z 1 z 2 | = |5 − 14i| = √ 5 2 +(−14) 2 = √ 221.|z 1 | = √ 3 2 +2 2 = √ 13,|z 2 | = √ (−1) 2 +(−4) 2 = √ 17,and hence|z 1 ||z 2 | = √ 13 √ 17 = √ 221 = |z 1 z 2 |. ◭We now examine the effect on a complex number z of multiplying it by ±1and ±i. These four multipliers have modulus unity and we can see immediatelyfrom (3.10) that multiplying z by another complex number of unit modulus givesa product with the same modulus as z. We can also see from (3.11) that if we88