Multiplication of Complex Numbers
Let z1=a+ib and z2=c+id. Their product, written z1⋅z2, is found by expanding like two binomials and then using i2=−1 to simplify:
z1⋅z2=(a+ib)(c+id)=ac+adi+bci+i2bd=ac+(ad+bc)i−bd
z1⋅z2=(ac−bd)+(ad+bc)i
Worked examples.
- z1=2+3i, z2=3−2i: z1z2=(2+3i)(3−2i)=2(3−2i)+3i(3−2i)=6−4i+9i−6i2=6−4i+9i+6=12+5i.
- z1=2−7i, z2=4−3i, z3=1+i: then (2z1)⋅z2⋅z3=2(2−7i)⋅(4−3i)⋅(1+i)=(4−14i)⋅[4+4i−3i−3i2]=(4−14i)⋅[7+i]=28+4i−98i−14i2=42−94i — with three factors, multiply two of them first, simplify, and then multiply by the third.
Properties of multiplication. For any complex numbers z1,z2,z3:
- (i) z1⋅z2=z2⋅z1 (commutative)
- (ii) (z1⋅z2)⋅z3=z1⋅(z2⋅z3) (associative)
- (iii) z⋅1=1⋅z=z (the complex number 1 is the multiplicative identity)
- (iv) (z1z2)=zˉ1⋅zˉ2 — the conjugate of a product is the product of the conjugates. …