College Algebra Complex Numbers Operations with Complex Numbers
Multiplication Concepts 1

|  |  | Definition |  |
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Let z1=x1+iy1 and z2=x2+iy2 . Then, by definition, the product z1· z2 is given by z1·z2 = ( x1· x2 - y1·y2) + i( x1· y2 + x2· y1 ).

Example 1. If z1 = 2 + 3i and z2 = 4 + 5i, find z1 · z2.
Solution.We have x1 = 2, y1 = 3, x2 = 4, y2 = 5. Hence, by definition,

z1· z2 | = | (2· 4 - 3· 5) + (2· 5+ 4· 3)i
| = | -7 + 22i.

Note. Observe that the right side of this equation can be obtained by formally manipulating the terms on the left as if they involved only real numbers and by replacing i2 by -1 when it occurs.
Illustration (compare with Example 1). If z1 = 2 + 3i and z2 = 4 + 5i, find z1 · z2.
Solution

z1· z2 | = | (2+3i)· ( 4+5i)
| = | 2· 4 + 2· 5i + 3i· 4 + 3i· 5i
| = | 8 + 10i +12i + 15· i2
| = | 8 + 22i + 15· (-1)
| = | -7 + 22i.

Properties
z1· z2 = z2· z1 | Commutative law
(z1· z2) · z3 = z1· ( z2· z3) | Associative law
(z1 + z2) · z3 = z1· z3 + z2· z3 | Distributive law
z·1 = z |

Conjugate of the product of complex numbers. The conjugate of the product of complex numbers equals to the product of the conjugate of these complex numbers. In formula

The non-negative integer powers of i .

i0 = 1 | i4 = 1 | i8 = 1
i1 = i | i5 = i | i9 = i
i2 = -1 | i6 = - 1 | i10 = -1
i3 = -i | i7 = -i | i11 = -i

The successive non-negative integer powers of i have only four different values: 1, i, -1, -i, repeating in regular order. To raise number i to non-negative integer power n, find the remainder, m, of dividing n by 4 and raise i to the power m.

Example 2. Find i722.
Solution
722 = 4· 180 + 2.
The remainder equals 2. Hence i722 = i2 = -1.

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