2.2. Complex Arithmetic
In the previous section we defined complex numbers as an extension of real numbers. How does one perform arithmetic with complex numbers? Addition and subtraction are easy; one simply adds or subtracts the respective real and imaginary parts of each complex number. In other words, if z1 = x1 + y1 i and z2 = x2 + y2 i are two arbitrary complex numbers, then their sum is given by
- (2.2.1) z1 + z2 = (x1 + x2) + (y1 + y2) i
and their difference by
- (2.2.2) z1 - z2 = (x1 - x2) + (y1 - y2) i.
Multiplication and division of complex numbers are more complicated. To define complex multiplication, we make use of the distributive property, i.e. the equations
- (2.2.3a) (a + b)c = ac + bc
- (2.2.3b) a(b + c) = ab + ac.
By applying both of these rules in turn, it is straightforward to derive the following additional property, sometimes known as the foil rule:
- (2.2.4) (a + b)(c + d) = ac + ad + bc + bd.
We leave the proof as an exercise.
Now let us apply the foil rule to compute the product of the complex numbers z1 and z2 given above. Letting a = x1, b = y1 i, c = x2, and d = y2 i, we find z1 z2 = (x1 + y1 i)(x2 + y2 i) = x1 x2 + x1 y2 i + y1 x2 i + y1 y2 i2. Collecting terms and noting that i2 = -1, we find
- (2.2.5) z1 z2 = (x1 x2 - y1 y2) + (x1 y2 + y1 x2) i.
This formula allows us to multiply any two complex numbers.
A very important identity, which is easily verified using the above definition of complex multiplication, is the following:
In other words, the norm of a complex number is equal to the square root of the product of the number and its complex conjugate. This definition will allow us to define complex division.
Division is the most complicated arithmetic operation on complex numbers. How does one divide two complex numbers? Consider the quotient z1/z2, where z1 and z2 are arbitrary complex numbers with z2 nonzero. If we multiply the numerator and denominator by the complex conjugate of z2, we obtain (z1/z2)(z2/z2) = z1z2 / |z2|2.
But we know how to multiply z1 and z2 and compute the norm of z2. Thus, we know how to divide two complex numbers. The result, in terms of the real and imaginary parts of z1 and z2, is
- Example 1: Consider the complex numbers z1 = 4 + 7i and z2 = 5 - 3i. Compute z1 + z2, z1 - z2, z1z2, and z1 / z2.
- Solution: Using (2.2.1), we find the sum of z1 and z2 is given by z1 + z2 = (4+5) + (7-3)i = 9 + 4i. Using (2.2.2), we find the difference of z1 and z2 is given by z1 - z2 = (4-5) + (7-(-3))i = -1 + 10i. Using (2.2.5), we find the product of z1 and z2 is given by z1z2 = ((4)(5) - (7)(-3)) + ((4)(-3) + (7)(5))i = (20+21) + (-12+35)i = 41 + 23i. Finally, using (2.6), we find the quotient of z1 and z2 is given by