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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 z_{1} = x_{1} + y_{1} i and z_{2} = x_{2} + y_{2} i are two arbitrary complex numbers, then their sum is given by
 (2.2.1) z_{1} + z_{2} = (x_{1} + x_{2}) + (y_{1} + y_{2}) i
and their difference by
 (2.2.2) z_{1}  z_{2} = (x_{1}  x_{2}) + (y_{1}  y_{2}) 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
and
 (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 z_{1} and z_{2} given above. Letting a = x_{1}, b = y_{1} i, c = x_{2}, and d = y_{2} i, we find z_{1} z_{2} = (x_{1} + y_{1} i)(x_{2} + y_{2} i) = x_{1} x_{2} + x_{1} y_{2} i + y_{1} x_{2} i + y_{1} y_{2} i^{2}. Collecting terms and noting that i^{2} = 1, we find
 (2.2.5) z_{1} z_{2} = (x_{1} x_{2}  y_{1} y_{2}) + (x_{1} y_{2} + y_{1} x_{2}) 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:
 (2.2.6)
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 z_{1}/z_{2}, where z_{1} and z_{2} are arbitrary complex numbers with z_{2} nonzero. If we multiply the numerator and denominator by the complex conjugate of z_{2}, we obtain (z_{1}/z_{2})(z_{2}/z_{2}) = z_{1}z_{2} / z_{2}^{2}.
But we know how to multiply z_{1} and z_{2} and compute the norm of z_{2}. Thus, we know how to divide two complex numbers. The result, in terms of the real and imaginary parts of z_{1} and z_{2}, is

(2.2.7)
 Example 1: Consider the complex numbers z_{1} = 4 + 7i and z_{2} = 5  3i. Compute z_{1 }+ z_{2}, z_{1 } z_{2}, z_{1}z_{2}, and z_{1 }/ z_{2}.
 Solution: Using (2.2.1), we find the sum of z_{1} and z_{2} is given by z_{1} + z_{2} = (4+5) + (73)i = 9 + 4i. Using (2.2.2), we find the difference of z_{1} and z_{2} is given by z_{1}  z_{2} = (45) + (7(3))i = 1 + 10i. Using (2.2.5), we find the product of z_{1} and z_{2} is given by z_{1}z_{2} = ((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 z_{1} and z_{2} is given by
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