Now, if two complex numbers are equal, then their real parts have to be equal and their imaginary parts have to be equal. In order that *zw* = 1, we’ll need

That gives us two equations. The first says that the real parts are equal:

and the second says that the imaginary parts are equal:

Now, in our case, *z* was given and *w* was unknown, so in these two equations *x* and *y* are given, and *u* and *v* are the unknowns to solve for. You can fairly easily solve for *u* and *v* in this pair of simultaneous linear equations. When you do, you’ll find

So, the reciprocal of *z* = *x* + *yi* is the number *w* = *u* + *vi* where *u* and *v* have the values just found. In summary, we have the following reciprocation formula:

This means the length of 1/*z* is the reciprocal of the length of *z.* For example, if |*z*| = 2, as in the diagram, then
|1/*z*| = 1/2. It also means the argument for 1/*z* is the negation of that for *z.* In the diagram, arg(*z*) is about 65° while arg(1/*z*) is about –65°.

You can see in the diagram another point labelled with a bar over *z.* That is called the *complex conjugate* of *z.* It has the same real component *x,* but the imaginary component is negated. Complex conjugation negates the imaginary component, so as a transformation of the plane **C** all points are reflected in the real axis (that is, points above and below the real axis are exchanged). Of course, points on the real axis don’t change because the complex conjugate of a real number is itself.

Complex conjugates give us another way to interpret reciprocals. You can easily check that a complex number *z* = *x* + *yi* times its conjugate *x* – *yi* is the square of its absolute value |*z*|^{2}.

Therefore, 1/*z* is the conjugate of *z* divided by the square of its absolute value |*z*|^{2}.

In the figure, you can see that 1/|*z*| and the conjugate of *z* lie on the same ray from 0, but 1/|*z*| is only one-fourth the length of the conjugate of *z* (and |*z*|^{2} is 4).

Incidentally, complex conjugation is an amazingly “transparent” operation. It commutes with all the arithmetic operations: the conjugate of the sum, difference, product, or quotient is the sum, difference, product, or quotient, respectively, of the conjugates. Such an operation is called a *field isomorphism.*

Next, we have an expression in complex variables that uses complex conjugation and division by a real number.

Both formulations are useful and well worth knowing and understanding.