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). It has the same real component x, but the imaginary component is negated. That is called the complex conjugate of z. You can see in the diagram another point labelled with a bar over z. In the diagram, arg( z) is about 65° while arg(1/ z) is about ≦5°. It also means the argument for 1/ z is the negation of that for z. For example, if | z| = 2, as in the diagram, then This means the length of 1/ z is the reciprocal of the length of z. If z and w are reciprocals, then zw = 1, so the product of their absolute values is 1, and the sum of their arguments (angles) is 0. Reciprocals done geometrically, and complex conjugates.įrom what we know about the geometry of multiplication, we can determine In summary, we have the following reciprocation formula: So, the reciprocal of z = x + yi is the number w = u + vi where u and v have the values just found. You can fairly easily solve for u and v in this pair of simultaneous linear equations. 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. The first says that the real parts are equal:Īnd the second says that the imaginary parts are equal: Now, if two complex numbers are equal, then their real parts have to be equal and their imaginary parts have to be equal. We’ll use the product formula we developed in the section on multiplication. By now, we can do that both algebraically and geometrically. In other words, given a complex number z = x + yi, find another complex number w = u + vi such that zw = 1. So we set ourselves the problem of finding 1/ z given z. Just as subtraction can be compounded from addition and negation, division can be compounded from multiplication and reciprocation. We’ve studied addition, subtraction, and multiplication.
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