Unformatted text preview: hieved by writing out z = a + bi and
w = c + di for real numbers a, b, c, and d. Next, we compute the left and right hand side
of each equation and check to see that they are the same. The proof of the ﬁrst property is
a very quick exercise.6 To prove the second property, we compare z + w and z + w. We have
z + w = a + bi + c + di = a − bi + c − di. To ﬁnd z + w, we ﬁrst compute
z + w = (a + bi) + (c + di) = (a + c) + (b + d)i
z + w = (a + c) + (b + d)i = (a + c) − (b + d)i = a − bi + c − di
As such, we have established z + w = z + w. The proof for multiplication works similarly. The proof
that the conjugate works well with powers can be viewed as a repeated application of the product
rule, and is best proved using a technique called Mathematical Induction.7 The last property is a
characterization of real numbers. If z is real, then z = a + 0i, so z = a − 0i = a = z . On the other
hand, if z = z , then a + bi = a − bi which means b = −b so b = 0. Hence, z = a + 0i = a and is real.
We now retu...
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