A First Course in Linear Algebra, 2017a

6.1. Complex Numbers 329
Hence, both |z|−|w| and |w|−|z| are no larger than |z − w|. This proves the second version because
||z|−|w|| is one of |z|−|w| or |w|−|z|.
♠
With this definition, it is important to note the following. You may wish to take the time to verify this
remark.
√
Let z = a + bi and w = c + di. Then|z − w| = (a − c) 2 +(b − d) 2 . Thus the distance between the
point in the plane determined by the ordered pair (a,b) and the ordered pair (c,d) equals |z − w| where z
and w are as just described.
For example, consider the distance between (2,5) and (1,8).Lettingz = 2+5i and w = 1+8i, z−w =
1 − 3i, (z − w)(z − w)=(1 − 3i)(1 + 3i)=10 so |z − w| = √ 10.
Recall that we refer to z = a + bi as the standard form of the complex number. In the next section, we
examine another form in which we can express the complex number.
Exercises
Exercise 6.1.1 Let z = 2 + 7i and let w = 3 − 8i. Compute the following.
(a) z + w
(b) z − 2w
(c) zw
(d) w z
Exercise 6.1.2 Let z = 1 − 4i. Compute the following.
(a) z
(b) z −1
(c) |z|
Exercise 6.1.3 Let z = 3 + 5i and w = 2 − i. Compute the following.
(a) zw
(b) |zw|
(c) z −1 w
Exercise 6.1.4 If z is a complex number, show there exists a complex number w with |w| = 1 and wz = |z|.