Answer:
Number of units possible in S are 4.
Explanation:
Given S is a set of complex number of the form [tex]a+bi[/tex] where a and b are integers.
[tex]z\in S[/tex] is a unit if [tex]w\in z[/tex] exists such that [tex]zw=1[/tex].
To find:
Number of units possible = ?
Solution:
Given that:
[tex]zw = 1[/tex]
Taking modulus both sides:
[tex]|zw| = |1|[/tex]
Using the property that modulus of product of two complex numbers is equal to their individual modulus multiplied.
i.e.
[tex]|z_1z_2|=|z_1|.|z_2|[/tex]
So,
[tex]|zw| = |1|\\\Rightarrow |zw| =|z|.|w| =1\\\Rightarrow |z|=\dfrac{1}{|w|}[/tex]......... (1)
Let [tex]z=a+bi[/tex]
Then modulus of z is [tex]|z| = \sqrt{a^2+b^2}[/tex]
Given that a and b are integers, so the equation (1) can be true only when [tex]|z| = |w| =1[/tex] (Reciprocal of 1 is 1). Modulus can be equal only when one of the following is satisfied:
(a = 1, b = 0) , (a = -1, b = 0), (a = 0, b = 1) OR (a = 0, b = -1)
So, the possible complex numbers can be:
[tex]1.\ 1 + 0i = 1\\2.\ -1 + 0i = -1\\3.\ 0+ 1i = i\\4.\ 0 -1i = -i[/tex]
Hence, number of units possible in S are 4.
