Answer:
Number of units possible in S are 4.
Explanation:
Given <em>S</em> is a set of complex number of the form
where <em>a</em> and <em>b</em> are integers.
is a unit if
exists such that
.
To find:
Number of units possible = ?
Solution:
Given that:
![zw = 1](https://tex.z-dn.net/?f=zw%20%3D%201)
Taking modulus both sides:
![|zw| = |1|](https://tex.z-dn.net/?f=%7Czw%7C%20%3D%20%7C1%7C)
Using the property that modulus of product of two complex numbers is equal to their individual modulus multiplied.
i.e.
![|z_1z_2|=|z_1|.|z_2|](https://tex.z-dn.net/?f=%7Cz_1z_2%7C%3D%7Cz_1%7C.%7Cz_2%7C)
So,
......... (1)
Let ![z=a+bi](https://tex.z-dn.net/?f=z%3Da%2Bbi)
Then modulus of z is ![|z| = \sqrt{a^2+b^2}](https://tex.z-dn.net/?f=%7Cz%7C%20%3D%20%5Csqrt%7Ba%5E2%2Bb%5E2%7D)
Given that a and b are <em>integers</em>, so the equation (1) can be true only when
(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:
![1.\ 1 + 0i = 1\\2.\ -1 + 0i = -1\\3.\ 0+ 1i = i\\4.\ 0 -1i = -i](https://tex.z-dn.net/?f=1.%5C%201%20%2B%200i%20%3D%201%5C%5C2.%5C%20-1%20%2B%200i%20%3D%20-1%5C%5C3.%5C%200%2B%201i%20%3D%20i%5C%5C4.%5C%200%20-1i%20%3D%20-i)
Hence, number of units possible in S are 4.