Let $S$ be the set of complex numbers of the form $a + bi,$ where $a$ and $b$ are integers. We say that $z \in S$ is a unit if there exists a $w \in S$ such that $zw = 1.$ Find the number of units in $S.$

Number of units possible in S are 4.

Explanation:

Given S is a set of complex number of the form where a and b are integers.

is a unit if exists such that .

To find:

Number of units possible = ?

Solution:

Given that:

Taking modulus both sides:

Using the property that modulus of product of two complex numbers is equal to their individual modulus multiplied.

i.e.

So,

......... (1)

Let

Then modulus of z is  

Given that a and b are integers, 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:

Hence, number of units possible in S are 4.

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