Question

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 $S.$

Answer 1

Answer:

Number of units possible in S are 4.

Explanation:

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

$z\in S$ is a unit if $w\in z$ exists such that $zw=1$.

To find:

Number of units possible = ?

Solution:

Given that:

$zw = 1$

Taking modulus both sides:

$|zw| = |1|$

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

i.e.

$|z_1z_2|=|z_1|.|z_2|$

So,

$|zw| = |1|\\\Rightarrow |zw| =|z|.|w| =1\\\Rightarrow |z|=\dfrac{1}{|w|}$......... (1)

Let $z=a+bi$

Then modulus of z is   $|z| = \sqrt{a^2+b^2}$

Given that a and b are integers, so the equation (1) can be true only when $|z| = |w| =1$ (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:

$1.\ 1 + 0i = 1\\2.\ -1 + 0i = -1\\3.\ 0+ 1i = i\\4.\ 0 -1i = -i$

Hence, number of units possible in S are 4.