Question

# If $\omega$ is the imaginary root of unity and a, b, c are natural numbers such that $(a - b)(b - c)(c - a) \ne 0$. Let $z = a + b\omega + c{\omega ^2}$, then the least value of $[2|z|]$ is (where $[.]$ denotes the greatest integer function)

Hint: Use the property of imaginary root of unity, $1 + \omega + {\omega ^2} = 0$. This formula can be used to find an equation for ${\omega ^2}$which can be substituted in $z = a + b\omega + c{\omega ^2}$. Use one of the values of i.e. $\left( { - \dfrac{1}{2} + \dfrac{{\sqrt 3 }}{2}i} \right)$ to make z in imaginary number notation. Next, use the formula $|z| = \sqrt {{x^2} + {y^2}}$and calculate it. Then find minimum value of $|z|$ by using the equations $a = b = k$ and $c = k + 1$. Finally, find the value of $[2|z|]$.

Complete step by step solution: Find equation for ${\omega ^2}$
Let $z = a + b\omega + c{\omega ^2}$
We know that $1 + \omega + {\omega ^2} = 0$.
We will use this to find the value of ${\omega ^2}$to form a linear equation that we can substitute in $z = a + b\omega + c{\omega ^2}$and solve it like we solve linear equations. This is known as substitution method.
Now,
$\begin{array}{l} 1 + \omega + {\omega ^2} = 0\\ \Rightarrow {\omega ^2} = - 1 - \omega \end{array}$
Substitute the value of ${\omega ^2}$in $z$.
$\begin{array}{l} z = a + b\omega + c{\omega ^2}\\ \Rightarrow z = a + b\omega + c( - 1 - \omega )\\ \Rightarrow z = (a - c) + (b - c)\omega \end{array}$
Put value of $\omega$ in $z$
For the equation $z = a + b\omega + c{\omega ^2}$, we know that one of the values of is $\left( { - \dfrac{1}{2} + \dfrac{{\sqrt 3 }}{2}i} \right)$.
Putting the value of $\omega$ in $z$, we get
$\begin{array}{l} z = (a - c) + (b - c)\left( { - \dfrac{1}{2} + \dfrac{{\sqrt 3 }}{2}i} \right)\\ z = \left( {a - \dfrac{b}{2} - \dfrac{c}{2}} \right) + \dfrac{{\sqrt 3 }}{2}(b - c)i \end{array}$
Compare with formula for $|z|$
We know, $|z| = \sqrt {{x^2} + {y^2}}$
So, here, $|z| = \sqrt {{{\left( {a - \dfrac{b}{2} - \dfrac{c}{2}} \right)}^2} + \dfrac{3}{4}{{(b - c)}^2}}$
$|z| = \sqrt {\dfrac{1}{2}[{{(a - b)}^2} + {{(b - c)}^2} + {{(c - a)}^2}}$
Find minimum value of $|z|$
To find minimum value of $|z|$, we will put, $a = b = k$ and $c = k + 1$
Then, $|z| = \sqrt {\dfrac{1}{2}(0 + {1^2} + {1^2}} = 1$
So, minimum value of $|z|$ will be 1
Find value of $[2|z|]$
$\begin{array}{l} 2|z| = 2 \times 1 = 2\\ [2|z|] = 2 \end{array}$

Note: Complex numbers can be a difficult topic for beginners. Learn about the imaginary roots of $\omega$ and its properties. Also remember the values of $\omega$ to solve questions easily. Make proper use of formulas of modulus and complex numbers.