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)

Answer 1

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,

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.