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

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.