Question

For any complex number $Z$, the minimum value of \[|Z| + |Z - 1|\] is
A. 0
B. 1
C. 2
D. -1

Answer 1

Hint: A complex number is a simple representation of addition of two numbers, i.e., real number and an imaginary number. One part of it is purely real and the other part is purely imaginary. Here, we are given an complex number Z and we need to find the minimum value of \[|Z| + |Z - 1|\] . For this we will use the triangular inequalities which is \[|{Z_1} + {Z_2}| \leqslant |{Z_1}| + |{Z_2}|\] where $Z_1$ and $Z_2$ are two complex numbers. Also we know that, \[|Z| = | - Z|\] . Thus, using both these, we will solve the given equation and get the final output.

Complete step by step answer:
Given that, Z is a complex number. The value of
\[|Z| + |Z - 1|\]
We know that, \[|Z| = | - Z|\] and applying this, we will get,
\[|Z| + | - ( - 1 - Z)|\]
On evaluating this, we will get,
\[|Z| + |1 - Z|\]
We know that,
\[|{Z_1} + {Z_2}| \leqslant |{Z_1}| + |{Z_2}|\]
where $Z_1$ and $Z_2$ are complex numbers.

This means that, \[|{Z_1}| + |{Z_2}| \geqslant |{Z_1} + {Z_2}|\] and so applying this, we will get,
\[|Z| + |1 - Z| \geqslant |Z + 1 - Z|\]
\[ \Rightarrow |Z| + |1 - Z| \geqslant |1|\]
\[ \Rightarrow |Z| + |1 - Z| \geqslant 1\]
Again we will use, \[|Z| = | - Z|\] and we will get,
\[ \Rightarrow |Z| + | - (Z - 1)| \geqslant 1\]
On evaluating this, we will get,
\[ \Rightarrow |Z| + |Z - 1| \geqslant 1\]
\[ \Rightarrow |Z| + |Z - 1| = 1\]
Alternatively,
\[1 = |Z - (Z - 1)|\]
\[ \therefore |Z| + |Z - 1| \geqslant 1\]
Hence, for given Z is a complex number, the minimum value of \[|Z| + |Z - 1| = 1\] .

Thus, option B is the correct answer.

Note: Complex numbers are the numbers that are expressed in the form of a + ib where a, b are real numbers and ‘i’ is an imaginary number called “iota”. An imaginary number is usually represented by ‘i’ or ‘j’, which is equal to \[\sqrt { - 1} \]. Thus, the square of the imaginary number gives a negative value ( \[{i^2} = - 1\]). The main application of these numbers is to represent periodic motions such as water waves, alternating current, light waves, etc.