Question

The Modulus of a Complex Number is
a) Always Positive
b) Always Negative
c) Maybe Positive or Negative
d) $0$

Answer 1

Hint: First, we should know about the Complex numbers. The Complex Number is the combination of a real Part and an Imaginary Part. We know that the general form of Complex Form is $z = x + iy$ Where $x$ is a real Part and $y$ is an imaginary Part.
Real part $x$ values may be positive or negative. Imaginary part $y$ values may be positive or negative.
Next, we have to learn about the Modulus of the Complex Number. The modulus of $Z$ is a non-negative real number. The Modulus of $Z$ is denoted by $|Z|$. The formula to find the Modulus of $|Z|$ is
\[|Z| = \sqrt {{a^2} + {b^2}} \]
Where $a$ is the real part of the Complex Number and $b$ is the imaginary part of the Complex Number.

Complete step by step solution:
In this Problem, we should prove that the Modulus of a Complex Number may be always positive or negative, or Zero. We are going to solve this problem by using some Examples. Take any three Complex Numbers. Take the Sample Complex numbers with different Signs in the real and imaginary parts.
Example $1$:
Take a complex Number with a positive Real part and a positive imaginary part.
$Z = 3 + 5i$
The Formula for finding the Modulus of a Complex Number
\[|Z| = \sqrt {{a^2} + {b^2}} \]
Substitute $a$ as $3$ and $b$ as $5$
\[|3 + 5i| = \sqrt {{3^2} + {5^2}} \]
$ = \sqrt {9 + 25} $
$|3 + 5i| = \sqrt {34} $
Example $2$:
Next, take a complex Number with a positive Real part and a negative imaginary part.
$Z = 9 - 12i$
Let us discover the Modulus of this Complex Number
$
|Z| = \sqrt {{a^2} + {b^2}} \\
|9 - 12i| = \sqrt {{9^2} + {{( - 12)}^2}} \\
$
\[
= \sqrt {81 + 144} \\
= \sqrt {225} \\
\]
Example $3$:
Next, take a complex Number with a Negative Real part and a positive imaginary part.
$Z = - 12 + 5i$
Next, find the Modulus of this Complex Number
$
|Z| = \sqrt {{a^2} + {b^2}} \\
| - 12 + 5i| = \sqrt {{{( - 12)}^2} + {{( - 5)}^2}} \\
$
$
= \sqrt {144 + 25} \\
= \sqrt {169} \\
= 13 \\
$
In the above Equation, all the Modulus values are Positive. So we made a Conclusion that the modulus of a Complex Number is always positive.
Hence, the correct option is (A).

Note:
If the Complex number is a real number then the Modulus of the complex number is equal to the Modulus of the real part. A complex number's modulus is always a non-negative real value. Hence, the modulus is always positive for all complex numbers.