Question

What is Modulus Z?

Answer 1

In order to know about Modulus Z, we must know about complex numbers. Complex numbers are represented by $z = a + ib$ where a, b are the real numbers, but $ib$ together is the imaginary part. The $'i'$ represents iota which is equal to $\sqrt { - 1}$. A complex number becomes an imaginary number when $a = 0$, similarly, it becomes a real number when $b = 0$.

Complete answer:
Suppose, we have a complex number $z = x + iy$, where $x$ is the real part of the number and $y$ is the imaginary part of the number and $i$ is iota.
According to complex number theory, Modulus of a complex number is nothing but the square root of the sum of the square of the real and the imaginary part of a complex number.
And, Modulus is numerically represented as:
$\left| z \right| = \sqrt {{x^2} + {y^2}}$
Let’s take an example of a complex number $z = 3 + i4$.
Comparing $z = 3 + i4$ with $z = x + iy$, we obtain the real part as $x = 3$ and imaginary part as $y = 4$
Since, we know that modulus is the square root of the sum of the square of the real and imaginary part, so substituting the values of x and y in $\left| z \right| = \sqrt {{x^2} + {y^2}}$, we get:
$\left| z \right| = \sqrt {{3^2} + {4^2}}$
Solving the radicands:
$\left| z \right| = \sqrt {9 + 16}$
$\Rightarrow \left| z \right| = \sqrt {25}$
$\Rightarrow \left| z \right| = 5$
Therefore, the modulus of $z = 3 + i4$ is $5$.

Note:
Since, we know that $\sqrt {25}$ can be $\pm 5$, but we wrote only $5$ as because modulus is the absolute value of the complex number z and absolute number is nothing but the distance between 0 and the number, on number line either on the left side or right side, and distance is always positive. Therefore, the absolute value is always positive.