Question

How do you find the absolute value of $3 + 4i$ ?

Answer 1

In this question, we have a complex number and we need to find the absolute value of a complex number. For finding the absolute value of a complex number, we will identify the coefficient of the real and imaginary part of the complex number and then calculate its resultant.

Complete step by step answer:
In this question, we have a complex number, whose absolute value is to be found. As we know that the complex number is defined as the combination of a number which has real number and imaginary number.it is written in the form of $a + bi$. Where $a$ and $b$ are the real numbers and $i$ is an imaginary unit.
$\Rightarrow {i^2} = - 1$
The above equation is not satisfied by any real number, so it is called an imaginary number.
Now according to the question, a complex number is given below.
$\Rightarrow 3 + 4i$
Where,
$a = 3$
$b = 4$
Then we find the absolute value of this number by using the above formula.
Hence the formula is.
$\left| {a + bi} \right| = \sqrt {\left( {{a^2} + {b^2}} \right)}$
Now we will put the value of $a$ and $b$ in the above formulas
$\Rightarrow \left| {3 + 4i} \right| = \sqrt {\left( {{3^2} + {4^2}} \right)}$
Now, we will simplify the above expression as,
$\Rightarrow \left| {3 + 4i} \right| = \sqrt {9 + 16}$
$\Rightarrow \left| {3 + 4i} \right| = \sqrt {25}$
After simplification we will get,
$\therefore \left| {3 + 4i} \right| = 5$

Therefore, the absolute value of $3 + 4i$ is $5$.

Note:
As we know that the absolute value of a complex number, it is also called the “modulus”. The absolute value of a complex number is defined as the distance between origins and the coordinate point in which the real part of the complex number denotes the x-axis and the coefficient of the imaginary part denotes the y-axis point. If the origin is $\left( {0,\;0} \right)$ and the point is $\left( {a,\;b} \right)$ in the complex plane then the absolute value of a complex number is expressed as below.
$\Rightarrow \left| {a + bi} \right| = \sqrt {\left( {{a^2} + {b^2}} \right)}$