Question

How do you find the absolute value of \[3 + 4i\] ?

Answer 1

Hint: 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.
\[ \Rightarrow \left| {a + bi} \right| = \sqrt {\left( {{a^2} + {b^2}} \right)} \]