Question

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

Answer 1

Hint: We use a variable to denote the given complex number. Compare the complex number with the general complex number and write values of ‘x’ and ‘y’. Use the formula of modulus or absolute value of a complex number to calculate the absolute value.
* In a complex number \[z = x + iy\], the real part is x and the imaginary part is y.
* A complex number is said to be purely real if it has an imaginary part equal to zero and purely imaginary if it has a real part equal to zero.
* Absolute value of \[z = x + iy\] is given by \[\left| z \right| = \sqrt {{x^2} + {y^2}} \]

Complete step-by-step answer:
We are given a complex number \[4 + 3i\]
Let us denote the complex number by ‘z’
Then we can write \[z = 4 + 3i\]
Compare the given complex number with general complex number i.e. \[z = x + iy\]
Then \[x = 4,y = 3\]
Now we know absolute value of a complex number \[z = x + iy\] is given by the formula \[\left| z \right| = \sqrt {{x^2} + {y^2}} \]
Then absolute value of complex number \[z = 4 + 3i\] will be
\[ \Rightarrow \left| z \right| = \sqrt {{4^2} + {3^2}} \]
Calculate the square of numbers under the square root in right hand side of the equation
\[ \Rightarrow \left| z \right| = \sqrt {16 + 9} \]
Add the numbers under the square root in right hand side of the equation
\[ \Rightarrow \left| z \right| = \sqrt {25} \]
Since we know that \[25 = {5^2}\], substitute the value of \[25 = {5^2}\] under the square root in right hand side of the equation
\[ \Rightarrow \left| z \right| = \sqrt {{5^2}} \]
Cancel square root by square power in right hand side of the equation
\[ \Rightarrow \left| z \right| = 5\]
Here we ignore the negative value because modulus or absolute value tells us the distance or length and that cannot be negative.

\[\therefore \]The absolute value of \[4 + 3i\] is 5.

Note:
Many students make the mistake of writing both the values i.e. \[ + 5\] and \[ - 5\] in the final answer which is wrong as the absolute value or modulus is always positive as it denotes the distance between the complex number and the origin.