Question

How do you find \[\left| 12-5i \right|\]?

Answer 1

Hint: This type of question is based on the concept of complex numbers. Here, \[\left| 12-5i \right|\] means modulus of the complex number 12-5i. We know that, for a complex number \[a+ib\] the modulus is \[\left| a+ib \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}\]. Here, a=12 and b=(-5).Then, substitute a and b in \[\sqrt{{{a}^{2}}+{{b}^{2}}}\] and find the squares of 12 and -5, that is, 144 and 25. Then, substitute these values in the formula. And find the value of \[\left| 12-5i \right|\] by making necessary calculations and thus we get the final required answer.

Complete step by step answer:
According to the question, we are asked to find the value of \[\left| 12-5i \right|\].
We have been given the function is \[\left| 12-5i \right|\]. --------(1)
First, let us consider 12-5i.
We know that, for a complex number \[a+ib\]
\[\left| a+ib \right|\] is the modulus of \[a+ib\] which is the magnitude of \[a+ib\].
Therefore, \[\left| a+ib \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}\].
Now, let us compare \[a+ib\] with \[12-5i\].
Here, we get, a=12 and b=(-5).
\[\Rightarrow \left| 12-5i \right|=\sqrt{{{\left( 12 \right)}^{2}}+{{\left( -5 \right)}^{2}}}\]
We know that \[{{\left( 12 \right)}^{2}}=144\] and \[{{\left( -5 \right)}^{2}}=25\].
Substituting this in the above expression, we get,
\[\Rightarrow \left| 12-5i \right|=\sqrt{144+25}\]
On further simplification, we get
\[\left| 12-5i \right|=\sqrt{169}\]
But we know that \[{{13}^{2}}=169\] and \[\sqrt{169}=13\].
We get,
\[\left| 12-5i \right|=13\]
Therefore, \[\left| 12-5i \right|=13\].
Hence, the modulus of 12-5i is 13.

Note:
We must not get confused with the composite of complex numbers. We should avoid calculation mistakes based on sign conventions. It is advisable to substitute the value of ‘a’ and ‘b’ in the formula to get an accurate answer and avoid confusion. Also, we never get a negative term under the root while calculating the modulus of a complex number. Always note that the modulus of any function is the magnitude of the function. We should know the square of basic numbers to find the answer easily.