Question

Find the modulus of the complex number \[2-5i\]

Answer 1

Hint: We solve this problem by using the direct formula for finding the modulus of a complex number.
The modulus of a complex number is given as the square root of the sum of squares of a real part and the imaginary part of the complex number.
The modulus of complex number \[a+ib\] is given as
\[\left| a+ib \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}\]

Complete step by step answer:
We are asked to find the modulus of complex number \[2-5i\]
Let us assume that the given complex number as
\[\Rightarrow z=2-5i\]
We know that the modulus of a complex number is given as the square root of the sum of squares of a real part and imaginary part of the complex number.
That is the modulus of complex number \[a+ib\] is given as
\[\left| a+ib \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}\]
By using the above formula to given complex number then we get
\[\begin{align}
  & \Rightarrow \left| z \right|=\sqrt{{{2}^{2}}+{{\left( -5 \right)}^{2}}} \\
 & \Rightarrow \left| z \right|=\sqrt{4+25} \\
 & \Rightarrow \left| z \right|=\sqrt{29} \\
\end{align}\]
Therefore, we can conclude that the modulus of given complex number is \[\sqrt{29}\]

Note:
 We can solve this problem by using the property of complex numbers.
We have the property of complex numbers as
\[{{\left| z \right|}^{2}}=z\times \bar{z}\]
Where, \[\bar{z}\] is the conjugate of\[z\]
If \[z=a+ib\] then \[z=a-ib\]
We are asked to find the modulus of complex number \[2-5i\]
Let us assume that the given complex number as
\[\Rightarrow z=2-5i\]
By using the conjugate of a complex number let us take the conjugate of given number as
\[\Rightarrow \bar{z}=2+5i\]
Now, by using the above property of complex numbers then we get
\[\Rightarrow {{\left| z \right|}^{2}}=\left( 2-5i \right)\left( 2+5i \right)........equation(i)\]
We know that the formula of algebra that is
\[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]
We also know that the imaginary number as
\[\begin{align}
  & \Rightarrow i=\sqrt{-1} \\
 & \Rightarrow {{i}^{2}}=-1 \\
\end{align}\]
By using the required formulas in equation (i) then we get
\[\begin{align}
  & \Rightarrow {{\left| z \right|}^{2}}={{\left( 2 \right)}^{2}}-{{\left( 5i \right)}^{2}} \\
 & \Rightarrow {{\left| z \right|}^{2}}=4-25\left( -1 \right) \\
 & \Rightarrow \left| z \right|=\sqrt{29} \\
\end{align}\]
Therefore, we can conclude that the modulus of given complex number is \[\sqrt{29}\]