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Question
AnswersProve that for any complex number \[z\],
\[|\operatorname{Re} (z)| + |\operatorname{Im} (z)| \leqslant |z|\sqrt 2 \]
or \[|x| + |y| \leqslant \sqrt 2 |x + iy|\]
Answer
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Hint:- \[\operatorname{Re} (z)\]is the real part of \[z\] and \[\operatorname{Im} (z)\] is the imaginary part of \[z\].
If \[z\] is any complex number then it can be written as,
\[ \Rightarrow z = x + iy = r{e^{i\theta }} = r(\cos \theta + i\sin \theta )\] (1)
As we know that \[\operatorname{Re} (z) = x,\operatorname{Im} (z) = y{\text{ and }}|z| = r\]
Using equation 1 we can write,
\[ \Rightarrow |x| + |y| = r[|\cos \theta | + |\sin \theta |]\] (2)
Now, to prove the given relation.
On squaring equation 2 we get,
\[ \Rightarrow {[|x| + |y|]^2} = {r^2}[{\cos ^2}\theta + {\sin ^2}\theta + |2\sin \theta \cos \theta |]\]
As we know that, \[{\cos ^2}\theta + {\sin ^2}\theta = 1\].
So, solving above equation it becomes,
\[ \Rightarrow {[|x| + |y|]^2} = {r^2}[1 + |2\sin \theta \cos \theta |]\]
As we know that according to trigonometric identities,
\[2\sin \theta \cos \theta = \sin 2\theta \]
So, on solving the above equation. It becomes,
\[ \Rightarrow {[|x| + |y|]^2} = {r^2}[1 + |\sin 2\theta |]\]
Now, as we know that, (3)
\[ \Rightarrow {\text{|}}\sin x| \leqslant 1\]
So, the maximum value of \[{\text{sin2}}\theta = 1\].
Now , equation 3 becomes,
\[ \Rightarrow {[|x| + |y|]^2} \leqslant 2{r^2}\]
Now, taking square roots to both sides of the above equation. We get,
\[ \Rightarrow |x| + |y| \leqslant \sqrt 2 r\]
So, above equation can be written as,
\[ \Rightarrow |\operatorname{Re} (z)| + |\operatorname{Im} (z)| \leqslant \sqrt 2 |z|\]
Using equation 1 above equation can be written as,
\[ \Rightarrow |x| + |y| \leqslant \sqrt 2 |x + iy|\]
Hence, \[|\operatorname{Re} (z)| + |\operatorname{Im} (z)| \leqslant |z|\sqrt 2 {\text{ }}\]or \[|x| + |y| \leqslant \sqrt 2 |x + iy|\]
Note:- Whenever you came up with this type of problem then easiest and efficient way
is to write complex number \[z\], in polar form \[\left( {z = r{e^{i\theta }}} \right)\], in terms of \[{\text{sin}}\theta \]and \[\cos \theta \]\[\left( {z = r(\cos \theta + i\sin \theta )} \right)\], or in terms of x and y \[\left( {z = x + iy} \right)\] as per required result to be proved.
If \[z\] is any complex number then it can be written as,
\[ \Rightarrow z = x + iy = r{e^{i\theta }} = r(\cos \theta + i\sin \theta )\] (1)
As we know that \[\operatorname{Re} (z) = x,\operatorname{Im} (z) = y{\text{ and }}|z| = r\]
Using equation 1 we can write,
\[ \Rightarrow |x| + |y| = r[|\cos \theta | + |\sin \theta |]\] (2)
Now, to prove the given relation.
On squaring equation 2 we get,
\[ \Rightarrow {[|x| + |y|]^2} = {r^2}[{\cos ^2}\theta + {\sin ^2}\theta + |2\sin \theta \cos \theta |]\]
As we know that, \[{\cos ^2}\theta + {\sin ^2}\theta = 1\].
So, solving above equation it becomes,
\[ \Rightarrow {[|x| + |y|]^2} = {r^2}[1 + |2\sin \theta \cos \theta |]\]
As we know that according to trigonometric identities,
\[2\sin \theta \cos \theta = \sin 2\theta \]
So, on solving the above equation. It becomes,
\[ \Rightarrow {[|x| + |y|]^2} = {r^2}[1 + |\sin 2\theta |]\]
Now, as we know that, (3)
\[ \Rightarrow {\text{|}}\sin x| \leqslant 1\]
So, the maximum value of \[{\text{sin2}}\theta = 1\].
Now , equation 3 becomes,
\[ \Rightarrow {[|x| + |y|]^2} \leqslant 2{r^2}\]
Now, taking square roots to both sides of the above equation. We get,
\[ \Rightarrow |x| + |y| \leqslant \sqrt 2 r\]
So, above equation can be written as,
\[ \Rightarrow |\operatorname{Re} (z)| + |\operatorname{Im} (z)| \leqslant \sqrt 2 |z|\]
Using equation 1 above equation can be written as,
\[ \Rightarrow |x| + |y| \leqslant \sqrt 2 |x + iy|\]
Hence, \[|\operatorname{Re} (z)| + |\operatorname{Im} (z)| \leqslant |z|\sqrt 2 {\text{ }}\]or \[|x| + |y| \leqslant \sqrt 2 |x + iy|\]
Note:- Whenever you came up with this type of problem then easiest and efficient way
is to write complex number \[z\], in polar form \[\left( {z = r{e^{i\theta }}} \right)\], in terms of \[{\text{sin}}\theta \]and \[\cos \theta \]\[\left( {z = r(\cos \theta + i\sin \theta )} \right)\], or in terms of x and y \[\left( {z = x + iy} \right)\] as per required result to be proved.