Notes On Modulus and Conjugate of a Complex Number - CBSE Class 11 Maths
Modulus of a complex number   Modulus of a real number is its absolute value.   The modulus of a number is the value of the number excluding its sign.   |7| = 7, |– 21| = 21, | – ½ | = ½   Consider a complex number z = a + ib, where a is the real part and b the imaginary part of z.   a = Re z, b = Im z Modulus or absolute value of z = |z|   |z| = $\sqrt{{\text{a}}^{\text{2}}\text{+}{\text{b}}^{\text{2}}}$   Since a and b are real, the modulus of the complex number will also be real.   Ex: Find the modulus of z = 3 – 4i.   |z| = |3 – 4i| =                      = $\sqrt{\text{25}}$                        = 5   Comparison of complex numbers   Consider two complex numbers z1 = 2 + 3i, z2 = 4 + 2i.   Since these complex numbers have imaginary parts, it is not possible to find out the greater complex number between them.   Thus, the ordering relation (greater than or less than) of complex numbers, that is greater than or less than, is meaningless.   Ordering relations can be established for the modulus of complex numbers, because they are real numbers.   Some important properties of the modulus of complex numbers |Z| = 0 ⇔ z = 0 i.e, Re(z) = 0 and Im (z) = 0 |z| = |$\stackrel{_}{\text{z}}$| = |–z| – |z| ≤ Re(z) ≤ |z| and -|z| ≤ Im(z) ≤ |z| z-1 = $\frac{\text{1}}{\text{z}}$  = $\frac{\stackrel{_}{\text{z}}}{{\left|\text{z}\right|}^{\text{2}}}$ , z ≠ 0     z. $\stackrel{_}{\text{z}}$ = ${\left|\text{z}\right|}^{\text{2}}$  ,  $\left|{\text{z}}^{\text{2}}\right|$  =  ${\left|\stackrel{_}{\text{z}}\right|}^{\text{2}}$   $\left|{\text{z}}_{1}\text{}\mathrm{\text{.}}\text{}{\text{z}}_{2}\right|$= $\left|{\text{z}}_{1}\text{|}\right|{\text{z}}_{2}\text{|}$ $\text{|}\mathrm{\text{}}\frac{{\text{z}}_{1}}{{\text{z}}_{2}}\mathrm{\text{|=}}\mathrm{\text{}}\frac{\left|{\text{z}}_{1}\right|}{\left|{\text{z}}_{2}\right|}$ ${\left|{\text{z}}_{\text{!}}\text{}\mathrm{\text{+}}\text{}{\text{z}}_{2}\right|}^{2}\text{}\mathrm{\text{=}}{\left|{\text{z}}_{\text{1}}\right|}^{\text{2}}\mathrm{\text{+}}{\left|{\text{z}}_{\text{2}}\right|}^{\text{2}}\mathrm{\text{+ 2 Re (}}{\text{z}}_{\text{1}}\mathrm{\text{}}\stackrel{_}{{\text{z}}_{\text{2}}}\mathrm{\text{)}}$ $\mathrm{\text{≤}}$ $\mathrm{\text{}}$$\mathrm{\text{≥}}$$\mathrm{\text{}}$ ${\left|{\text{z}}_{1}\mathrm{\text{+}}{\text{z}}_{2}\right|}^{2}\mathrm{\text{+}}{\left|{\text{z}}_{1}\mathrm{\text{-}}{\text{z}}_{2}\right|}^{2}\mathrm{\text{=}}\text{2(}{\left|{\text{z}}_{1}\right|}^{2}\mathrm{\text{+}}{\left|{\text{z}}_{2}\right|}^{2}\text{)}$ ${\left|\text{b}{\text{z}}_{1}\mathrm{\text{+}}\text{a}{\text{z}}_{2}\right|}^{2}\mathrm{\text{+}}{\left|\text{a}{\text{z}}_{1}\mathrm{\text{-}}\text{b}{\text{z}}_{2}\right|}^{2}\mathrm{\text{=}}\text{(}{\text{a}}^{2}\mathrm{\text{+}}{\text{b}}^{2}\text{)(}{\left|{\text{z}}_{1}\right|}^{2}\mathrm{\text{+}}{\left|{\text{z}}_{2}\right|}^{2}\text{)}$, where a and b are real

Summary

Modulus of a complex number   Modulus of a real number is its absolute value.   The modulus of a number is the value of the number excluding its sign.   |7| = 7, |– 21| = 21, | – ½ | = ½   Consider a complex number z = a + ib, where a is the real part and b the imaginary part of z.   a = Re z, b = Im z Modulus or absolute value of z = |z|   |z| = $\sqrt{{\text{a}}^{\text{2}}\text{+}{\text{b}}^{\text{2}}}$   Since a and b are real, the modulus of the complex number will also be real.   Ex: Find the modulus of z = 3 – 4i.   |z| = |3 – 4i| =                      = $\sqrt{\text{25}}$                        = 5   Comparison of complex numbers   Consider two complex numbers z1 = 2 + 3i, z2 = 4 + 2i.   Since these complex numbers have imaginary parts, it is not possible to find out the greater complex number between them.   Thus, the ordering relation (greater than or less than) of complex numbers, that is greater than or less than, is meaningless.   Ordering relations can be established for the modulus of complex numbers, because they are real numbers.   Some important properties of the modulus of complex numbers |Z| = 0 ⇔ z = 0 i.e, Re(z) = 0 and Im (z) = 0 |z| = |$\stackrel{_}{\text{z}}$| = |–z| – |z| ≤ Re(z) ≤ |z| and -|z| ≤ Im(z) ≤ |z| z-1 = $\frac{\text{1}}{\text{z}}$  = $\frac{\stackrel{_}{\text{z}}}{{\left|\text{z}\right|}^{\text{2}}}$ , z ≠ 0     z. $\stackrel{_}{\text{z}}$ = ${\left|\text{z}\right|}^{\text{2}}$  ,  $\left|{\text{z}}^{\text{2}}\right|$  =  ${\left|\stackrel{_}{\text{z}}\right|}^{\text{2}}$   $\left|{\text{z}}_{1}\text{}\mathrm{\text{.}}\text{}{\text{z}}_{2}\right|$= $\left|{\text{z}}_{1}\text{|}\right|{\text{z}}_{2}\text{|}$ $\text{|}\mathrm{\text{}}\frac{{\text{z}}_{1}}{{\text{z}}_{2}}\mathrm{\text{|=}}\mathrm{\text{}}\frac{\left|{\text{z}}_{1}\right|}{\left|{\text{z}}_{2}\right|}$ ${\left|{\text{z}}_{\text{!}}\text{}\mathrm{\text{+}}\text{}{\text{z}}_{2}\right|}^{2}\text{}\mathrm{\text{=}}{\left|{\text{z}}_{\text{1}}\right|}^{\text{2}}\mathrm{\text{+}}{\left|{\text{z}}_{\text{2}}\right|}^{\text{2}}\mathrm{\text{+ 2 Re (}}{\text{z}}_{\text{1}}\mathrm{\text{}}\stackrel{_}{{\text{z}}_{\text{2}}}\mathrm{\text{)}}$ $\mathrm{\text{≤}}$ $\mathrm{\text{}}$$\mathrm{\text{≥}}$$\mathrm{\text{}}$ ${\left|{\text{z}}_{1}\mathrm{\text{+}}{\text{z}}_{2}\right|}^{2}\mathrm{\text{+}}{\left|{\text{z}}_{1}\mathrm{\text{-}}{\text{z}}_{2}\right|}^{2}\mathrm{\text{=}}\text{2(}{\left|{\text{z}}_{1}\right|}^{2}\mathrm{\text{+}}{\left|{\text{z}}_{2}\right|}^{2}\text{)}$ ${\left|\text{b}{\text{z}}_{1}\mathrm{\text{+}}\text{a}{\text{z}}_{2}\right|}^{2}\mathrm{\text{+}}{\left|\text{a}{\text{z}}_{1}\mathrm{\text{-}}\text{b}{\text{z}}_{2}\right|}^{2}\mathrm{\text{=}}\text{(}{\text{a}}^{2}\mathrm{\text{+}}{\text{b}}^{2}\text{)(}{\left|{\text{z}}_{1}\right|}^{2}\mathrm{\text{+}}{\left|{\text{z}}_{2}\right|}^{2}\text{)}$, where a and b are real

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