## Complex Numbers and Quadratic Equations

# The Modulus and the Conjugate of a Complex Number

**Tips:**

- Conjugate of a complex number a + ib is defined as a – ib
- Geometrically the conjugate of ‘z’ is the reflection or point image of ‘z’ in the real axis.
- \tt z = \overline{z} if and only if ‘z’ is purely real
- if and only if ‘z’ is purely imaginary.
- \tt \overline{z^{n}} = (\overline{z})^{n} \ and \ \overline{z_{1}\cdot z_{2}} = \overline{z_{1}}\cdot \overline{z_{2}}
- \tt z_{1}\overline{z_{2}} + \overline{z_{1}} \ z_{2} = 2 \ Real \ (\overline{z_{1}} \cdot z_{2}) = 2 \ Real (z_{1}\overline{z_{2}}))
- \tt z \cdot \overline{z} = |z|^{2} ; |z|^{n} = |z|^{n}
- |z
_{1}± z_{2}| ≤ |z_{1}| + |z_{2}| - |z
_{1}+ z_{2}|^{2}= |z_{1}|^{2}+ |z_{2}|^{2}+ 2 Real \tt (z_{1}\overline{z_{2}}) - |z
_{1}− z_{2}|^{2}= |z_{1}|^{2}+ |z_{2}|^{2}− 2 Real \tt (z_{1}\overline{z_{2}}) - \mid z_{1} + z_{2}\mid^{2} = |z_{1}|^{2} + |z_{2}|^{2} \Rightarrow \frac{z_{1}}{z_{2}} is purely imaginary (or) Real \tt \left(\frac{z_{1}}{z_{2}}\right) = 0

**Tricks:**

- To put the complex number \tt \frac{a + ib}{c + id} in the form A + iB we should multiply the numerator and the denominator by the conjugate of the denominator.
- Most of the complex numbers equations are solved using the property \tt z\overline{z} = |z|^{2}

### Part1: View the Topic in this video From 09:52 To 51:50

### Part2: View the Topic in this video From 00:40 To 56:40

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**Properties of Modulus**

(i) \mid z \mid \geq 0

(ii) If \mid z \mid =0, then *z* = 0 i.e., Re(*z*) = 0 = Im(*z*)

(iii) −|*z*| ≤ Re(z) ≤ |*z*| and −|*z*| ≤ Im (*z*) ≤ |*z*|

(iv) \mid z \mid =\mid\bar{z}\mid=\mid-z\mid=\mid-\bar{z}\mid

(v) z\bar{z}=\mid z\mid^{2}

(vi) |*z*_{1}*z*_{2}| = |*z*_{1}||*z*_{2}|

In general |*z*_{1}*z*_{2}*z*_{3 }....*z*_{n}| = |*z*_{1}||*z*_{2}||*z*_{3}| .... |*z*_{n}|

(vii) \begin{vmatrix}\frac{z_{1}}{z_{2}}\end{vmatrix}=\frac{|z_{1}|}{|z_{2}|}, provided z_{2} ≠ 0

(viii) |*z*_{1 }± *z*_{2}| ≤ |*z*_{1}| + |*z*_{2}|

In general |*z*_{1 }± *z*_{2 }± *z*_{3 }± .....± *z*_{n}| ≤ |*z*_{1}| + |*z*_{2}| + |*z*_{3}|+ .... +|*z*_{n}|

(ix) |*z*_{1 }± *z*_{2}| ≥ |*z*_{1}| − |*z*_{2}|

(x) |*z*^{n}| = |*z*|^{n}

(xi) \mid z_{1}\pm z_{2}\mid^{2}=(z_{1}+z_{2})(\bar{z}_{1}\pm \bar{z}_{2})=\mid z_{1}\mid^{2}+\mid z_{2}\mid^{2}\pm (z_{1}\bar{z}_{2}+\bar{z}_{1}z_{2}) or \mid z_{1}\mid^{2}+\mid z_{2}\mid^{2}\pm 2 \ Re(z_{1}\bar{z}_{2})

(xii) \mid z_{1}+z_{2}\mid^{2}=\mid z_{1}\mid^{2}+\mid z_{2}\mid^{2}\Rightarrow\frac{z_{1}}{z_{2}} is purely imaginary or Re\left(\frac{z_{1}}{z_{2}}\right)=0

(xiii) \mid z_{1}+z_{2}\mid^{2}+\mid z_{1}-z_{2}\mid^{2}=2\left\{\mid z_{1}\mid^{2}+\mid z_{2}\mid^{2}\right\} (Law of parallelogram)

**Properties of Conjugate:**

(i) \overline{(\bar{z})}=z

(ii) z+\bar{z}\Leftrightarrow z is purely real

(iii) z-\bar{z}\Leftrightarrow z is puely imaginary

(iv) Re(z)=\frac{z+\bar{z}}{2}

(v) Im(z)=\frac{z-\bar{z}}{2i}

(vi) \overline{z_{1}+z_{2}}=\bar{z}_{1}+\bar{z}_{2}

(vii) \overline{z_{1}-z_{2}}=\bar{z}_{1}-\bar{z}_{2}

(viii) \overline{z_{1}\cdot z_{2}}=\bar{z}_{1}\cdot \bar{z}_{2}

(ix) \overline{\left(\frac{z_{1}}{z_{2}}\right)}=\frac{\bar{z}_{1}}{\bar{z}_{2}},z_{2}\neq0

(x) (\bar{z})^n=\overline{\left(z^{n}\right)}