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| '''Answer''': The conjugate of <math>\frac{(3-2i)(2+3i)}{(1+2i)(2-i)}</math><math>=\frac{63}{25}+\frac{16i}{25}</math> | | '''Answer''': The conjugate of <math>\frac{(3-2i)(2+3i)}{(1+2i)(2-i)}</math><math>=\frac{63}{25}+\frac{16i}{25}</math> |
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| | === Properties of Conjugate of a Complex Number === |
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| | *<math>\overline{z_1z_2}=\overline{z_1} \ \overline{z_2}</math> |
| | * <math>\overline{z_1 \pm z_2}=\overline{z_1} \pm \overline{z_2}</math> |
| | *<math>\overline{\left [\frac{z_1}{z_2} \right]}=\frac{\overline{z_1}}{ \overline{z_2}}</math>provided <math> |
| | \left\vert z_2 \right\vert \ne 0</math> |
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| == Modulus of a Complex Number == | | == Modulus of a Complex Number == |
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| <math>\left\vert z \right\vert=\frac{\sqrt{13}}{2}</math> | | <math>\left\vert z \right\vert=\frac{\sqrt{13}}{2}</math> |
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| === Properties of Modulus of Complex Number === | | === Properties of Modulus of a Complex Number === |
| * <math>\left\vert z1z2 \right\vert=\left\vert z1 \right\vert | | * <math>\left\vert z_1z_2 \right\vert=\left\vert z_1 \right\vert |
| \left\vert z2 \right\vert</math> | | \left\vert z_2 \right\vert</math> |
| * <math>\left\vert \frac{z1}{z2} \right\vert=\frac{\left\vert z1 \right\vert}{ | | * <math>\left\vert \frac{z_1}{z_2} \right\vert=\frac{\left\vert z_1 \right\vert}{ |
| \left\vert z2 \right\vert}</math> provided <math> | | \left\vert z_2 \right\vert}</math> provided <math> |
| \left\vert z2 \right\vert \ne 0</math> | | \left\vert z_2 \right\vert \ne 0</math> |
| * | | * |
Revision as of 12:56, 12 November 2023
The modulus of a complex number gives the distance of the complex number from the origin in the Argand plane, whereas the conjugate of a complex number gives the reflection of the complex number about the real axis in the Argand plane.
Conjugate of a Complex Number
Fig 1 - Conjugate of a complex number
The representation of a complex number and its conjugate in the Argand plane are, respectively , the points and
Geometrically , the point is the mirror image of the point on the real axis (Fig 1) .Here is the real axis.
Example
Find the conjugate of
Answer: The conjugate of
Properties of Conjugate of a Complex Number
- provided
Modulus of a Complex Number
Fig 2 - Modulus of a complex number
In the Argand plane, the modulus of the complex number is
is the distance between the point representing the complex number and representing the origin (Fig 2).
Example
Find the modulus of the complex number
Properties of Modulus of a Complex Number
- provided