mathematics-class-11-unit-05-chapter-09-complex-numbers-l-9-15-hwfqn3amc0s

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Conjugate of complex number</B></Primary></p>
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<li>$\bar z$ is mirror image of $z$ with respect to x-axis.</li>
<li>e.g: $z=2+3i \quad \bar z=2-3i$</li>
<li>$ z=5 \quad \bar z=5$</li>
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Complex number $\to$ Remark  $\to$ Notations $\to$<span style="color:blue"> Conjugate of complex number</span> $\to$ Properties  $\to$ Problem $\to$ Solution $\to$ Identities $\to$ Exercise $\to$ Modulus of a complex number
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Properties</B></Primary></p>
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<p><strong>(1)</strong> $\ z=\bar z \ \text{iff} \ z \in \mathbb{R} \ \text{(or)} \ \text{Im}(z)=0$</p>
<p>Let $z=a+ib$. Suppose $ z=\bar z \ \Leftrightarrow a+ib=a-ib$</p>
<p>$\Leftrightarrow 2ib=0$</p>
<p>$\Leftrightarrow b=0$</p>
<p>i.e., Im$(z)=0$</p>
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Complex number $\to$ Remark  $\to$ Notations $\to$ Conjugate of complex number $\to$ <span style="color:blue">Properties</span>  $\to$ Problem $\to$ Solution $\to$ Identities $\to$ Exercise $\to$ Modulus of a complex number
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Properties</B></Primary></p>
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<p><strong>(2)</strong> $\ z= \bar{\bar z}$</p>
<p><strong>Proof:</strong> Let $z=a+ib \Rightarrow \bar z=a-ib=a+i(-b)$</p>
<p>$\bar z=a+i[-(-b)]$</p>
<p>$=a+ib$</p>
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Complex number $\to$ Remark  $\to$ Notations $\to$ Conjugate of complex number $\to$ <span style="color:blue">Properties</span>  $\to$ Problem $\to$ Solution $\to$ Identities $\to$ Exercise $\to$ Modulus of a complex number
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Properties</B></Primary></p>
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<p><strong>(3)</strong> $ \ \overline {z_1+z_2}=\bar z_1+ \bar z_2$</p>
<p><strong>Proof:</strong>   $ z_1=a+ib \quad z_2=c+id$</p>
<p>$\overline {z_1+z_2}= \overline {(a+c)+i(b+d)}=(a+c)-i(b+d)$</p>
<p>$=a-ib+c-id$</p>
<p>$=\bar z_1+\bar z_2$</p>
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Complex number $\to$ Remark  $\to$ Notations $\to$ Conjugate of complex number $\to$ <span style="color:blue">Properties</span>  $\to$ Problem $\to$ Solution $\to$ Identities $\to$ Exercise $\to$ Modulus of a complex number
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Properties</B></Primary></p>
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<p><strong>(4)</strong> $\quad$  For every $z \in \mathbb{C}$</p>
<p>$z \bar z $ is a non-negative real number.</p>
<p>$(a+ib)(a-ib)=a^2+b^2 \geq0$</p>
<p><strong>Remark</strong></p>
<ul>
<li>$z_1z_2 \neq 0$</li>
<li>Then $z_1 \neq 0 \ and \ z_2 \neq 0 \ \ (??)$</li>
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Complex number $\to$ Remark  $\to$ Notations $\to$ Conjugate of complex number $\to$ <span style="color:blue">Properties</span>  $\to$ Problem $\to$ Solution $\to$ Identities $\to$ Exercise $\to$ Modulus of a complex number
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Properties</B></Primary></p>
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<p><strong>(5)</strong> $\quad \overline {z_1.z_2}=\bar z_1.\bar z_2$</p>
<p>$\overline {z_1.z_2}=\overline {(a+ib)(c+id)}=(ac-bd)-i(ad+bc)$</p>
<p>$=(a-ib).(c-id)$</p>
<p>$=\bar z_1.\bar z_2$</p>
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Complex number $\to$ Remark  $\to$ Notations $\to$ Conjugate of complex number $\to$ <span style="color:blue">Properties</span>  $\to$ Problem $\to$ Solution $\to$ Identities $\to$ Exercise $\to$ Modulus of a complex number
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Properties</B></Primary></p>
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<p><strong>(6)</strong> $ \ \bar z^{-1}=(\bar z)^{-1}, z \neq 0$</p>
<p><strong>Proof:</strong>  $z.z^{-1}=1$</p>
<p>$ \overline {z.z^{-1}}=1 \ \Rightarrow \bar z.\bar z^{-1}=1$</p>
<p>$\Rightarrow (\bar z)^{-1}=\overline z^{-1}$</p>
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Complex number $\to$ Remark  $\to$ Notations $\to$ Conjugate of complex number $\to$<span style="color:blue"> Properties</span>  $\to$ Problem $\to$ Solution $\to$ Identities $\to$ Exercise $\to$ Modulus of a complex number
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Properties</B></Primary></p>
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<p><strong>(7)</strong> $ \ \overline{ (\frac {z_1}{z_2})}=\frac{\bar z_1}{\bar z_2}, z_2 \neq 0$</p>
<p><strong>Proof:</strong>  From 5 and 6,</p>
<p>$\overline{ (\frac {z_1}{z_2})} = \overline { ({z_1. \frac {1}{z_2}})} = \bar z_1.\bar z_2^{-1}$</p>
<p>$ = \bar z_1.(\bar z_2)^{-1}$</p>
<p>$\overline{ (\frac {z_1}{z_2})}=\bar z_1.(\bar z_2)^{-1}=\frac{\bar z_1}{\bar z_2}$</p>
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Complex number $\to$ Remark  $\to$ Notations $\to$ Conjugate of complex number $\to$ <span style="color:blue">Properties</span>  $\to$ Problem $\to$ Solution $\to$ Identities $\to$ Exercise $\to$ Modulus of a complex number
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Properties</B></Primary></p>
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<p><strong>(8)</strong> $ \ \text{Re}(z) = \frac{z+\bar z}{2} \;\ \text{Im}(z)=\frac{z-\bar z}{2i}$</p>
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<li>
<p>$z = \text{Re}(z)+i \ \text{Im}(z)$</p>
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<li>
<p>$z = \text{Re}(z) - i \text{Im}(z)$</p>
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Complex number $\to$ Remark  $\to$ Notations $\to$ Conjugate of complex number $\to$ <span style="color:blue">Properties</span>  $\to$ Problem $\to$ Solution $\to$ Identities $\to$ Exercise $\to$ Modulus of a complex number
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Problem 1</B></Primary></p>
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<p>If  $x+iy=\sqrt {\frac{a+ib}{c+id}}, $ then show that $(x^2+y^2)^2=\frac{a^2+b^2}{c^2+d^2}$</p>
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
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<p>Let $z_1,z_2 \in \mathbb{C}$. Show that $z_1.\bar z_2+\bar z_1.z_2$ is a real number.</p>
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Problem 2</B></Primary></p>
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<p>If the sum and product of three complex numbers are real then which of the following is/are possible for some $z_1,z_2,z_3 \in \mathbb{C}$:</p>
<p>(a) Exactly one of the three number is non-real.</p>
<p>(b) Exactly two of the three numbers are non-real.</p>
<p>(c) All three numbers are non-real.</p>
<p>(d) All three numbers are pure imaginary and non-zero.</p>
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Complex number $\to$ Remark  $\to$ Notations $\to$ Conjugate of complex number $\to$ Properties  $\to$ <span style="color:blue">Problem</span> $\to$ Solution $\to$ Identities $\to$ Exercise $\to$ Modulus of a complex number
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p>
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<p><strong>(b)</strong> $z_1+z_2+z_3 \in \mathbb{R}$</p>
<p>$z_1.z_2.z_3 \in \mathbb{R}$</p>
<p>$z_1=1+i \ z_2=1-i \ z_3=1$</p>
<p>Yes</p>
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Complex number $\to$ Remark  $\to$ Notations $\to$ Conjugate of complex number $\to$ Properties  $\to$ Problem $\to$ <span style="color:blue">Solution</span> $\to$ Identities $\to$ Exercise $\to$ Modulus of a complex number
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p>
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<p><strong>(c)</strong>  Yes $\quad  z_1=1-i, \ z_2=1-i, \ z_3=2i$</p>
<p>$z_1z_2=1^2-1-2i$</p>
<p>$=-2i$</p>
<p><strong>(d)</strong> Exercise</p>
<p>No</p>
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Complex number $\to$ Remark  $\to$ Notations $\to$ Conjugate of complex number $\to$ Properties  $\to$ Problem $\to$ <span style="color:blue">Solution</span> $\to$ Identities $\to$ Exercise $\to$ Modulus of a complex number
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Modulus of a complex number</B></Primary></p>
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<ul>
<li>Let $a \in \mathbb{R}$</li>
<li></li>
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<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">|</mo><mi>a</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">|</mo></mrow><mo>:=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"><mtr><mtd><mi>a</mi></mtd><mtd><mi>a</mi><mo>≥</mo><mn>0</mn></mtd></mtr><mtr><mtd><mo>−</mo><mi>a</mi></mtd><mtd><mi>a</mi><mo><</mo><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow><mo>=</mo><mo data-mjx-texclass="OP" movablelimits="true">max</mo><mo fence="false" stretchy="false">{</mo><mi>a</mi><mo>,</mo><mo>−</mo><mi>a</mi><mo fence="false" stretchy="false">}</mo></math></p>
<ul>
<li>There is no ‘<’ in $\mathbb{C}$:</li>
<li>If there is a ‘<’ relation in $\mathbb{C}$.
<ul>
<li>Either $i < 0$ or $i > 0$</li>
<li>$i^2 > 0 $ but $-1 < 0 $</li>
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</li>
</ul>
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