mathematics-class-11-unit-05-chapter-10-complex-numbers-l-10-15-qzfisxlgpbo

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number: lecture 3</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Properties of modulus </B></Primary></p>
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<p><strong>(3)</strong>  $|z|=|-z|=|\bar z|$</p>
<p><strong>(4)</strong>  $z. \bar z=|z|^2$</p>
<p><strong>(5)</strong> $ |z_1 \cdot z_2|=|z_1|\cdot|z_2|$</p>
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<span style="color:blue">Property of modulus</span> $\to$ Triangle inequality $\to$ Proof $\to$ Condition for equality $\to$ General inequality for complex number $\to$ Parallelogram law $\to$ Problem $\to$ Solution $\to$ Remark $\to$ Observation $\to$ Theorem $\to$ Unimodular
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number: lecture 3</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Properties of modulus </B></Primary></p>
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<p><strong>Proof:</strong></p>
<p>Consider $\ |z_1 \cdot z_2|^2=(z_1 \cdot z_2) \overline {(z_1 \cdot z_2)}$</p>
<p>$=(z_1 \cdot z_2) \bar {|z_1|}\cdot \bar{|z_2|}$</p>
<p>$=|z_1|^2|z_2|^2$</p>
<p>$|z_1 \cdot z_2|=|z_1||z_2|$</p>
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<span style="color:blue">Property of modulus</span> $\to$ Triangle inequality $\to$ Proof $\to$ Condition for equality $\to$ General inequality for complex number $\to$ Parallelogram law $\to$ Problem $\to$ Solution $\to$ Remark $\to$ Observation $\to$ Theorem $\to$ Unimodular
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number: lecture 3</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Proof</B></Primary></p>
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<p>$|z_1+z_2|^2=(z_1+z_2) \cdot \overline {(z_1+z_2)} \quad \{\because |z|^2=z \cdot \bar z\}$</p>
<p>$=(z_1+z_2)( \bar z_1+ \bar z_2)$</p>
<p>$=z_1 \cdot \bar z_1+z_1 \bar z_2+z_2 \bar z_1+z_2\cdot \bar z_2$</p>
<p>$=|z_1|^2+z_1 \bar z_2+\overline{z_1 \bar z_2}+|z|^2$</p>
<p>$=|z_1|^2+2Re(z_1 \bar z_2)+|z_2|^2$</p>
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Property of modulus $\to$ Triangle inequality $\to$ <span style="color:blue">Proof</span> $\to$ Condition for equality $\to$ General inequality for complex number $\to$ Parallelogram law $\to$ Problem $\to$ Solution $\to$ Remark $\to$ Observation $\to$ Theorem $\to$ Unimodular
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number: lecture 3</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Proof</B></Primary></p>
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<p>$|z_1+z_2|^2 =|z_1|^2+2Re(z_1 \bar z_2)+|z_2|^2$ <br>
We know that,<br>
$|Re(z)| \leq |z|$</p>
<p>$\leq |z_1|^2+2|z_1 \bar z_2|+|z_2|^2$</p>
<p>$\leq |z_1|^2+2|z_1|\cdot|z_2|+|z_2|^2$</p>
<p>$\leq (|z_1|+|z_2|)^2$</p>
<p>Thus,</p>
<p>$|z_1+z_2| \leq |z_1|+|z_2| $</p>
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Property of modulus $\to$ Triangle inequality $\to$ <span style="color:blue">Proof</span> $\to$ Condition for equality $\to$ General inequality for complex number $\to$ Parallelogram law $\to$ Problem $\to$ Solution $\to$ Remark $\to$ Observation $\to$ Theorem $\to$ Unimodular
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number: lecture 3</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Remark</B></Primary></p>
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<p>$ \quad |z|=1 \ \Rightarrow z\bar z=1 \ \Rightarrow \bar z=\frac{1}{z}$</p>
<p>$\quad |\bar z|=\frac{1}{|z|}=1.$</p>
<p><strong>Explanation:</strong><br>
Consider $U =\{z \in \mathbb{C} : |z|=1\}$<br>
$|z|=1$<br>
$ z=x+iy \quad \quad |z|^2=x^2+y^2=1$<br>
$|z|=1 \ \text{then} \ \bar z= \frac{1}{z}$</p>
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Property of modulus $\to$ Triangle inequality $\to$ Proof $\to$ Condition for equality $\to$ General inequality for complex number $\to$ Parallelogram law $\to$ Problem $\to$ Solution $\to$ <span style="color:blue">Remark</span> $\to$ Observation $\to$ Theorem $\to$ Unimodular
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number: lecture 3</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Observation</B></Primary></p>
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<p>$(1) z_1,z_2 \in U \ \Rightarrow z_1.z_2 \in U$</p>
<p>$(2) z \in U \Rightarrow z^{-1} \in U, z^{-1}= \bar z$</p>
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Property of modulus $\to$ Triangle inequality $\to$ Proof $\to$ Condition for equality $\to$ General inequality for complex number $\to$ Parallelogram law $\to$ Problem $\to$ Solution $\to$ Remark $\to$ <span style="color:blue">Observation</span> $\to$ Theorem $\to$ Unimodular
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number: lecture 3</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Unimodular</B></Primary></p>
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<p>$z$ is said to be unimodular if $|z|=1.$</p>
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Property of modulus $\to$ Triangle inequality $\to$ Proof $\to$ Condition for equality $\to$ General inequality for complex number $\to$ Parallelogram law $\to$ Problem $\to$ Solution $\to$ Remark $\to$ Observation $\to$ Theorem $\to$ <span style="color:blue">Unimodular</span>
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Complex number: lecture 3</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Problem 3</B></Primary></p>
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<p>Let $z_1,z_2 \in \mathbb{C}$. Suppose $ \frac{z_1-2z_2}{2-z_1 \bar z_2} $is unimodular and $z_2$ is not unimodular. Then, the point $z_1$ lies on a</p>
<p>(a) straight line parallel to x-axis.</p>
<p>(b) straight line parallel to y-axis.</p>
<p>(c) circle of radius $2 \ \ (\text{i.e.}, \ |z_1|=2)$</p>
<p>(d) circle of radius $\sqrt{2}.$</p>
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Property of modulus $\to$ Triangle inequality $\to$ Proof $\to$ Condition for equality $\to$ General inequality for complex number $\to$ Parallelogram law $\to$ <span style="color:blue">Problem</span> $\to$ Solution $\to$ Remark $\to$ Observation $\to$ Theorem $\to$ Unimodular
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