{"id":9122,"date":"2025-03-08T16:49:48","date_gmt":"2025-03-08T11:19:48","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=9122"},"modified":"2025-05-30T22:59:10","modified_gmt":"2025-05-30T17:29:10","slug":"complex-numbers-shorts-notes-and-formula-for-jee-main-and-advanced","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/complex-numbers-shorts-notes-and-formula-for-jee-main-and-advanced\/","title":{"rendered":"Complex Numbers Formulas Sheet &#8211; Complete Guide PDF"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Complex Numbers<\/h2>\n\n\n\n<p>Complex numbers are the numbers of the form $a+ ib$ where $i= \\sqrt {-1}$&nbsp; and a and b are real numbers<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Algebra for i<\/h2>\n\n\n\n<p>$i= \\sqrt {-1}$<br>$i^2= -1$<br>$i^3 =i \\times i^2=  -i$<br>$i^4= i^2 \\times i^2 = 1$<br>$i^{4n} =1$<br>$i^n + i^{n+1} + i^{n+2} + i^{n+3} =i^n( 1 + i +i^2 + i^3 ) = i^b(1 + i -1 -i)=0$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Addition of Complex Numbers<\/h2>\n\n\n\n<p>Let z<sub>1<\/sub> = x<sub>1<\/sub> + iy<sub>1&nbsp; \u00ad<\/sub> and z<sub>2<\/sub> = x<sub>2<\/sub> + iy<sub>2&nbsp;<\/sub> z<sub>1<\/sub> +z<sub>2<\/sub> = (x<sub>1<\/sub> + x<sub>2<\/sub>) + i(y<sub>1<\/sub> + y<sub>2<\/sub>)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Subtraction of Complex Numbers<\/h2>\n\n\n\n<p>Let z<sub>1<\/sub> = x<sub>1<\/sub> + iy<sub>1&nbsp; \u00ad<\/sub> and z<sub>2<\/sub> = x<sub>2<\/sub> + iy<sub>2&nbsp;<\/sub> z<sub>1<\/sub> -z<sub>2<\/sub> = (x<sub>1<\/sub> &#8211; x<sub>2<\/sub>) + i(y<sub>1<\/sub> &#8211; y<sub>2<\/sub>)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Multiplication of Complex numbers<\/h2>\n\n\n\n<p>Let z<sub>1<\/sub> = x<sub>1<\/sub> + iy<sub>1&nbsp; \u00ad<\/sub> and z<sub>2<\/sub> = x<sub>2<\/sub> + iy<sub>2&nbsp;<\/sub><\/p>\n\n\n\n<p>z<sub>1<\/sub> z<sub>2<\/sub> = (x<sub>1<\/sub> + iy<sub>1 <\/sub>) (x<sub>2<\/sub> + iy<sub>2&nbsp; <\/sub>) =(x<sub>1<\/sub>x<sub>2<\/sub> \u2013 y<sub>1<\/sub>y<sub>2<\/sub>) +i(x<sub>1<\/sub>y<sub>2<\/sub> +x<sub>2<\/sub>y<sub>1<\/sub>)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Multiplicative Inverse<\/h2>\n\n\n\n<p>Let z= a+ ib , Then Multiplicative inverse is given as<br>$\\frac {1}{z}=z^{-1} = \\frac {1}{a+ ib} = \\frac {a}{a^2 + b^2} &#8211; \\frac {ib}{a^2 + b^2}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Division of Complex Numbers<\/h2>\n\n\n\n<p>Let z<sub>1<\/sub> = x<sub>1<\/sub> + iy<sub>1&nbsp; \u00ad<\/sub> and z<sub>2<\/sub> = x<sub>2<\/sub> + iy<sub>2&nbsp;<\/sub><br>z<sub>1<\/sub> \/z<sub>2<\/sub> = (x<sub>1<\/sub> + iy<sub>1 <\/sub>) \/(x<sub>2<\/sub> + iy<sub>2&nbsp; <\/sub>) =z<sub>1<\/sub> z<sub>2<\/sub><sup>-1<\/sup><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Conjugate of Complex Number<\/h2>\n\n\n\n<p>If z= x+iy then Conjugate is given by $\\bar{z} = x-iy$<\/p>\n\n\n\n<p><strong>Properties of Conjugate of Complex Number<\/strong><\/p>\n\n\n\n<p>If z= x + iy<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$\\bar {\\left( \\bar{z} \\right)}= z$<\/li>\n\n\n\n<li>$ z + \\bar{z} = 2 Re(z) = 2x$<\/li>\n\n\n\n<li>$ z &#8211; \\bar{z} = 2 Im(z) = 2iy$<\/li>\n\n\n\n<li>If $ z + \\bar{z} = =0$, then $z=-\\bar{z} $ and z is purely Imaginary<\/li>\n\n\n\n<li>$\\overline {z_1 + z_2} = \\bar{z_1} + \\bar{z_2}$<\/li>\n\n\n\n<li>$\\overline {z_1 &#8211; z_2} = \\bar{z_1} &#8211; \\bar{z_2}$<\/li>\n\n\n\n<li> $\\overline {z_1  z_2} = \\bar{z_1}  \\bar{z_2}$<\/li>\n\n\n\n<li>$\\overline {z^n} = \\left(  \\bar{z} \\right)^n$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Modulus of complex Number<\/h2>\n\n\n\n<p>If z= z + iy, then modulus is defined<\/p>\n\n\n\n<p>$|z| = \\sqrt {x^2 + y^2}$<\/p>\n\n\n\n<p><strong>Properties of Modulus of complex Number<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$|z| \\geq 0$<\/li>\n\n\n\n<li>If |z| = 0 , then z=0<\/li>\n\n\n\n<li>|z| &gt; 0, then $z \\ne 0$<\/li>\n\n\n\n<li>$ -|z| \\leq Re(z) \\leq |z|$<\/li>\n\n\n\n<li>$ -|z| \\leq Img(z) \\leq |z|$<\/li>\n\n\n\n<li>$z \\bar{z} =|z|^2$<\/li>\n\n\n\n<li>$|z|=|-z|$<\/li>\n\n\n\n<li>$|z_1 z_2| = |z_1||z_2|$<\/li>\n\n\n\n<li>$|\\frac {z_1}{z_2}| = \\frac {|z_1|}{|z_2|}$<\/li>\n\n\n\n<li>$|z_1 + z_2| \\leq |z_1| + |z_2|$<\/li>\n\n\n\n<li>$|z_1 + z_2| \\geq ||z_1| &#8211; |z_2||$<\/li>\n\n\n\n<li>$ ||z_1| &#8211; |z_2|| \\leq  |z_1 + z_2|   \\leq |z_1| + |z_2|$<\/li>\n\n\n\n<li>$|z_1 &#8211; z_2| \\leq |z_1| &#8211; |z_2|$<\/li>\n\n\n\n<li>$|z^n| = |z|^n$<\/li>\n\n\n\n<li>$|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + (z_1 \\bar{z_2} + z_2 \\bar{z_1})$<\/li>\n\n\n\n<li>$|z_1 &#8211; z_2|^2 = |z_1|^2 + |z_2|^2 &#8211; (z_1 \\bar{z_2} + z_2 \\bar{z_1})$<\/li>\n\n\n\n<li>$|z_1 + z_2|^2 + |z_1 &#8211; z_2|^2 =2 |z_1|^2 +2 |z_2|^2$<\/li>\n\n\n\n<li>$|az_1 +b z_2|^2 + |bz_1 -a z_2|^2 =(a^2 + b^2)( |z_1|^2 + |z_2|^2)$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Argand Diagram<\/h2>\n\n\n\n<p>Complex number $z=a +ib$ can be represented by a point P (a, b) on a x-y plane. The x -axis represent the real part while the y-axis represent the imaginary part<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2024\/04\/argand_plane.png\"><img loading=\"lazy\" decoding=\"async\" width=\"268\" height=\"205\" src=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2024\/04\/argand_plane.png\" alt=\"\" class=\"wp-image-9131\"\/><\/a><\/figure>\n\n\n\n<p><strong>Important points<\/strong><br>(1) The length OP is called the module of z and is denoted by $|z| =\\sqrt {x^2 + y^2}$<br>(2) Purely real number lies on X-axis while purely imaginary number lies on y-axis<br>(3) The Line OP makes an angle&nbsp;$\\theta$&nbsp;with the positive direction of x-axis in anti-clockwise sense is called the argument or amplitude of z<br>It is given by <br>$arg (z) = \\tan^{-1} \\frac {y}{x}$<br>The unique value of $\\theta$&nbsp;such that&nbsp;$-\\pi &lt; \\theta \\leq \\pi$ &nbsp;is called the principal value of the amplitude or principal argument<br>(4) The principal argument of the complex number is find using the below steps<br>Step 1) for z=a+ib , find the acute angle value of&nbsp;$\\theta =tan^{-1} \\frac {|y|}{|x|}$<br>Step 2) Look for the values of a ,b<br>if (a, b) lies in First quadrant then Argument=$\\theta$<br>if (a, b) lies in second quadrant then Argument =$\\pi  -\\theta$<br>if (a, b) lies in third quadrant then Argument =$-\\pi + \\theta$<br>if (a, b) lies in Fourth quadrant then Argument =$-\\theta$<br><\/p>\n\n\n\n<p><strong>Properties of Argument of Complex Numbers<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$arg (z_1z_2)=  arg(z_1) + arg (z_2)  + 2k \\pi$<\/li>\n\n\n\n<li>$arg (\\frac {z_1}{z_2})=  arg(z_1) &#8211; arg (z_2)  + 2k \\pi$<\/li>\n\n\n\n<li>$arg (\\frac {z}{\\bar{z}})=  2 arg(z)   + 2k \\pi$<\/li>\n\n\n\n<li>$arg (\\bar{z})=  &#8211; arg(z) $<\/li>\n\n\n\n<li>$arg (z^n)=  n arg(z) + 2k\\pi $<\/li>\n<\/ul>\n\n\n\n<p>Where k is a integer  such that Arg lies between $-\\pi$ and $\\pi$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Polar Representation<\/h2>\n\n\n\n<p>$z= |z| (cos (arg) + i sin(arg)$<\/p>\n\n\n\n<p>Here arg is the principal argument of the complex number<\/p>\n\n\n\n<p>we can write this as<\/p>\n\n\n\n<p>$z= r(cos \\theta + \\sin \\theta)$<br>where  r =|z| and $\\theta$ is the principle argument<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Euler form<\/h2>\n\n\n\n<p>For any real number x, e<sup>ix<\/sup> = cos x + i sin x <br>Let z be a non zero complex number; we can write z in the polar form as,<br>$z = r(cos \\theta + i sin \\theta ) = re^{i \\theta}$, where r is the modulus and $\\theta$  is argument of z.<\/p>\n\n\n\n<p><strong>Important point<\/strong><\/p>\n\n\n\n<p>(1) Multiplying a complex number z with $e^{i \\alpha}$ gives<\/p>\n\n\n\n<p>$z \\times e^{i \\alpha} = r e^{i \\theta} \\times e^{i \\alpha} = r e^{i(\\alpha + \\theta)}$<\/p>\n\n\n\n<p>(2) $e^{i \\theta} + e^{-i \\theta} = 2 \\cos \\theta$<\/p>\n\n\n\n<p>(3) $e^{i \\theta} &#8211; e^{-i \\theta} = 2 i\\sin \\theta$<\/p>\n\n\n\n<p>(4) $e^{i \\theta} + e^{-i \\alpha} = e^{i (\\theta + \\alpha)\/2} 2 \\cos (\\theta &#8211; \\alpha)\/2$<\/p>\n\n\n\n<p>(5) $e^{i \\theta} + 1 = e^{i \\theta \/2} 2 \\cos \\theta \/2$<\/p>\n\n\n\n<p>(6) $e^{i \\theta} &#8211; e^{-i \\alpha} = e^{i (\\theta &#8211; \\alpha)\/2} 2 i\\sin (\\theta &#8211; \\alpha)\/2$<\/p>\n\n\n\n<p>(7)$e^{i \\theta} &#8211; 1 = e^{i \\theta \/2} i 2 \\sin \\theta \/2$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Logarithm of the Complex Number<\/h2>\n\n\n\n<p>$log _{e} z = log _ {e} |z| e^{i\\theta} = log_{e} |z| + i \\theta$<\/p>\n\n\n\n<p>General Form<\/p>\n\n\n\n<p>$log _{e} z = log_{e}  z + 2n \\pi i $<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">De Moivre&#8217;s theorem<\/h2>\n\n\n\n<p>De Moivre&#8217;s theorem states following cases<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Case I It states that for any integer n,<br><br>$(cos \\theta + i sin \\theta)^n = cos (n \\theta) + i sin (n\\theta)$<br><\/li>\n\n\n\n<li>Case II if n is of the form p\/q where p, q are integers and q &gt; 0<br>then<br>$(cos \\theta + i sin \\theta)^n =cos (2k \\pi+ \\theta)p\/q + sin (2k \\pi+ \\theta)p\/q$<br>Where k=0,1,2,&#8230;q-1<\/li>\n<\/ul>\n\n\n\n<p><strong>Important points<\/strong><\/p>\n\n\n\n<p>$(cos \\theta _1 + i sin \\theta_1) (cos \\theta _2 + i sin \\theta_2) (cos \\theta _3 + i sin \\theta_3) =cos (\\theta _1 + \\theta _2  + \\theta _3) + i sin (\\theta _1 + \\theta _2 + \\theta _3)$<\/p>\n\n\n\n<p>$(cos \\theta &#8211; i sin \\theta)^n = \\cos n \\theta &#8211; i \\sin n\\theta$<\/p>\n\n\n\n<p>$(sin \\theta + i cos \\theta)^n = (-1)^n (\\cos n \\theta &#8211; i \\sin n\\theta )$<\/p>\n\n\n\n<p>$(sin \\theta &#8211; i cos \\theta)^n = (-1)^n (\\cos n \\theta + i \\sin n\\theta)$<\/p>\n\n\n\n<p>$\\frac {1}{cos \\theta + i sin \\theta} = cos \\theta &#8211; i sin \\theta $<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">How to find the roots  of the equation<\/h2>\n\n\n\n<p>$(a+ ib)^{p\/q}$ where p and q are integers and $q \\ne  0$<\/p>\n\n\n\n<p>$(a + ib) = r (\\cos \\theta + i \\sin \\theta)$<br>then $(a+ ib)^{p\/q} = r^{p\/q} ( \\cos \\theta + i \\sin \\theta)^{p\/q}$<\/p>\n\n\n\n<p>$=r^{p\/q}(\\cos \\frac {(2k \\pi+ \\theta)p}{q} + \\sin \\frac {(2k \\pi+ \\theta)p}{q} )$<\/p>\n\n\n\n<p>Where k=0,1,2,&#8230;q-1<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Cube Root of Unity<\/h2>\n\n\n\n<p>$z^3 -1=0$<br>$(z-1)(z^2 + z +1)=0$<br>$z = 1, \\frac {-1 \\pm i \\sqrt 3}{2}$<\/p>\n\n\n\n<p>We define them as <\/p>\n\n\n\n<p>$\\omega =  \\frac {-1 + i \\sqrt 3}{2}$<br>$\\omega ^2 =  \\frac {-1 &#8211; i \\sqrt 3}{2}$<\/p>\n\n\n\n<p><strong>Important points<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$1 + \\omega + \\omega ^2 =0$<\/li>\n\n\n\n<li>$\\omega ^3 =1$<\/li>\n\n\n\n<li>$\\omega ^4 = \\omega$<\/li>\n\n\n\n<li>$\\omega ^{3n +1} = \\omega$<\/li>\n\n\n\n<li>$\\omega ^{3n +2} = \\omega ^2$<\/li>\n\n\n\n<li>$a^2 + ab + b^2 = (a- b \\omega)(a &#8211; b\\omega ^2)$<\/li>\n\n\n\n<li>$a^2 &#8211; ab + b^2 = (a+ b \\omega)(a + b\\omega ^2)$<\/li>\n\n\n\n<li>$a^3 + b^3= (a+b) (a + b \\omega) (a + b \\omega ^2)$<\/li>\n\n\n\n<li>$a^3 &#8211; b^3= (a-b) (a &#8211; b \\omega) (a &#8211; b \\omega ^2)$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">nth root of Unity<\/h2>\n\n\n\n<p>$z^n -1 =0$<br>$z^n =1$<br>$z = 1^{1\/n} = (cos 2 k \\pi + i \\sin 2k \\pi)^{1\/n}$<br>$= cos \\frac {2 k \\pi}{n} + i sin  \\frac {2 k \\pi}{n}$<\/p>\n\n\n\n<p>where k=0,1,2,&#8230;n-1<\/p>\n\n\n\n<p>If $\\alpha = cos \\frac {2  \\pi}{n} + i sin  \\frac {2  \\pi}{n}$<\/p>\n\n\n\n<p>(A) Sum of the nth Roots of Unity<\/p>\n\n\n\n<p>$1 + \\alpha + \\alpha ^2 + \\alpha ^3 + ..+ \\alpha ^{n-1} = \\frac {1 (1 &#8211; \\alpha ^n)}{1 &#8211; \\alpha}$<br>Now as $\\alpha ^n =1$, Hence<br>$1 + \\alpha + \\alpha ^2 + \\alpha ^3 + ..+ \\alpha ^{n-1} =0$<br>So sum of the roots are the nth root of unit is zero<\/p>\n\n\n\n<p>(B) Product of the nth Roots of Unity<\/p>\n\n\n\n<p>$1  \\times \\alpha \\times \\alpha ^2 \\times \\alpha ^3 \\times ..\\times \\alpha ^{n-1} = \\alpha^{ 1 + 2 +&#8230;(n-1)}=\\alpha ^{n(n-1)\/2}$<br> $= (-1)^{n-1}$<br><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Cube root of Any Number<\/h2>\n\n\n\n<p>$x^3=a$  is given as<\/p>\n\n\n\n<p>$a^{1\/3} , a^{1\/3} \\omega , a^{1\/3} \\omega ^2$<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Complex Numbers Complex numbers are the numbers of the form $a+ ib$ where $i= \\sqrt {-1}$&nbsp; and a and b are real numbers Algebra for i $i= \\sqrt {-1}$$i^2= -1$$i^3 =i \\times i^2= -i$$i^4= i^2 \\times i^2 = 1$$i^{4n} =1$$i^n + i^{n+1} + i^{n+2} + i^{n+3} =i^n( 1 + i +i^2 + i^3 ) = [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[24],"tags":[],"class_list":["post-9122","post","type-post","status-publish","format-standard","hentry","category-general"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Complex Numbers Formulas Sheet PDF - Class 11 &amp; 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