{"id":221457,"date":"2025-05-29T14:52:37","date_gmt":"2025-05-29T14:52:37","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=221457"},"modified":"2025-05-29T14:52:39","modified_gmt":"2025-05-29T14:52:39","slug":"express-the-following-complex-numbers-in-the-form-a-ib","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/29\/express-the-following-complex-numbers-in-the-form-a-ib\/","title":{"rendered":"Express the following complex numbers in the form a + ib"},"content":{"rendered":"\n

Express the following complex numbers in the form a + ib.<\/p>\n\n\n\n

    \n
  1. A: 2 e^(3 ipi \/2)<\/li>\n\n\n\n
  2. B: 4?(3 e^(ipi \/4))<\/li>\n\n\n\n
  3. C: e^(ipi)<\/li>\n\n\n\n
  4. D: ((sqrt 5)\/(1+sqrt2)) (e^(ipi \/4))<\/li>\n<\/ol>\n\n\n\n

    The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

    To express the given complex numbers in the form a+iba + ib, we convert the exponential form using Euler\u2019s formula<\/strong>: ei\u03b8=cos\u2061(\u03b8)+isin\u2061(\u03b8)e^{i\\theta} = \\cos(\\theta) + i\\sin(\\theta)<\/p>\n\n\n\n

    \n\n\n\n

    A: 2e3i\u03c022e^{\\frac{3i\\pi}{2}}<\/strong><\/h3>\n\n\n\n

    e3i\u03c02=cos\u2061(3\u03c02)+isin\u2061(3\u03c02)=0\u2212i=\u2212ie^{\\frac{3i\\pi}{2}} = \\cos\\left(\\frac{3\\pi}{2}\\right) + i\\sin\\left(\\frac{3\\pi}{2}\\right) = 0 – i = -i \u21d22e3i\u03c02=2(\u2212i)=0\u22122i\\Rightarrow 2e^{\\frac{3i\\pi}{2}} = 2(-i) = \\boxed{0 – 2i}<\/p>\n\n\n\n

    \n\n\n\n

    B: 43ei\u03c044\\sqrt{3}e^{\\frac{i\\pi}{4}}<\/strong><\/h3>\n\n\n\n

    ei\u03c04=cos\u2061(\u03c04)+isin\u2061(\u03c04)=22+i22e^{\\frac{i\\pi}{4}} = \\cos\\left(\\frac{\\pi}{4}\\right) + i\\sin\\left(\\frac{\\pi}{4}\\right) = \\frac{\\sqrt{2}}{2} + i\\frac{\\sqrt{2}}{2} \u21d243ei\u03c04=43(22+i22)=26+i26=26+2i6\\Rightarrow 4\\sqrt{3}e^{\\frac{i\\pi}{4}} = 4\\sqrt{3}\\left(\\frac{\\sqrt{2}}{2} + i\\frac{\\sqrt{2}}{2}\\right) = 2\\sqrt{6} + i2\\sqrt{6} = \\boxed{2\\sqrt{6} + 2i\\sqrt{6}}<\/p>\n\n\n\n

    \n\n\n\n

    C: ei\u03c0e^{i\\pi}<\/strong><\/h3>\n\n\n\n

    ei\u03c0=cos\u2061(\u03c0)+isin\u2061(\u03c0)=\u22121+0i=\u22121e^{i\\pi} = \\cos(\\pi) + i\\sin(\\pi) = -1 + 0i = \\boxed{-1}<\/p>\n\n\n\n

    \n\n\n\n

    D: 51+2ei\u03c04\\frac{\\sqrt{5}}{1+\\sqrt{2}} e^{\\frac{i\\pi}{4}}<\/strong><\/h3>\n\n\n\n

    We simplify: ei\u03c04=22+i22e^{\\frac{i\\pi}{4}} = \\frac{\\sqrt{2}}{2} + i\\frac{\\sqrt{2}}{2} So: 51+2(22+i22)=522(1+2)(1+i)\\text{So: } \\frac{\\sqrt{5}}{1+\\sqrt{2}} \\left(\\frac{\\sqrt{2}}{2} + i\\frac{\\sqrt{2}}{2}\\right) = \\frac{\\sqrt{5}\\sqrt{2}}{2(1+\\sqrt{2})}(1 + i)<\/p>\n\n\n\n

    Now rationalize the denominator: 102(1+2)=10(1\u22122)2(12\u2212(2)2)=10(1\u22122)2(1\u22122)=\u221210(1\u22122)2\\frac{\\sqrt{10}}{2(1+\\sqrt{2})} = \\frac{\\sqrt{10}(1-\\sqrt{2})}{2(1^2 – (\\sqrt{2})^2)} = \\frac{\\sqrt{10}(1-\\sqrt{2})}{2(1 – 2)} = \\frac{-\\sqrt{10}(1-\\sqrt{2})}{2} \u21d2\u221210(1\u22122)2(1+i)\\Rightarrow \\boxed{\\frac{-\\sqrt{10}(1-\\sqrt{2})}{2}(1 + i)}<\/p>\n\n\n\n

    \n\n\n\n

    Explanation <\/strong><\/h3>\n\n\n\n

    To convert complex numbers from exponential (polar) form to rectangular (Cartesian) form a+iba + ib, we use Euler’s formula: ei\u03b8=cos\u2061(\u03b8)+isin\u2061(\u03b8)e^{i\\theta} = \\cos(\\theta) + i\\sin(\\theta)<\/p>\n\n\n\n

    This formula links trigonometry and exponential functions, enabling the transformation of expressions involving angles and magnitudes into real and imaginary parts.<\/p>\n\n\n\n

    For A<\/strong>, the angle is 3\u03c02\\frac{3\\pi}{2}, where cosine is 0 and sine is -1. Multiplying by 2 yields 0\u22122i0 – 2i.<\/p>\n\n\n\n

    For B<\/strong>, we multiply 434\\sqrt{3} by ei\u03c0\/4e^{i\\pi\/4}, whose cosine and sine values are both 22\\frac{\\sqrt{2}}{2}. Multiplying yields 26+2i62\\sqrt{6} + 2i\\sqrt{6}.<\/p>\n\n\n\n

    C<\/strong> involves the identity ei\u03c0=\u22121e^{i\\pi} = -1, which is a famous result from Euler’s formula and demonstrates the connection between ee, ii, and \u03c0\\pi.<\/p>\n\n\n\n

    D<\/strong> combines a real coefficient with an exponential. After expressing the exponential using cosine and sine, we simplify and rationalize the denominator to express it as a complex number. Rationalization ensures the final form doesn’t contain irrational numbers in the denominator, which is standard in mathematics.<\/p>\n\n\n\n

    These conversions are crucial in fields like electrical engineering, quantum mechanics, and signal processing, where complex numbers are often used in polar form for convenience in multiplication and division, but rectangular form is preferred for calculations involving addition and visualization.<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Express the following complex numbers in the form a + ib. The Correct Answer and Explanation is: To express the given complex numbers in the form a+iba + ib, we convert the exponential form using Euler\u2019s formula: ei\u03b8=cos\u2061(\u03b8)+isin\u2061(\u03b8)e^{i\\theta} = \\cos(\\theta) + i\\sin(\\theta) A: 2e3i\u03c022e^{\\frac{3i\\pi}{2}} e3i\u03c02=cos\u2061(3\u03c02)+isin\u2061(3\u03c02)=0\u2212i=\u2212ie^{\\frac{3i\\pi}{2}} = \\cos\\left(\\frac{3\\pi}{2}\\right) + i\\sin\\left(\\frac{3\\pi}{2}\\right) = 0 – i = -i […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"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":"","ast-disable-related-posts":"","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":"default","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":[25],"tags":[],"class_list":["post-221457","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221457","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=221457"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221457\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=221457"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=221457"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=221457"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}