{"id":65103,"date":"2025-08-18T09:48:52","date_gmt":"2025-08-18T06:48:52","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65103"},"modified":"2025-08-18T09:48:52","modified_gmt":"2025-08-18T06:48:52","slug":"complex-conjugate","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/complex-conjugate\/","title":{"rendered":"Complex Conjugate Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 class=\"text-lg font-display font-bold\">Find the Conjugate of a Complex Number<\/h2><\/div><div class=\"relative group inline-block\">\n  <button class=\"favorite\" id=\"favorite\" data-favorite-id=\"65103\" data-favorite-title=\"Complex Conjugate\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-conjugate\/\" data-favorite-excerpt=\"This calculator finds the &lt;strong&gt;complex conjugate&lt;\/strong&gt; of any complex number. It\u2019s a useful tool in algebra, engineering, signal processing, and complex analysis, where conjugates are used to simplify expressions or solve equations involving complex numbers.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-algebra\">\n    <span class=\"icon icon-shape-star-empty\"><\/span>\n  <\/button>\n  <div class=\"absolute right-full -translate-y-1\/2 top-1\/2 mr-2 w-max max-w-xs px-3 py-2 bg-gray-800 text-white text-xs rounded shadow-lg opacity-0 group-hover:opacity-100 transition-opacity duration-200 z-10 pointer-events-none\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-conjugate\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65103\" data-post=\"65103\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Real part<\/label><div class=\"term\">Re<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"real_part\" id=\"field-1\" step=\"any\" value=\"3\"\/><small>Enter the real part of the complex number<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Imaginary part<\/label><div class=\"term\">Im<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"imaginary_part\" id=\"field-2\" step=\"any\" value=\"4\"\/><small>Enter the imaginary part of the complex number<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65103\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Complex Conjugate (a - bi)<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Absolute Value |a + bi|<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"65103\" aria-live=\"polite\"><h2 class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/h2><div class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><ins class=\"adsbygoogle\"\n     style=\"display:block; text-align:center; margin: 32px 0;\"\n     data-ad-layout=\"in-article\"\n     data-ad-format=\"fluid\"\n     data-ad-client=\"ca-pub-1721569815777345\"\n     data-ad-slot=\"6317458308\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Complex Conjugate Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n$$\\text{If } z = a + bi,\\ \\text{then the complex conjugate is } \\bar{z} = a &#8211; bi$$\n<\/div>\r\n\t<\/figure>\r\n\t<br \/>\n<strong>Explanation:<\/strong><br \/>\nTo find the conjugate of a complex number, simply change the sign of the imaginary part. This operation reflects the number across the real axis on the complex plane.<br \/>\r\n<\/section>\n<section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Complex Conjugate \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<p>\nz = 5 + 3i<br \/>\nConjugate: 5 &#8211; 3i<\/p>\n<p>z = -2 &#8211; 7i<br \/>\nConjugate: -2 + 7i<\/p>\n\r\n<\/section>\n<p>Complex conjugates are used in rationalizing denominators, finding moduli, and solving complex equations. In electrical engineering and physics, they simplify calculations involving impedance, power, and waveforms. This calculator provides fast, accurate conjugates for any complex input.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This calculator finds the <strong>complex conjugate<\/strong> of any complex number. It\u2019s a useful tool in algebra, engineering, signal processing, and complex analysis, where conjugates are used to simplify expressions or solve equations involving complex numbers.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[11],"class_list":["post-65103","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-algebra"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65103","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=65103"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65103\/revisions"}],"predecessor-version":[{"id":65105,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65103\/revisions\/65105"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65103"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65103"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65103"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}