{"id":493,"date":"2019-06-29T22:02:00","date_gmt":"2019-06-29T22:02:00","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=493"},"modified":"2020-01-03T23:51:38","modified_gmt":"2020-01-03T23:51:38","slug":"i-to-the-i-is-a-real-number","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/i-to-the-i-is-a-real-number\/","title":{"rendered":"i to the i is a Real Number"},"content":{"rendered":"\n<p>If you are familiar with\u00a0complex numbers, the &#8220;imaginary&#8221; number i has the property that the square of i is -1. It is a rather curious fact that i raised to the i-th power is actually a real number!<\/p>\n\n\n\n<p>In fact, its value is approximately 0.20788.<\/p>\n\n\n\n<p><strong>Presentation&nbsp;Suggestions:<\/strong><br>This makes a great exercise after learning the basics about complex numbers.<\/p>\n\n\n\n<p><strong>The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong><br>From\u00a0Euler&#8217;s formula, we know that exp(i*x) = cos(x) + i*sin(x), where &#8220;exp(z)&#8221; is the exponential function\u00a0<em>e<\/em><sup>z<\/sup>. Then<br>exp(i*Pi\/2) = cos(Pi\/2) + i*sin(Pi\/2) = i.<br>Raising both sides to i-th power, we see that the right side is the desired quantity i<sup>i<\/sup>, while the left side becomes exp(i*i*Pi\/2), or exp(-Pi\/2), which is approximately .20788.<\/p>\n\n\n\n<p>(Actually, this is one of many possible values for i to the i, because, for instance, exp(5i*Pi\/2)=i. In\u00a0complex analysis, one learns that exponentiation with respect to i is a\u00a0<em>multi-valued<\/em>\u00a0function.)<\/p>\n\n\n\n<p><strong>How to Cite this Page:<\/strong>&nbsp;<br>Su, Francis E., et al. &#8220;i to the i is a Real Number.&#8221;&nbsp;<em>Math Fun Facts<\/em>. &lt;http:\/\/www.math.hmc.edu\/funfacts&gt;.<\/p>\n\n\n\n<p><strong>References:<\/strong><br>Paul Nahin, <a href=\"https:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0691027951\/ref=nosim\/mathfunfacts-20\"><em>An Imaginary Tale<\/em><\/a>.<\/p>\n\n\n\n<p><strong>Fun Fact suggested by:   <\/strong><br>Ed Poncin <\/p>\n","protected":false},"excerpt":{"rendered":"<p>If you are familiar with\u00a0complex numbers, the &#8220;imaginary&#8221; number i has the property that the square of i is -1.&#46;&#46;&#46;<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[20,7,126,125,124,127,4],"class_list":["post-493","page","type-page","status-publish","hentry","tag-analysis","tag-calculus","tag-i-raised-to-the-i","tag-i-to-the-i-th-power","tag-ii","tag-imaginary-power","tag-medium"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/493","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=493"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/493\/revisions"}],"predecessor-version":[{"id":1707,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/493\/revisions\/1707"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=493"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=493"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}