{"id":436,"date":"2019-06-27T17:47:24","date_gmt":"2019-06-27T17:47:24","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=436"},"modified":"2019-12-11T17:09:08","modified_gmt":"2019-12-11T17:09:08","slug":"products-of-sums-of-two-squares","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/products-of-sums-of-two-squares\/","title":{"rendered":"Products of Sums of Two Squares"},"content":{"rendered":"\n<p>Here&#8217;s a nice theorem due to\u00a0Fibonacci, in 1202.<\/p>\n\n\n\n<p>Theorem. If integers N and M can each be written as the sum of two squares, so can their product!<\/p>\n\n\n\n<p>Example: since 2=1<sup>2<\/sup>+1<sup>2<\/sup>&nbsp;and 34=3<sup>2<\/sup>+5<sup>2<\/sup>, their product 68 should be expressible as the sum of two squares. In fact, 68=8<sup>2<\/sup>+2<sup>2<\/sup>. Is there an easy way to figure out what squares the product will be made of?<\/p>\n\n\n\n<p>Yes! This all follows from the very cool formula:<\/p>\n\n\n\n<p style=\"text-align:center\">(a<sup>2<\/sup>+b<sup>2<\/sup>) (c<sup>2<\/sup>+d<sup>2<\/sup>) = (ac+bd)<sup>2<\/sup>\u00a0+ (ad-bc)<sup>2<\/sup>.<\/p>\n\n\n\n<p><strong>Presentation&nbsp;Suggestions:<\/strong><br>Do a few more numerical examples before showing the cool formula.<\/p>\n\n\n\n<p><strong>The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong><br>The formula above can be checked trivially. But there are other ways to see why it is true. If you know some linear algebra, take a look at the Fun Fact\u00a0Really Complex Matrices, and take the determinant of both sides of the matrix equation there. You will get the formula above!<\/p>\n\n\n\n<p>Or, if you know about\u00a0complex numbers, the left hand side is the squared modulus of two complex numbers and the right side is the squared modulus of their product!<\/p>\n\n\n\n<p>To see which numbers can be written as the sum of two squares, see\u00a0Sums Of Two Squares.<\/p>\n\n\n\n<p><strong>How to Cite this Page:<\/strong>&nbsp;<br>Su, Francis E., et al. &#8220;Products of Sums of Two Squares.&#8221;&nbsp;<em>Math Fun Facts<\/em>. &lt;http:\/\/www.math.hmc.edu\/funfacts&gt;.<\/p>\n\n\n\n<p><strong>Fun Fact suggested by: &nbsp; <\/strong><br>Lesley Ward<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here&#8217;s a nice theorem due to\u00a0Fibonacci, in 1202. Theorem. If integers N and M can each be written as the&#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":[73,3,16,4,10,12,152],"class_list":["post-436","page","type-page","status-publish","hentry","tag-complex-numbers","tag-easy","tag-matrix","tag-medium","tag-numtheory","tag-other","tag-sum-of-squares"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/436","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=436"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/436\/revisions"}],"predecessor-version":[{"id":1581,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/436\/revisions\/1581"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=436"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=436"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}