{"id":7708,"date":"2026-08-08T00:02:38","date_gmt":"2026-08-08T07:02:38","guid":{"rendered":"https:\/\/c-for-dummies.com\/blog\/?p=7708"},"modified":"2026-08-08T10:05:17","modified_gmt":"2026-08-08T17:05:17","slug":"ramanujan-nested-radicals-solution","status":"publish","type":"post","link":"https:\/\/c-for-dummies.com\/blog\/?p=7708","title":{"rendered":"Ramanujan Nested Radicals – Solution"},"content":{"rendered":"

The topic for this month’s Exercise<\/a> is to code a recursive function that solves the Ramanujan Nested Radical, illustrated in Figure 1. Remember that Ramanujan created and solved this puzzle in his head. #genius<\/span>
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Figure 1. One of Ramanujan’s more famous nested radicals also called an infinite identity.<\/p><\/div><\/p>\n

In past Exercises on this blog, I’ve looked at continued fractions and nested radicals such as this one and recursion immediately comes to mind as a way to code it. Of course, the issue with recursion is how to unwind the thing.<\/p>\n

For my solution, the root()<\/em> function includes an integer argument count<\/code>. This variable decrements each time the function calls itself until the value of count<\/code> is zero. At this point, the recursion unwinds.<\/p>\n

2026_08-Exercise.c<\/a><\/h3>\n
\r\n#include <stdio.h>\r\n#include <math.h>\r\n\r\nfloat root(float a,int count)\r\n{\r\n    while( count-- )\r\n        return( sqrt(1.0+(a+1.0) * root(a+1.0,count)) );\r\n    return(a);\r\n}\r\n\r\nint main()\r\n{\r\n    float a = 0.0;\r\n\r\n    a = root(1.0,25);\r\n    printf(\"%f\\n\",a);\r\n\r\n    return 0;\r\n}<\/pre>\n
In the main()<\/em> function, the recursive root()<\/em> function is initially called with values of one and 25. The one is written as 1.0<\/code>, which the compiler identifies as a real number; 25 is an integer. I found that 25 repetitions is adequate to reach the result of 3.0 (at least on my computers).<\/p>\n
My C language version of the Ramanujan Nested Radical thingy appears in the return<\/em> statement in the root()<\/em> function:<\/p>\n
sqrt(1.0+(a+1.0) * root(a+1.0,count))<\/code><\/p>\n
Variable a<\/code> represents the incrementing value in the nested radical, increasing by one each time the root(<\/em>) function is called. The result is finally returned once the value of count<\/code> is zero, which happens in the while<\/em> statement: while( count-- )<\/code><\/p>\n
Remember to add the -lm<\/code> switch when building this code in Linux at the terminal prompt. This switch brings in the math library, which is required for the sqrt()<\/em> function to behave.<\/p>\n
Here is output from a sample run:<\/p>\n
3.000000<\/pre>\n
I hope your solution met with success!<\/p>\n","protected":false},"excerpt":{"rendered":"
The topic for this month’s Exercise is to code a recursive function that solves the Ramanujan Nested Radical, illustrated in Figure 1. Remember that Ramanujan created and solved this puzzle in his head. #genius<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-7708","post","type-post","status-publish","format-standard","hentry","category-solution"],"_links":{"self":[{"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/7708","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7708"}],"version-history":[{"count":4,"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/7708\/revisions"}],"predecessor-version":[{"id":7721,"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/7708\/revisions\/7721"}],"wp:attachment":[{"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7708"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7708"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7708"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}