{"id":46,"date":"2025-09-13T14:19:33","date_gmt":"2025-09-13T14:19:33","guid":{"rendered":"https:\/\/gsama.cc\/?p=46"},"modified":"2025-09-13T14:20:50","modified_gmt":"2025-09-13T14:20:50","slug":"guess-pi","status":"publish","type":"post","link":"https:\/\/gsama.cc\/?p=46","title":{"rendered":"can you guess the value \u03c0?"},"content":{"rendered":"\n

The word “guess” is loosely used here. <\/p>\n\n\n\n

Can we determine the value of \u03c0 by generating random number? <\/p>\n\n\n\n

As of 2024 humans have calculated 105 trillion digits of \u03c0. It feels like a stretch of imagination to think we could get even a fair approximation using only random numbers.<\/p>\n\n\n\n

But consider this: if we generate random points inside a square, and check how many fall inside an inscribed quarter circle, we can estimate \u03c0.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Explanation: The probability of such randomly generated points to fall within the blue coloured quarter circle is <\/p>\n\n\n\n

P(A) = \\frac{\\text{Area of Quarter Circle}}{\\text{Area of Square}}<\/pre><\/div>\n\n\n\n
P(A) = \\frac{\\tfrac{1}{4}\\pi r^2}{r^2} = \\frac{\\pi}{4}<\/pre><\/div>\n\n\n\n
So if you have a large enough sample of points<\/p>\n\n\n\n
\\pi = 4 \\cdot P(A)<\/pre><\/div>\n\n\n\n
or<\/p>\n\n\n\n
\\pi = \\lim_{N_{\\text{total}} \\to \\infty} 4 \\cdot \\frac{N_{\\text{inside}}}{N_{\\text{total}}}<\/pre><\/div>\n\n\n\n
So next time you see \u03c0, remember \u2014 even randomness, given patience, knows its way home.<\/p>\n\n\n\n
\n
Run Simulation<\/a><\/div>\n<\/div>\n\n\n\n
Note
\u2022 Using randomness to solve deterministic problems is called the Monte Carlo method<\/a>.
\u2022 Demo created with AI tools \u2014 the code could use some polishing.
\u2022 The simulation speed is deliberately slowed down for effect. Left to the computer\u2019s clock speed, it would approximate \u03c0 to practical accuracy within seconds.
\u2022 Computers don\u2019t generate true randomness \u2014 only pseudo-random numbers.<\/a>
<\/p>\n","protected":false},"excerpt":{"rendered":"
The word “guess” is loosely used here. Can we determine the value of \u03c0 by generating random number? As of 2024 humans have calculated 105 trillion digits of \u03c0. It feels like a stretch of imagination to think we could get even a fair approximation using only random numbers. But consider this: if we generate […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[11,15],"class_list":["post-46","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-math-2","tag-pi"],"_links":{"self":[{"href":"https:\/\/gsama.cc\/index.php?rest_route=\/wp\/v2\/posts\/46","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gsama.cc\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gsama.cc\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gsama.cc\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gsama.cc\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=46"}],"version-history":[{"count":2,"href":"https:\/\/gsama.cc\/index.php?rest_route=\/wp\/v2\/posts\/46\/revisions"}],"predecessor-version":[{"id":49,"href":"https:\/\/gsama.cc\/index.php?rest_route=\/wp\/v2\/posts\/46\/revisions\/49"}],"wp:attachment":[{"href":"https:\/\/gsama.cc\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=46"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gsama.cc\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=46"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gsama.cc\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=46"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}