{"id":155446,"date":"2022-03-26T00:39:19","date_gmt":"2022-03-25T19:09:19","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/infinitesimal-history-properties-uses-and-faqs\/"},"modified":"2022-12-31T14:59:09","modified_gmt":"2022-12-31T09:29:09","slug":"infinitesimal-history-properties-uses-and-faqs","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/infinitesimal\/","title":{"rendered":"Infinitesimal &#8211; History, Properties, Uses and FAQs"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/infinitesimal\/#What_is_Infinitesimal_Meaning\" title=\"What is Infinitesimal Meaning?\">What is Infinitesimal Meaning?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/infinitesimal\/#Define_Infinitesimally_Small\" title=\"Define Infinitesimally Small\">Define Infinitesimally Small<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/infinitesimal\/#History_of_Infinitesimal\" title=\"History of Infinitesimal\">History of Infinitesimal<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/infinitesimal\/#First_Order_Properties_of_Infinitesimal\" title=\"First Order Properties of Infinitesimal\">First Order Properties of Infinitesimal<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/infinitesimal\/#Laurent_Series\" title=\"Laurent Series\">Laurent Series<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/infinitesimal\/#The_Levi-Civita_Field\" title=\"The Levi-Civita Field\">The Levi-Civita Field<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/infinitesimal\/#Surreal_Numbers\" title=\"Surreal Numbers\">Surreal Numbers<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/infinitesimal\/#Hyperreals\" title=\"Hyperreals\">Hyperreals<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/infinitesimal\/#Superreals\" title=\"Superreals\">Superreals<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/infinitesimal\/#Smooth_Infinitesimal_Analysis\" title=\"Smooth Infinitesimal Analysis\">Smooth Infinitesimal Analysis<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_is_Infinitesimal_Meaning\"><\/span>What is Infinitesimal Meaning?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The infinitesimal calculus is a branch of mathematics that deals with the properties of infinitely small quantities. It is a powerful tool for solving problems in physics and engineering.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-155445 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/infinitesimal-history-properties-uses-and-faqs.jpg\" alt=\"\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/infinitesimal-history-properties-uses-and-faqs.jpg?v=1648235355 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/infinitesimal-history-properties-uses-and-faqs-300x212.jpg?v=1648235355 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Define_Infinitesimally_Small\"><\/span>Define Infinitesimally Small<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The term infinitesimally small is used to describe something that is extremely tiny. It can be used to describe the size of an object or the distance between two objects.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"History_of_Infinitesimal\"><\/span>History of Infinitesimal<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Calculus<\/p>\n<p>The history of infinitesimal calculus is long and complex. It is often said to have begun with the ancient Greek mathematician Archimedes, who developed a method for finding the area of a curved shape. This method involved breaking the shape down into a series of smaller and smaller shapes, each of which had a known area. Archimedes was not able to calculate the exact area of the original shape, but he could calculate the area of each smaller shape and then add them all up.<\/p>\n<p>In the 16th century, the Italian mathematician Leonardo Fibonacci developed a similar method for finding the volume of a curved object. His method involved breaking the object down into a series of smaller and smaller cubes, and then calculating the volume of each cube.<\/p>\n<p>In the 17th century, the German mathematician Gottfried Wilhelm Leibniz developed a new way of thinking about mathematics that made it possible to calculate the area and volume of a curved object using infinitesimals. An infinitesimal is a number that is so small that it is impossible to calculate its exact value. Leibniz&#8217;s method involved dividing a curved shape into an infinite number of infinitely small pieces. He was then able to calculate the area and volume of the shape by adding up the areas and volumes of all the pieces.<\/p>\n<p>Leibniz&#8217;s method was controversial, and many other mathematicians disagreed with his approach. It was not until the 19th century that<\/p>\n<h2><span class=\"ez-toc-section\" id=\"First_Order_Properties_of_Infinitesimal\"><\/span>First Order Properties of Infinitesimal<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Infinitesimal numbers are numbers that are smaller than any other number. They are used in mathematics to help describe the properties of other numbers. Some of the most important properties of infinitesimal numbers are that they are:<\/p>\n<p>1. Nonzero<\/p>\n<p>2. Negative or positive<\/p>\n<p>3. Less than any other number<\/p>\n<p>4. Cannot be divided into smaller numbers<\/p>\n<p>5. Cannot be expressed in terms of other numbers<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Laurent_Series\"><\/span>Laurent Series<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In mathematics, a Laurent series is a power series representation of a function in a neighborhood of a point in the complex plane. In other words, a Laurent series is a representation of a function as a sum of terms involving powers of a complex variable, with some of the terms negative.<\/p>\n<p>The Laurent series for a function &#8220;f&#8221; at a complex number &#8220;z&#8221; is the power series<\/p>\n<p>The coefficients &#8220;a&#8221; are called the Laurent coefficients.<\/p>\n<p>The Laurent series is named after Jules Fran\u00e7ois Antoine Laurent.<\/p>\n<p>The Laurent series is a special case of a more general formula, the Puiseux series.<\/p>\n<p>The Laurent series can be used to calculate the function near the point &#8220;z&#8221;. For example, the Laurent series for the function &#8220;f&#8221;(&#8220;x&#8221;) = &#8220;x&#8221; near the point &#8220;z&#8221; = 0 is<\/p>\n<p>The Laurent series can also be used to calculate the function&#8217;s derivative near the point &#8220;z&#8221;. For example, the Laurent series for the derivative of &#8220;f&#8221;(&#8220;x&#8221;) = &#8220;x&#8221; near the point &#8220;z&#8221; = 0 is<\/p>\n<p>The Laurent series can also be used to calculate the function&#8217;s integral near the point &#8220;z&#8221;. For example, the Laurent series for the integral of &#8220;f&#8221;(&#8220;x&#8221;) = &#8220;x&#8221; near the point &#8220;z&#8221; = 0 is<\/p>\n<p>The Laurent series can also be used to calculate the function&#8217;s complex conjugate near the point &#8220;z&#8221;.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"The_Levi-Civita_Field\"><\/span>The Levi-Civita Field<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In mathematics, the Levi-Civita field is a differential field associated to a Lie group. It is a rank-1 tensor field that is divergence-free and curl-free. It is often used in the study of Lie groups and Lie algebras.<\/p>\n<p>The Levi-Civita field can be defined in terms of the Levi-Civita symbol:<\/p>\n<p>where &#8220;g&#8221; is a Lie group and &#8220;a&#8221; is an element of &#8220;g&#8221;. The Levi-Civita field is then the vector field associated to the Levi-Civita symbol.<\/p>\n<p>The Levi-Civita field is divergence-free and curl-free because the Levi-Civita symbol is divergence-free and curl-free.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Surreal_Numbers\"><\/span>Surreal Numbers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A surreal number is a number that is not real, but can be constructed using certain procedures.<\/p>\n<p>To construct a surreal number, first create a list of all the real numbers. Next, create a new list that includes all the pairs of real numbers that are next to each other on the original list. Finally, create a new list that includes all the triplets of real numbers that are next to each other on the original list.<\/p>\n<p>The first surreal number is the number that is the sum of all the numbers in the first list. The second surreal number is the number that is the sum of all the numbers in the second list. The third surreal number is the number that is the sum of all the numbers in the third list.<\/p>\n<p>The fourth surreal number is the number that is the sum of all the numbers in the first list, plus the number that is the sum of all the numbers in the second list, plus the number that is the sum of all the numbers in the third list.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Hyperreals\"><\/span>Hyperreals<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Hyperreals are a subset of the reals that are larger than the reals, but still countable. The hyperreals are denoted by the symbol \u211d*.<\/p>\n<p>The hyperreals can be thought of as the &#8220;real numbers plus infinity&#8221;. Just as the natural numbers are a subset of the integers, the hyperreals are a subset of the reals.<\/p>\n<p>The hyperreals are used in mathematical analysis to deal with quantities that are too large or too small to be handled by the reals. For example, the hyperreal number \u03c0 represents the ratio of the circumference of a circle to its diameter, which is a quantity that is too large to be handled by the reals.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Superreals\"><\/span>Superreals<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Superreals are a subset of the reals that includes all real numbers and all infinitesimal numbers.<\/p>\n<p>Superreals are used in mathematics to provide a more complete description of the real number line than is possible with just the reals. In addition to the real numbers, the superreal number line includes all infinitesimal numbers, which allows for a more accurate representation of the continuum.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Smooth_Infinitesimal_Analysis\"><\/span>Smooth Infinitesimal Analysis<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Smooth infinitesimal analysis is a branch of mathematics that deals with the behavior of functions and their derivatives near points where the derivatives vanish. This area of mathematics was developed in the early 1800s by the mathematicians Cauchy and Weierstrass.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What is Infinitesimal Meaning? The infinitesimal calculus is a branch of mathematics that deals with the properties of infinitely small [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Infinitesimal -History Properties","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn about infinitesimal topic of Maths in details explained by subject experts on infinitylearn.com. Register free for online.","custom_permalink":"maths\/infinitesimal\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Infinitesimal - History, Properties, Uses and FAQs - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"Learn about infinitesimal topic of Maths in details explained by subject experts on infinitylearn.com. 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