{"id":93,"date":"2017-12-17T09:51:46","date_gmt":"2017-12-17T09:51:46","guid":{"rendered":"https:\/\/borovik.net\/selecta\/?p=93"},"modified":"2025-12-17T09:59:37","modified_gmt":"2025-12-17T09:59:37","slug":"what-are-the-real-life-applications-of-abstract-algebra","status":"publish","type":"post","link":"https:\/\/borovik.net\/selecta\/2017\/12\/17\/what-are-the-real-life-applications-of-abstract-algebra\/","title":{"rendered":"What are the real life applications of abstract algebra?"},"content":{"rendered":"
\n
\n
\n
\n
\n
\n
\n
My answer to a question on Quora: <\/span><\/div>\n
\n
What are the real life applications of abstract algebra?<\/span><\/div>\n<\/blockquote>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n
\n
\n
\n
\n
Let me start with an example that I found in some high-profile mathematics education policy document of recent times. Perhaps you would agree that it is about \u201creal life\u201d applications of mathematics.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n

Modeling the cost of a sandwich.\u00a0The food operations controller of a catering company that supplies sandwiches and lunches both through mobile vans and as special orders for external customers has developed a spreadsheet that
\nenables the cost of sandwiches and similar items to be calculated.<\/p>\n<\/blockquote>\n

Please notice the word \u201cspreadsheet\u201d. Anyone who ever did a spreadsheet of complexity of a sandwich should know that the key mathematical skill needed is awareness of the role of brackets in arithmetic expressions and an intuitive feel of how brackets are manipulated, something that is sometimes called “structural arithmetic\u201d or \u201cpre-algebra\u201d. At a slightly more advanced level working with spreadsheets requires understanding of the concept of functional dependency in its\u00a0algebraic\u00a0aspects (frequently ignored in pre-calculus), but very prominent in abstract algebra (and in computer programming, say, in the form of\u00a0polymorphism<\/a>\u00a0in C++ and other programming language).<\/p>\n

\n
<\/div>\n<\/div>\n

To illustrate this point, I prepared a very simple spreadsheet in\u00a0Apache OpenOffice<\/a>\u00a0Calc\u00a0(it uses essentially the same interface as\u00a0Microsoft Excel).<\/p>\n

Look at the picture above: if the content of cell C14 is SUM(C8:C13) and you copy cell C14 into cell D14 (look at the next picture),<\/p>\n

\n
<\/div>\n<\/div>\n

the content of cell D14 becomes SUM(D8:D13) and thus involves\u00a0change of variables. What is copied is a\u00a0structure of the algebraic expression, not even an algebraic expression itself. And of course this is no copying of the\u00a0value\u00a0of this expression: please notice that the value\u00a085\u00a0becomes\u00a0130\u00a0when moved from cell C14 to cell D14!<\/p>\n

At a very elementary level, abstract algebra provides intuition about such things as a\u00a0structure of an algebraic expression.<\/p>\n

Intuitive understanding that SUM(C8:C13) is in a sense the same as SUM(D8:D13) is best achieved by exposing a student to a variety of algebraic problems which convince him\/her that a polynomial of kind\u00a0x^<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>+<\/span><\/span>2<\/span><\/span>x<\/span><\/span>+<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span>\u00a0is, from an algebraic point of view, the same as\u00a0z^<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>+<\/span><\/span>2<\/span><\/span>z<\/span><\/span>+<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span>, and that in a similar vein, the sum<\/p>\n

\n

C8 + C9 + C10 + C11 + C12 + C13<\/p>\n<\/blockquote>\n

is in some sense the same as<\/p>\n

\n

D8 + D9 + D10 + D11 + D12 + D13 .<\/p>\n<\/blockquote>\n

In the terminology of abstract algebra, it is called\u00a0isomorphism. Abstract algebra studies structure of algebraic entities\u00a0up to isomorphisms. I do not claim that everyone who uses a spreadsheet should know this terminology, but it is desirable to have some\u00a0intuition\u00a0about what is going on. Even if the number-crunching is passed to the computer, use of spreadsheets still requires mastering, at an intuitive or semi-intuitive level, some mathematical concepts, like understanding that arithmetic expressions have certain structure, or developing some basic intuition of functional dependency.\u00a0And the most acute problem of mathematical education in our time is that this intuitive component of mathematics is being lost.<\/p>\n

My former student who now works as a project manager in a serious engineering company once told me that many his colleagues cannot handle macroses in Excel spreadsheets with time-dependent entries (a basic tool of project management, an IT version of the proverbial clipboard) because they suffer from \u201cbrackets overload\u201d \u2014 I love this formulation!<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

My answer to a question on Quora: What are the real life applications of abstract algebra? Let me start with an example that I found in some high-profile mathematics education policy document of recent times. Perhaps you would agree that it is about \u201creal life\u201d applications of mathematics. Modeling the cost of a sandwich.\u00a0The food […]<\/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":[],"class_list":["post-93","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/93","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/comments?post=93"}],"version-history":[{"count":1,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/93\/revisions"}],"predecessor-version":[{"id":94,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/93\/revisions\/94"}],"wp:attachment":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/media?parent=93"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/categories?post=93"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/tags?post=93"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}