{"id":39480,"date":"2025-05-15T13:45:00","date_gmt":"2025-05-15T13:45:00","guid":{"rendered":"https:\/\/www.lifeandnews.com\/articles\/?p=39480"},"modified":"2025-05-16T06:50:39","modified_gmt":"2025-05-16T06:50:39","slug":"algebra-is-more-than-alphabet-soup-its-the-language-of-algorithms-and-relationships","status":"publish","type":"post","link":"https:\/\/www.lifeandnews.com\/articles\/algebra-is-more-than-alphabet-soup-its-the-language-of-algorithms-and-relationships\/","title":{"rendered":"Algebra is more than alphabet soup \u2013 it\u2019s the language of algorithms and relationships"},"content":{"rendered":"\n
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Algebra often involves manipulating numbers or other objects using operations like addition and multiplication. Flavio Coelho\/Moment via Getty Images<\/a><\/figcaption><\/figure>\n\n\n\n

Courtney Gibbons<\/a>, Hamilton College<\/a><\/em><\/p>\n\n\n\n

You scrambled up a Rubik\u2019s cube<\/a>, and now you want to put it back in order. What sequence of moves should you make?<\/p>\n\n\n\n

Surprise: You can answer this question with modern algebra<\/a>.<\/p>\n\n\n\n

Most folks who have been through high school mathematics courses will have taken a class called algebra \u2013 maybe even a sequence of classes called algebra I and algebra II that asked you to solve for x<\/em><\/a>. The word \u201calgebra\u201d may evoke memories of complicated-looking polynomial equations like ax\u00b2<\/em> + bx<\/em> + c<\/em> = 0 or plots of polynomial functions like y<\/em> = ax\u00b2<\/em> + bx<\/em> + c<\/em>.<\/p>\n\n\n\n

You might remember learning about the quadratic formula to figure out the solutions to these equations and find where the plot crosses the x<\/em>-axis, too.<\/p>\n\n\n\n

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Graph of a quadratic equation and its roots via the quadratic formula. Jacob Rus<\/a>, CC BY-SA<\/a><\/figcaption><\/figure>\n\n\n\n

Equations and plots like these are part of algebra, but they\u2019re not the whole story. What unifies algebra is the practice of studying things \u2013 like the moves you can make on a Rubik\u2019s cube or the numbers on a clock face you use to tell time \u2013 and the way they behave when you put them together in different ways. What happens when you string together the Rubik\u2019s cube moves or add up numbers on a clock?<\/p>\n\n\n\n

In my work as a mathematician<\/a>, I\u2019ve learned that many algebra questions come down to classifying objects by their similarities.<\/p>\n\n\n\n

Sets and groups<\/h2>\n\n\n\n

How did equations like ax\u00b2<\/em> + bx<\/em> + c<\/em> = 0 and their solutions lead to abstract algebra?<\/p>\n\n\n\n

The short version of the story is that mathematicians found formulas that looked a lot like the quadratic formula for polynomial equations where the highest power of x<\/em> was three or four. But they couldn\u2019t do it for five. It took mathematician \u00c9variste Galois<\/a> and techniques he developed \u2013 now called group theory<\/a> \u2013 to make a convincing argument that no such formula could exist for polynomials with a highest power of five or more.<\/p>\n\n\n\n

So what is a group, anyway?<\/p>\n\n\n\n

It starts with a set<\/a>, which is a collection of things. The fruit bowl in my kitchen is a set, and the collection of things in it are pieces of fruit. The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 also form a set. Sets on their own don\u2019t have too many properties \u2013 that is, characteristics \u2013 but if we start doing things to the numbers 1 through 12, or the fruit in the fruit bowl, it gets more interesting.<\/p>\n\n\n\n

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In clock addition, 3 + 12 = 3. OpenStax<\/a>, CC BY-SA<\/a><\/figcaption><\/figure>\n\n\n\n

Let\u2019s call this set of numbers 1 through 12 \u201cclock numbers.\u201d Then, we can define an addition function for the clock numbers using the way we tell time. That is, to say \u201c3 + 11 = 2\u201d is the way we would add 3 and 11. It feels weird, but if you think about it, 11 hours past 3 o’clock is 2 o’clock.<\/p>\n\n\n\n

Clock addition has some nice properties. It satisfies:<\/p>\n\n\n\n