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<\/p>\n<\/p>\n
You scrambled up a\u00a0Rubik\u2019s cube<\/a>, and now you want to put it back in order. What sequence of moves should you make?<\/p>\n Surprise: You can answer this question with\u00a0modern algebra<\/a>.<\/p>\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\u00a0solve for x<\/em><\/a>. The word \u201calgebra\u201d may evoke memories of complicated-looking polynomial equations like\u00a0ax\u00b2<\/em>\u00a0+\u00a0bx<\/em>\u00a0+\u00a0c<\/em>\u00a0= 0 or plots of polynomial functions like\u00a0y<\/em>\u00a0=\u00a0ax\u00b2<\/em>\u00a0+\u00a0bx<\/em>\u00a0+\u00a0c<\/em>.<\/p>\n You might remember learning about the quadratic formula to figure out the solutions to these equations and find where the plot crosses the\u00a0x<\/em>-axis, too.<\/p>\n Graph of a quadratic equation and its roots via the quadratic formula. Jacob Rus, CC BY-SA<\/p><\/div>\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 In my work\u00a0as a mathematician<\/a>, I\u2019ve learned that many algebra questions come down to classifying objects by their similarities.<\/p>\n How did equations like\u00a0ax\u00b2<\/em>\u00a0+\u00a0bx<\/em>\u00a0+\u00a0c<\/em>\u00a0= 0 and their solutions lead to abstract algebra?<\/p>\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\u00a0x<\/em>\u00a0was three or four. But they couldn\u2019t do it for five. It took mathematician\u00a0\u00c9variste Galois<\/a>\u00a0and techniques he developed \u2013 now called\u00a0group theory<\/a>\u00a0\u2013 to make a convincing argument that no such formula could exist for polynomials with a highest power of five or more.<\/p>\n So what is a group, anyway?<\/p>\n It starts with\u00a0a 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 In clock addition, 3 + 12 = 3. OpenStax, CC BY-SA<\/p><\/div>\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 Clock addition has some nice properties. It satisfies:<\/p>\n By satisfying all these properties, mathematicians can consider clock numbers with clock addition\u00a0a group<\/a>. In short, a group is a set with some way of combining the elements layered on top. The set of fruit in my fruit bowl probably can\u2019t be made into a group easily \u2013 what\u2019s a banana plus an apple? But we can make a set of clock numbers into a group by showing that clock addition is a way of taking two clock numbers and getting to a new one that satisfies the rules outlined above.<\/p>\n Along with groups, the two other\u00a0fundamental types of algebraic objects<\/a>\u00a0you would study in an introduction to modern algebra are rings and fields.<\/p>\n We could introduce a second operation for the clock numbers: clock multiplication, where 2 times 7 is 2, because 14 o’clock is the same as 2 o’clock. With clock addition and clock multiplication, the clock numbers meet the criteria for what mathematicians call\u00a0a ring<\/a>. This is primarily because clock multiplication and clock addition together satisfy a key component that defines a ring: the distributive property, where\u00a0a<\/em>(b<\/em>\u00a0+\u00a0c<\/em>) =\u00a0ab<\/em>\u00a0+\u00a0ac<\/em>. Lastly,\u00a0fields<\/a>\u00a0are rings that satisfy even more conditions.<\/p>\n At the turn of the 20th century, mathematicians\u00a0David Hilbert<\/a>\u00a0and\u00a0Emmy Noether<\/a>\u00a0\u2013 who were interested in understanding how the principles in Einstein\u2019s\u00a0relativity worked mathematically<\/a>\u00a0\u2013 unified algebra and showed the utility of studying groups, rings and fields.<\/p>\n Groups, rings and fields are abstract, but they have many useful applications.<\/p>\n For example, the symmetries of molecular structures are categorized by different\u00a0point groups<\/a>. A point group describes ways to move a molecule in space so that even if you move the individual atoms, the end result is indistinguishable from the molecule you started with.<\/p>\n The water molecule H\u2082O can be flipped horizontally and the end result is indistinguishable from the original position. Courtney Gibbons, CC BY-SA<\/p><\/div>\n But let\u2019s take a different example that uses rings instead of groups. You can set up a pretty complicated set of equations to\u00a0describe a Sudoku puzzle<\/a>: You need 81 variables to represent each place you can put a number in the grid, polynomial expressions to encode the rules of the game, and polynomial expressions that take into account the clues already on the board.<\/p>\n To get the spaces on the game board and the 81 variables to correspond nicely, you can use two subscripts to associate the variable with a specific place on the board, like using\u00a0x\u2083\u2085<\/em>\u00a0to represent the cell in the third row and fifth column.<\/p>\n The first entry must be one of the numbers 1 through 9, and we represent that relationship with (x\u2081\u2081<\/em>\u00a0– 1)(x\u2081\u2081<\/em>\u00a0– 2)(x\u2081\u2081<\/em>\u00a0– 3) \u22c5\u22c5\u22c5 (x\u2081\u2081<\/em>\u00a0– 9). This expression is equal to zero if and only if you followed the rules of the game. Since every space on the board follows this rule, that\u2019s already 81 equations just to say, \u201cDon\u2019t plug in anything other than 1 through 9.\u201d<\/p>\n The rule \u201c1 through 9 each appear exactly once in the top row\u201d can be captured with some sneaky pieces of algebraic thinking. The sum of the top row is going to add up to 45, which is to say\u00a0x\u2081\u2081<\/em>\u00a0+\u00a0x\u2081\u2082<\/em>\u00a0+ \u22c5\u22c5\u22c5 +\u00a0x\u2081\u2089<\/em>\u00a0– 45 will be zero, and the product of the top row is going to be the product of 1 through 9, which is to say\u00a0x<\/em>\u2081\u2081\u00a0x\u2081\u2082<\/em>\u00a0\u22c5\u22c5\u22c5\u00a0x\u2081\u2089<\/em>\u00a0– 9\u22c58\u22c57\u22c56\u22c55\u22c54\u22c53\u22c52\u22c51 will be zero.<\/p>\n If you\u2019re thinking that it takes more time to set up all these rules than it does to solve the puzzle, you\u2019re not wrong.<\/p>\n Turning Sudoku into algebra takes a fair bit of work. Courtney Gibbons<\/p><\/div>\n What do we get by doing this complicated translation into algebra? Well, we get to use\u00a0late-20th century algorithms<\/a>\u00a0to figure out what numbers you can plug into the board that satisfy all the rules and all the clues. These algorithms are based on describing the structure of the special ring \u2013 called\u00a0an ideal<\/a>\u00a0\u2013 these game board clues make within the larger ring. The algorithms will tell you if there\u2019s no solution to the puzzle. If there are multiple solutions, the algorithms will find them all.<\/p>\n This is a small example where setting up the algebra is harder than just doing the puzzle. But the techniques generalize widely. You can use algebra to tackle\u00a0problems in artificial intelligence<\/a>, robotics, cryptography, quantum computing and so much more \u2013 all with the same bag of tricks you\u2019d use to solve the Sudoku puzzle or Rubik\u2019s cube.<\/p>\n![\"\"](\"https:\/\/world.edu\/wp-content\/uploads\/2025\/05\/Graph.png\")
Sets and groups<\/strong><\/h4>\n
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Rings and fields<\/strong><\/h4>\n
It\u2019s all fun and games until you do the math<\/strong><\/h4>\n
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