You resolved a Rubik cube and now you want to order it. What sequence of movements should you do? Surprise: you can answer this question with modern algebra.
Most of the people who attended mathematics in high school will have taken a class called algebra, perhaps even a sequence of classes called algebra I and algebra II that asked them to solve x. The word “algebra” can evoke memories of complex -looking polynomial equations, such as ax² + bx + c = 0, or graphs of polynomial functions such as y = ax² + bx + c.
You may remember having learned about the quadratic formula to find the solutions of these equations and also to find where the graph intersects the X axis.
Equations and graphs like these are part of the algebra, but they are not everything. What unifies algebra is the practice of studying things – as the movements that can be made in a Rubik cube or numbers in the sphere of a clock that are used to measure the time – and how they behave when combining them in different ways. What happens when chaining Rubik’s cube movements or by adding the numbers of a clock?
Sets and groups
How equations such as ax² + bx + c = 0 and their solutions led to abstract algebra?
In summary, mathematicians found formulas very similar to the quadratic formula for polynomial equations where the greatest power of X was three or four. But they couldn’t do it for five. The mathematician was needed Évariste Galois and the techniques he developed – now called groups theory – to convincingly argue that there could not be such formula for polynomials with a maximum power of five or more.
So what is a group?
Start with a set, which is a collection of things. The fruit bowl of my kitchen is a set, and the collection of things that contains are fruit pieces. Numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 also form a set. The sets alone do not have too many properties – that is, characteristics – but if we begin to apply changes to the numbers from 1 to 12, or the fruit of the fruit bowl, the thing becomes more interesting.
Let’s call this set of numbers from 1 to 12 “clock numbers.” Then, we can define a sum function for watch numbers using the way we say the time. That is, saying “3 + 11 = 2” is the way we would add 3 and 11. It seems strange, but if you think about it, 11 hours after 3 are 2.
The sum of watches has some interesting properties
Satisfies: closure, where by adding elements of the set you get something more in the whole;
Identity, where there is an element that does not change the value of the other elements of the set by joining: add 12 to any number will result in that same number; associativity, where you can add any number that is desired in the whole; inverse, where any action of an element can be undone; Y Conmutatativity, where you can change the order of the clock numbers that are added without changing the result: A + B = B + a.
When satisfying all these properties, mathematicians can consider watch numbers with the sum of watches as a group. In short, a group is a set with some way of combining overlapping elements. The fruit set of my fruitman can probably not be easily grouped: what is an apple plus? But we can group a set of clock numbers demonstrating that the sum of watches is a way of taking two clock numbers and obtaining a new one that meets the rules described above.
Rings and bodies
Together with the groups, the other two fundamental types of algebraic objects that would be studied in an introduction to modern algebra are the rings and bodies.
We could introduce a second operation for clock numbers: the multiplication of watches, where 2 by 7 is 2, because 14 o’clock are the same as 2 o’clock. With the sum and multiplication of watches, watch numbers meet the criteria of what mathematicians call a ring. This is mainly due to the fact that the multiplication of watches and the sum of watches jointly satisfy a key component that defines a ring: distributive property, where A (B + C) = AB + AC. Finally, bodies are rings that meet even more conditions.
At the beginning of the 20th century, the mathematicians David Hilbert and Emmy Noether, interested in understanding the mathematical functioning of the principles of Einstein’s relativity, unified algebra and demonstrated the usefulness of studying groups, rings and fields.
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Everything is fun until the calculations are made
Groups, rings and fields are abstract, but have many useful applications.
For example, the symmetries of molecular structures are classified by different specific groups. A punctual group describes the ways of moving a molecule in space, so that, even moving individual atoms, the final result is indistinguishable from the initial molecule.
But let’s take a different example that uses rings instead of groups. A set of quite complex equations can be established to describe a sudoku: 81 variables are needed to represent each place of the grid where a number can be placed, polynomial expressions to encode the rules of the game and polynomial expressions that consider the clues already present on the board.
So that the spaces on the board and the 81 variables correspond correctly, two subscripts can be used to associate the variable with a specific place on the board, such as using X₃₅ to represent the cell in the third row and the fifth column.
The first entry must be one of the numbers from 1 to 9, and we represent that relationship with (x₁₁ – 1) (x₁₁ – 2) (x₁₁ – 3) ⋅⋅⋅ (x₁₁ – 9). This expression is equal to zero if and only if the rules of the game are followed. As each box of the board meets this rule, there are already 81 equations, only to say: “Numbers that are not from 1 to 9” should not be introduced.
The “1 to 9 rule appears exactly once in the upper row” can be summarized with a bit of algebraic reasoning. The sum of the upper row will be 45, that is, x₁₁ + x₁₂ + ⋅⋅ + x₁₉ – 45 will be zero, and the product of the upper row will be the product of 1 to 9, that is, x₁₁ x₁₂ ⋅⋅ got x₁₉ – 9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1 will be zero. If you think it takes longer to establish all these rules than to solve the puzzle, you are not wrong.
What do we get when doing this complex translation into algebra? Well, we can use algorithms of the late twentieth century to determine which numbers can be introduced into the board that satisfy all the rules and all the tracks. These algorithms are based on the description of the structure of the special ring – called ideal – that form these boards of the board inside the major ring. The algorithms will indicate if there is no solution to the puzzle. If there are multiple solutions, the algorithms will find them all.
This is a small example of how to establish algebra is more difficult than simply solving the puzzle. But techniques can be widely generalized. You can use algebra to address problems of artificial intelligence, robotics, cryptography, quantum computing and many more, all with the same tricks as you would use to solve the Sudoku or Rubik’s cube.
*Courtney Gibbons is a mathematics associated professor at Hamilton College
This article was originally published in The Conversation
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