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An open source ad-free resource, algebrarules.com provides all the most useful Rules of Algebra in one place: easy to understand, free, and accompanied by informative descriptions & examples.

These simple rules — applied with a pinch of imagination and a dash of arithmetic — can divide, conquer, and solve just about any algebra problem.


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    “we were looking for the fundamental rules of algebra concisely presented in one place. We couldn’t find such a place, so we made Algebrarules.com”

    As I understand it, mathematicians generally develop the “law of algebra” by formulating the laws of behavior of a series of distinct abstract algebraic objects - groups, rings, modules and more exotic things like monads and “algebras”. From these abstract objects, more concrete sets like the integers, the rational numbers, the real numbers, the complex numbers, the quaternions and beyond can be characterized. These latter more concrete sets can also characterized directly with various axioms.

    Which is a long way to say if it feels like “the fundamental rules of algebra” aren’t concisely presented in one place, that is because there are really a number of different groups of “fundamental rule”. There is a lot of intersection but they aren’t the same.

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      Agreed, this feels more like a list of those little things that are useful to know when working out something in the real numbers. Taking for example the first rule they present, it already implies the existence of two binary operations with one being leftdistributive over the other. Rule 13 implies the used operation to be commutative. There’s other assumptions that indicate the authors seem to only consider algebra as one might have seen in highschool.

      Which is not a problem per se, if that is the target audience. I see the authors are calling themselves autoditact math enthusiasts and having had formal education in the matter I can only encourage an interest in the subject. I just feel calling it “fundamental rules of algebra” is incorrect, these aren’t.

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      This website pertains to what’s commonly called “high-school algebra”, so the title may be misleading to anyone familiar with more exotic algebraic structures.

      Further, it is aimed at a strange goal—the listing of the “fundamental” rules of algebra… While listing a number of derivable laws and providing no organization to help demonstrate that this list is complete.

      A better name might be “handbook” as it does list a set of laws that may be useful to keep “on the tip of your tongue” while doing HSA manipulations.

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        Thanks for your excellent feedback. We agree these rules are far from fundamental in the mathematical sense: they don’t describe the foundations of number theory, and depend on assumptions about numbers and arithmetic.

        From the human perspective, though, they are fundamental in the sense that they are a needed foundation for greater understanding of and practical facility with mathematics.

        To avoid being misleading, we have edited the introduction and about text to remove the word “fundamental” and make it more clear that these are basic rules for practical use (not mathematical foundations).

        @Tel, do you have a suggestion as to how we could organize the content that would demonstrate what we have covered and what we haven’t (and perhaps how these fit into a more complete picture of mathematics)?

        We’re keen to improve the site and make it maximally and accurate. Thanks for your help!