As a relative beginner trying to understand math more deeply, I'm trying to learn more about the mathematical laws (the laws of the operations $+, -, \times, \div$)

For example, I know the basic laws (the ones that are just taken to be true) -- the commutative, associative, and distributive laws. What area of math (what books to read) would I find things like how subtraction is defined from addition (and how division is defined from multiplication), and the presentation of the various laws of subtraction and division as they relate to the basic laws and how to prove them. Some examples:

$n-(m+k) = (n-m)-k$


$\frac{\frac{n}{m}}{\frac{k}{l}}=\frac{n\times l}{m\times k}$

How to derive and prove such laws (and many others like them)?

Thank you

  • $\begingroup$ Subtraction is defined from addition only when the field $\mathbb{F}$ has an additive inverse. That's math-speak for, subtraction only exists when "0" (the additive identity for the integers) is in the set. $\endgroup$ – Don Larynx Sep 21 '13 at 13:05
  • $\begingroup$ The laws you described in example 1) Associative law, distributive law 2) closure law 3) not part of the axioms, but it can be proven $\endgroup$ – Don Larynx Sep 21 '13 at 13:08

The rules you speak about are the subject of elementary algebra.

It can be a bit difficult to find places where they are derived in rigorous detail, because elementary algebra is usually taught to children who don't particularly care for mathematical rigor, and so many textbooks will focus on indoctrinating students with the correct rules, and at best try to present informal arguments that they work rather than something that satisfies mathematical standards of proof.

There must surely be textbooks out there that do present proper derivations, but rather than searching for them, I would suggest you look for an introduction to abstract algebra instead. Abstract algebra studies rules of operation that act on things that are not ordinary numbers, but where the rules of operation satisfy laws that are like those of arithmetic to a greater or smaller extent. Among other things it provides a vocabulary for describing how "arithmetic-like" a particular set of operations are, and since it's working in a context where not all of the usual laws may be available, introductory texts will usually place some emphasis on how some laws can be derived from others, rigorously.


Generally, one would need a good experience with number systems to understand the properties of numbers thoroughly well. In order to begin understanding what makes up those laws, just read about them and see if you can make something of it.



I suggest you read "How to write proofs: A strategic approach" to get you started on proof-writing.

  • $\begingroup$ According to Google there is no book or any other web page other than this containing phrase "How to write proofs: A strategic approach". $\endgroup$ – Trismegistos Sep 23 '13 at 8:30

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