They’re completely different. Everything covered in Landau is in the first chapter of Rudin, I believe. I don’t know how approachable it is–all my professors told me that the German edition was easy to read even if you don’t know German, so I got the German edition, and I can’t read it.
I appreciate the responses, but I am now more confused then when I started!
Just, regular , -3, -2, -1, 0, 1, 2, 3, 1/2, 3/4, e, Pi, real numbers.
How is the commutative property of multiplication for real numbers proven?
Starting from sets, you construct the naturals, and you prove all the properties they have. Then you construct the integers from the naturals, and prove all their properties. Then you construct the rationals from the integers, and prove all their properties. Lastly, you construct the reals from the rationals, and prove all their properties. That’s how you get it.
Is there anywhere on the Internet or in a not too technical book that I can find this laid out step by step?
Muad’Dib, the first thing that has to be determined before a proof can be provided is “what are we allowed to assume?”. In my opinion, the starting place that would be most satisfying for you would be Peano’s Axioms. When we begin at this point, we assume the existence of the natural numbers, rather than having to construct them from sets. We don’t assume the existence of addition and multiplication, but we do assume the existence of a successorship operation, and some other “natural” properties that the natural numbers possess.
I did find this page, (excerpted from Landau’s Foundations of Analysis, which ultrafilter mentioned above) that lists some definitions and results built on Peano’s Axioms. Proofs aren’t provided, but, looking over it, it looks to me like the results are listed in the logical order in which they would need to be proved. Commutativity of multiplication is Theorem 29. Theorem 30 is distributivity. Even though proofs aren’t provided, this should at least give you the flavor of how these things are established. Right now I don’t have time to find a cite for the actual proofs, but you might be able to get some results using the search string Peano Axioms Commutative Multiplication, or something similar.
Now, this will only get you these properties for the natural numbers. Getting the properties for integers, rationals, reals, and complex numbers is mostly a matter of figuring out how to define these objects in terms of the natural numbers. Once this is done, the properties are (more or less) straightforwardly “inherited” from the natural numbers upon which they are constructed.
By the way, JS Princeton, I am also curious about how graph theory is used to prove the properties of multiplication. Can you provide some more details or references? Also, what is an “operational definition” in this context? I’m familiar with that term from fields like psychology, but not in math. Thanks.
Goodness. This is all much more complicated then I had first assumed. When was this first proven? Is it a recent thing?
Most of it was done a little bit before 1900, which is when the movement to axiomatize math took off.
Just accept that multiplication is nothing but a figment of your imagination.
Muad’Dib - you think it’s bad so far, check out Goedel’s Incompleteness Theorem, which states that some truths are impossible to prove, even in theory.