Section 3.1 Rings Part B: Analysis

Cartesian products are very interesting. They essentially take everything that has done thus far and combines it.  An example given is a cartesian product \mathbb{Z}_4\cross\mathbb{Z} where addition and multiplication are defined as \begin{equation*}(a,z)+(a',z')&=(a+a',z+z')\\ (a,z)(a',z')&=(aa',zz')\end{equation*}

The idea that this is valid for any two cartesian sets, given a few conditions, is fascinating. Yet I wonder how this would be used in computer programming. Thus far in studying Abstract Algebra I’ve seen application, more often than not, with computers (every now and then an application not computer related but for the most part the applications deal with computer usage and calculations). I wonder if this has any sort of application to the physical universe, that is this part of the world where we can make measurements based on theory. Following a quick search online I discovered that cartesian products are used with tensors and align Hilbert spaces!

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