ring theory – Multiplication of fractions is well-defined (Explanation)

I am studying the construction of the field of fractions from an integral domain. The multiplication operation on this field work as follows

$$(a,b)(c,d)=(ac,bd)$$

Also, $$(a,b)=(a_1,b_1)Leftrightarrow ab_1=ba_1$$

To show that the multiplication operation is well-defined in my book appears the following:

Suppose that $(a_1,b_1)=(a,b)$ and $(c_1,d_1)=(c,d)$. We have $(a_1,b_1)(c_1,d_1)=(a_1c_1,b_1d_1)$ and begin{eqnarray*} acb_1 d_1 -a_1c_1bd&=&(acb_1d_1-a_1cbd_1)+(a_1cbd_1-a_1c_1bd) \
&=& cd_1(ab_1-a_1b)+a_1b(cd_1-c_1d)\
&=& cd_1(0+a_1b(0)\
&=& 0end{eqnarray*}

My problem is that I do not know exactly where the term “$a_1cbd_1$” comes from. I have not been able to understand the proof at all for this detail, could you please help me?

fractions – Numbers and the period of their inverse

I am currently investigating the correlation between a natural number and the length of its inverse’s period. For example:

$$
frac{1}{3760} = 0.00026595744680851063829787234042553191489361702127 (period 46) \
frac{1}{1122} = 0.00089126559714795 (period 16)
$$

And so on.
There is a simple explanation for why does this happen mixing powers and modular arithmetic but it’s just an iterative process, which is troublesome for really big numbers.

Which conjectures, theories, hypothesis exists regarding the computation of the periodic length of any natural number's inverse?

How to interpret fractions as numbers in sheets?

You can convert text strings like 1/2 and 2/3 in column A2:A into decimal numbers like 0.5 and 0.666 like this:

=arrayformula( 
  iferror( 
    regexextract(trim(A2:A), "^(d+)") 
    / 
    regexextract(trim(A2:A), "(d+)$") 
  ) 
)

In the general case, there is not much point to try and implement a parser to evaluate a text string as a spreadsheet formula. It is much easier to use the initial = and let Google Sheets do the hard lifting.

haskell – Sum of fractions

Objective:

  • Create a function to sum a list of fractions (represented as pairs)

Rules

  • Only Prelude functions allowed

Notes

  • I was debating if I should create a Fraction data type. Is it worth “upgrading” from a simple pair?
  • Is the code easy to follow and well-structured? How are my function names?
  • Is a simple error call a good way to inform the caller of incorrect arguments?

Code

sumOfFractions :: ((Integer, Integer)) -> (Integer, Integer)
sumOfFractions () = error "empty list not allowed"
sumOfFractions fractions = reduce (numerator, lcd)
  where
    reduce (n, d) = (n `div` gcd_, d `div` gcd_)
    numerator = sum . map ((x, y) -> x * (lcd `div` y)) $ fractions
    lcd = foldr1 lcm (map snd fractions)
    gcd_ = gcd numerator lcd 

ag.algebraic geometry – Algebraic closure of field of fractions of multivariate polynomial ring over $mathbb{R}$

I am searching for good references on the topic of the behaviour of the elements in the algebraic closed field $(mathbb{R}(x_{1},dots,x_{n}))^{operatorname{alg}}.$ I imagine that, when these are seen as functions from $mathbb{C}^{n}tomathbb{C},$ they are some very wild functions on $(x_{1},dots,x_{n})$ but I suppose that some of their properties should be known and further studied. I am interested in introductory books, papers or advanced monographs describing these kind of fields and their geometric properties.

understanding solving for A and B in partial fractions

I’m trying to step myself through solving partial fractions in a year 10 book by Cambridge. This is a concept they’re introducing early for students who want to challenge themselves and it’s pretty light on the explanation.

For example: 7/( x+2 ) ( 2x-3 ) = A/2x-3 + B/x+2. I understand how to work this to the point where I reach 7=x(A+2B)+2A−3B. From there I’ve read that I need to do something called “equating coefficients. The coefficients near the like terms should be equal, so the following system is obtained: A+2B=0 2A−3B=7.

But I don’t understand WHY or how it is valid that we set these parts of the equation to these values. Why not A+2B=7 2A−3B=0 for instance. I’ve tried looking at YouTube and asking friends, but I can’t seem to get my head around it.

I can do it and I can solve for A and B using this method. But I’m really struggling to understand what it is I’m doing at that point in the process.