## 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?

## Is there a way to do Partial Fractions which don’t work with apart function?

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## 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?`

## fractions – What does it mean to add something half times?

I want to ask what does to mean to add something “half times”. Because, if we multiply fractions like $$16 times frac12$$, we are essentially adding $$16$$ one-half times, but it doesn’t make any sense.

So is there any way to explain the multiplication do fractions like this?

## Generating reduced fractions on Mathematica

The problem is how to generate the reduced fractions a/b in the interval (0,1) on MATHEMATICA. Where a and b are coprimes.

## 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.

## boolean algebra – XORing two fractions of integers

I was wondering if it is possible to apply the XOR operator between two fractions of integers. `2 ⨁ 3 = 0010 ⨁ 1101 = 1111 = 15`.

`2 = 6/3` and `13 = 26/2`. So how does one compute `6/3 ⨁ 26/2`?
Does the rules for summing two fractions apply for XORing two fractions? Because `6/3 ⨁ 26/2 != (12^78)/6`.

## 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.