More information about "parallel maths"

Question

I came across a version of maths on a site (it requires you to sign up to read the article), where addition is defined as the reciprocal of the sum of the reciprocals. In the below article, the author describes the basic operations in that variant (addition and subtraction) and proceeds to use those to get analogies of the derivative operator (and product, chain and quotient and sum rules), integral, Taylor series in that variant. I would like to learn more about this variant of maths, which the author calls "parallel maths" because it looks similar to the parallel addition of resistances. Also, have you seen this referred to with another name anywhere? If so, please do let me know.

https://www.cantorsparadise.com/a-fifth-fundamental-operation-of-arithmetic-and-the-beauty-of-parallel-calculus-93a2dfe28dda

If you would like to read the article, I have uploaded screenshots of the article on imgur. https://imgur.com/a/1NfJ8Xr

Answer 1

I think what you need is Group Theory or Field Theory. It allows for the systematic analysis of binary operations.

So what we have is $x,y \in \mathbb {R}$ and $x\oplus y=\frac{1}{1/x+1/y}=\frac{xy}{x+y}$. Though with the addition of $\pm \infty$.

To have a group, you need a closed, associative binary operation with inverses and an identity element. This new operation is closed under addition and it's associative. To get the other properties, you need to take some liberties.

Inverses are tricky. We need $e\in G$ so that $\frac{xe}{e+x}=x\implies x^2+ex=ex\implies x^2=0$.So no finite $e$ exists. $e=-\infty$ gets you there.

What about inverses? $\frac{xy}{x+y}=-\infty\implies y=-x $.

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To do calculus, you need invertible multiplication.

Because of the paywall, I don't know how the article defines multiplication. Repeated addition seems feasible.

$x\oplus x\oplus ...\oplus x$ n times is $x/n$. So $x \otimes n= x/n$. This suggests we use $n=1$ as the multiplicative identity.

$(x \otimes y)\otimes z = x/yz. x\otimes( y \otimes z) = x \otimes y/z=zx/y$ so it's not associative. So under this definition of multiplication we do not have a field.

An attempt at differentiation, incomplete.

$\frac{[(x\oplus h) \ominus x]}{h}=\frac{\frac{xh}{h+x}-x}{h}= \frac{-x^2}{h+x}\otimes (1/h)=? \frac{-x^2h}{h+x}$

In usual calculus we have $h\to 0$. Do we need $h\to 0$ or $h \to -\infty$ since the latter is the additive identity? In the first case, we get $0$, in the latter, $-x^2$.

So this is one example of analyzing the "new maths" you mention. It's a special case of Group/Field Theory.

Answer 2

What started small has become a treatise. I actually asked this same question back in 2016 here. There are several answers that are relevant over there. Below is a sort of "FAQ".

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What/how many sources discuss the operation?

The operation has had many different names/symbols. In my question I refer to a 3Blue1Brown video about the Triangle of Power in which he simply refers to the operation as "oplus" (denoted $\oplus$). The same useful properties of logs and roots studied in your article are mentioned there.

There is 1971 book by the name of Non-Newtonian Calculus by Grossman and Katz in which many different forms of differentiation and integration are considered. Among them is an early version of this unnamed operation (henceforth referred to as 'it') that uses $0$ instead of $\infty$. See Chapter 8 of the book, in which they refer to it as the "Harmonic Calculus", and the operation $\Delta$ as "harmonic addition," denoted $⊞$. There is also a dedicated website to their study of Non-Newtonian Calculus, which has not been regularly updated since 2021 but includes further references on this related subject.

The so-called "parallel" operator (same as yours, and denoted "$||$") has a wikipedia page with many details and further resources on the operation. It actually has been implemented in at least one calculator, as shown on the "Implementations" section of that page.

To summarize, it's been studied in many different places by many different people, for many different reasons. Many of them appear to be independent discoveries. This is not even an exhaustive list of the coverage of the operation. I'm actually working right now on digitizing notes (that I could share, if you like) of everything I know about the operation, some of which I figured out and I have not seen in sources. And, given that it's been independently studied by many people, there are likely more sources unknown to me or the English-speaking world.

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How should the operation referred to?

There simply isn't a single agreed-upon symbol for it, as (including the one shown in your article and the one I use) there are at least 5 different notations for it, none of which are nearly universally understood. Some have different uses than others. Any report of which is best will inevitably biased, so here I will be straightforward with my preferences.

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Which notation is best in practice?

(In order of my preference)

Notation 1

$a:b$

Pros:

• Not really used for other operations consistently, except ratios, which can be equivalently denoted by a $/$, and so less ambiguous.
• Takes up very little space.
• Commonly used in totally different contexts, so readily type-settable. It's basically like how - is used for subtraction and for hyphens and ranges of numbers. In short: not that big of a deal.

Cons: <--

• You can see one above
• Could be confused with a ratio (though not likely, given the contexts in which each are used).
• Not as noticeable (could be viewed as a positive in some contexts).

Notation 2

$a\Delta b$

Pros:

• It has an inverse symbol $\nabla$
• Looks pretty cool; not gonna lie.

Cons:

• Inverse operation $\nabla$ is used for gradients, and so almost certainly going to get ambiguous or confusing at some point (though that is a unary operation, which helps).
• $\Delta$ and the similar symbol $\triangle$ are used for a lot of different mathematical operations, from the "change in" operator to triangles; to the Laplace operator; to all sorts of things.

Notation 3

$a||b$

Pros:

• Looks cool.
• Been historically used the most.

Cons:

• Still very nonstandard.
• VERY easily confused with the magnitude operation $||\cdot||$. Game-ending weakness to me.

Notation 4

$a\oplus b$

Pros:

Cons:

• Quite clunky.
• Pretty lame-looking.
• Already used for literally dozens of different things, at least. In fact, it's one of the de-facto "arbitrary new operation" symbols and it's used for tensor sums.

Notation 5

$a⊞b$

Pros:

• Used by a book once.
• Oldest.

Cons:

• Gross-looking.

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Which name is best?

(in reverse order of my preference)

"Parallel Addition"/"Parallel Sum"

Pros:

• Arguably the most commonly understood.
• Sounds cool.
• Echos its physical meaning in resistors/springs/"working in parallel" etc.

Cons:

• Gives the impression that parallel resistors/springs are the only use

"Harmonic Addition"/"Harmonic Sum"

Pros:

• Sounds even cooler.
• Also echos the physical meaning of doing things "in tandem."
• The harmonic mean is double the harmonic sum.
• My favorite.

Cons:

• A little less obvious what the physical meaning is for resistors.

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Which pronunciation is best?

I like "a with b" a lot because it is short and sweet and not used for anything else. It also perfectly describes that the harmonic sum is the solution to the following problem:

If Jerzi and Alex can finish painting a wall alone in 4 minutes and 5 minutes, respectively, how long will they take to finish working with one another?

Saying "a parallel b" is also fine, but it's much longer despite conveying practically the same thing.