Which mathematical operations are atomic?

Question

That’s probably a stupid a newbie question and I’m not sure if I'm phrasing it in the right words so please feel free to correct me.

So there are only several atomic operations in math. By atomic I mean it can’t be further simplified. So any formula or equation, despite of how complex it is in the end is only a collection of these basic operations.

E.g. multiplying can be simplified as addition (as in the end it is the same result):

3 × 3 = 3 + 3 + 3 = 9

One can argue that subtracting can be also simplified (rewritten) as adding:

3 - 3 = 3 + (-3) = 0

I’m not sure how I feel about this. While it’s still the same result I would consider subtracting to be an atomic though.

The division, though, is interesting thing. It can't be rewritten as adding or subtracting because it answers completely different question: how many times a number can be subtracted from another number.

So would it be correct to say that:

• Addition
• Subtractoin
• Division

are atomic mathematical operations?

Answer 1

1. Multiplication can be broken into additions only if you restrict your argument to integers. Otherwise there is no clear way to rewrite $$\pi\times \sqrt{2}$$ using addition.

2. If you allow for $x-y$ to be seen as $x+(-y)$ then you should also allow for $\frac{x}{y}$ to be seen as $x\times\frac{1}{y}$. As far as this is concerned I would be tempted to consider the operation of inverting with respect to a group operation as atomic ($x\mapsto x^{-1}$), which encompasses both previous cases and rules out subtraction and division.

Answer 2

You could go even further and say addition isn't atomic. The Peano axioms define addition and multiplication for natural numbers. Addition is defined as $$ \begin{align} a + 0 &= a \\ a + S (b) &= S (a + b), \end{align} $$ where $S$ is the successor function, which adds one to its argument.

You could even define equality in terms of other functions: $$ \begin{align} \forall\ x\colon\ & x=x \\ \forall\ a,b,f\colon\ & a=b\ \wedge f(a) \implies f(b)\\ \end{align} $$

I'm writing a proof assistant in which I use second-order logic as atomic operations and use the axioms above.

You can come up with various systems for atomic operations. There is no single way of doing this. For example, you could include addition and the inverse and then define subtraction as the inverse of addition. But you could also include subtraction and define addition as the inverse of subtraction.

Important to know is that every system has its limitations. There is no system that can describe all mathematics due to Gödel's incompleteness theorems. (if I understood it correctly, complicated stuff)

Answer 3

Intuitively, Addition & subtraction. If you go down further, it's only numbers. Numbers themselves have properties and are interesting - like prime numbers, etc. But since we are concerned with only "operations" - its addition & subtraction. Take space, universe, human body, trees, nature, everything, time - it's all about growth / decay.