I was planning on sharing this with a math teacher friend to see what he thought. Maybe he would find something pedagogically useful in it. Maybe he would help me understand it better myself–he loved teaching and often wrote about what delight he found in helping a student “get it”. Unfortunately he passed away a couple of days ago. So, in memoriam, here are some idle thoughts on basic arithmetic. I wasn’t sure how to wind it up, so it ends sort of suddenly. He would have found that funny.

Given the set of all natural numbers greater than zero, ℕ₁, and the operation of addition, +, there are no values of variables a and b from the set of ℕ₁ such that the sum, a + b is not also a member of ℕ₁:

a ∈ ℕ₁, b ∈ ℕ₁
a + b ∈ ℕ₁

This notion is called “closure”. If we expand the set to the positive reals, ℝ⁺ we’d still wouldn’t be able to add two numbers escape the positive reals. We could flip it around and try it with the set of negative integers, ℤ⁻, but still, we can’t escape. If we reduce the set in some way things get a little interesting. Reduced to positive odd numbers, we can derive the evens, but reduced to positive evens we find ourselves confined to evens. Reduced to the set of primes, ℙ, it looks like we can derive many if not all composites (if Goldbach’s conjecture is true we can derive all the evens). But in all variants of one side of zero or the other, addition makes for a self-contained world.

All that changes if you trade addition for subtraction.

This fact has long amused and intrigued me, even before I really knew very much about algebra or what number classes were. Addition is commutative a + b = b + a but subtraction isn’t. Somehow the asymmetry of subtraction opens up new possibilities for numbers and makes symmetries possible. If you only had subtraction, you could invent addition.

Given the set of natural numbers greater than zero, ℕ₁ and the operation of subtraction -, you can derive zero–the additative identity, the negative numbers–the additive inverse, and of course, addition itself.

a ∈ ℕ₁, b ∈ ℕ₁
a - a = 0
a - 0 = a
0 - a = (a - a) - a
a + b = a - (0 - b)

We may have started out with counting numbers, but we discovered zero and soon derived all the rest of the set of integers, ℤ. I’m not versed enough in first order logic to really break it down, and others, I’m sure, already have. But taking what we have, how far can we go?

Lets skip ahead a bit an consider the case with multiplication.

Given the set of all natural numbers greater than zero, ℕ₁, and the operation of multiplication, ×, there are no values of a and b from the set of ℕ₁ such that the product, a × b is not also a member of ℕ₁.

a ∈ ℕ₁, b ∈ ℕ₁
a × b ∈ ℕ₁

The same enclosure of positivity around addition is also around multiplication. Within this enclosure is the fundamental theorem of arithmetic which is pretty great and all, but, again, if you want to get anywhere, you must trade multiplication for division.

a ∈ ℕ₁, b ∈ ℕ₁
a ÷ a = 1
a ÷ 1 = a
1 ÷ a = (a ÷ a) ÷ a
a × b = a ÷ (1 ÷ b)

You could constrain a and b further, to simply prime numbers, ℙ. Given division you can derive the multiplicative identity–1, the multiplicative inverses–the rationals ~ℝ~, and multiplication emerges there after, by the fundamental theorem of arithmetic everything else.

Can you derive or define division from subtraction? Maybe? If you’re confined to primes, you have to define 1 as 3 - 2.

But then you can start walking up to it:

a - (0 - a) = a ÷ (1 ÷ 2)
   a + a    =    a × 2

Generalizing this seems tedious, but perhaps there is an elegant way. I’m not an expert, but I know the arithmetic I’m using is too primitive to use for defining processes, and I’d have to be careful with process definition because many looping constructs for repeating actions could be seen as multiplications in disguise, and such a definition of division by repeating subtraction could therefore be seen as tautological.

But, supposing this is done, could we go further? what about powers, roots and logarithms complex and algebraic numbers?

Well it happens that this has really intrigued me in the past because it seems like the pattern of fundamental operations deriving new classes of numbers repeats with a couple of interesting wrinkles, however:

  1. My competence with these operations quickly wanes.
  2. You have to derive a product-over-sum operation, (a × b) ÷ (a + b), to really bring them together. Per #1 I’m still figuring out how to use this.
  3. The offical notation is inconvenient for showing the patterns, but whatever it’s faults are, it’s at least compact, and I found spelling things out in a linear way was complicated enough to lose the patterns too. There’s a couple of good attempts though, see 3Blue1Brown’s video “Triangle of Power” https://www.youtube.com/watch?v=sULa9Lc4pck