Operational algebra

I vaguely remember wondering — as a child of seven or eight — how new mathematics became established. I pictured a long line of people in front of a guy with a desk (or maybe a tent), waiting to register their latest idea — such as a new mathematical operation.

I had one such idea myself. I knew by this age that multiplication consisted of repeated addition, so I thought why not define something else in terms of repeated multiplication? Thus I invented integer exponentiation. (It’s an obvious idea: I blame the Childcraft books for arousing my interest.)

When I was older and learned that exponentiation actually existed, I was troubled by a curiosity of language. When people talk about 104, they say “ten to the power of four” — and yet 102, 103, 104 are collectively referred to as “powers of ten”. In one context, the number after the “of” is the exponent, and in the other context, it’s the base. Why is this? Isn’t mathematics supposed to be consistent?

If I could answer my childhood self, I’d point out that the same ambiguity crops up in everyday language. We say that x-ray vision is one of the powers of Superman, but we also say that Superman has the power of x-ray vision. When we use terms like “power of” we are translating mathematical notation into English, which has no use for consistency.

I not surprisingly came to speculate about an operation defined in terms of repeated exponentiation, another operation defined in terms of that, and so on ad infinitum. Some of you will find this familiar.

Let’s adopt the symbol ➀ for addition, ➁ for multiplication and ➂ for exponentiation, and let’s have ➊ for subtraction, ➋ for division and ➌ for the calculation of roots (although exponentiation actually has two inverse operations — the other being the calculation of logarithms — so we’ll leave that well alone).

Now we can specify operations using algebraic variables — and for that reason I referred to this system as operational algebra. I’ll use the symbol ✪ for “operation”, and write 2 [✪3] 5 to mean the same as 2 ➂ 5 which means the same as 25. For inverse operations I’ll insert a minus sign so that 2 [-✪2] 5 means the same as 2 ➋ 5 which means the same as 2/5. My original notation was handwritten, obviously, but this is a reasonable compromise. I won’t pretend it’s perfect.

For positive integers, higher-order operations can be defined as:

a [✪n+1] b = a [✪n] a [✪n] … [✪n] a {b terms}

But because exponentiation is non-associative, this presents us with a choice. If we bracket the expression from right to left — as in 2(2^(2^(2^…))) — then we are in the same realm as Knuth arrow notation (which I didn’t know about at the time). But if we bracket it from left to right — as in (((22)2)2) — then because (ab)c=abc, it follows under this definition that ➃ can be extended to non-integer operands using the formula a ➃ b = a(a^(b-1)), and then a ➄ b can be defined for non-integer a. I always used the latter (left-associative) definition.

As for higher operations in this sequence, I was convinced at the time that they could all be extended to non-integer operands, but my reasoning was faulty.

The fun wasn’t all at the high end, though. Early on I recorded the trivial fact that there is no zeroth operation. That is, no operation can be consistently defined such that a + b = a ✪ a ✪ … ✪ a {b terms}.

I don’t propose to share everything I did with this system. For one thing, a lot of it was wrong, and for another, I threw my notes away years ago. But here is something worth finishing on. Recall that an inverse operation can be defined such that:

(a [-✪n] b) [✪n] b = a

Now observe:

  • 2 ➊ 2 = 2 – 2 = 0
  • 2 ➋ 2 = 2 / 2 = 1
  • 2 ➌ 2 = sqrt(2) = 1.41421…
  • 2 ➍ 2 = [solution of nn = 2] = 1.55961…
  • 2 ➎ 2 = [solution of (n^(n^(n-1)) = 2] = 1.6498…

Based on this, it seems intuitively likely that if we could extend all higher operations to non-integer operands, then as n approaches infinity, we’d find that 2 [-✪n] 2 approaches 2, giving us a glimpse into the infinite operation. This intrigued me.

I abandoned my speculations in operational algebra around the time I left school and started university. I am vaguely aware that everything I explored (except the wrong bits) exists in established mathematics, if often subsumed under much more abstract generalisations. This makes no difference to the fact that a teenager had fun messing about.

As a final word: please don’t take from this blog post the idea that I’m good at maths. I lack the rigour to be good at it; I am merely playful.

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