If you can avoid rounding then your answers will be more accurate. But that's almost entirely separate from how big your "1" is.

If you set "1" to be "2^40 dollars (~1.1 trillion)", then $1 is represented as 2^-40. Adding that up 100 times will have no rounding, and give you exactly the same result as a deposit of $100.

On the opposite side of things, setting "1" to be "3 dollars" or "70 cents" would cause rounding errors all over, even though that's barely different in scale.

Okay, I think we are basically on the same page.

But since I'm finding this helpful ... (:

We've been talking about addition so far, and relative scales between numbers. But suppose we just consider a single number, and multiply it by itself some times.

Certainly if that number is 1, we can keep doing it forever without error.

But the further we get away from 1 (either 1e+X or 1e-X), the more quickly error will be generated from that sequence, eventually hitting infinity or zero.

I'm just trying to express through this example that there is still something "special" about 1 in scale (likewise 0, in offset), where you want to be "close to" it, in the face of doing some arbitrary math, in order to produce better results. It doesn't even need to involve relative sizes between 2 different numbers.

It depends on what math you're doing. Very often you're likely to find that a base number of 1e-6 makes you less likely to hit your exponent limits than a base number of 1.

1 is special in that half the positive floats are above it and half are below. That doesn't mean your use case wants half.

Then would it be fair to say that if you don't know what calculations might be coming, all other things being equal, 1 is a good choice since it is "unbiased" in this sense?

It's perfectly fine, but because addition is very common it's unlikely to be optimal.