Nice, thanks!

I’d let my daughter (she was in 3rd grade at the time) lead herself to the then discomfiting conclusion that 0.99.. =1. (Let 0.999… = x, then 10x = 9.999…, then 10x minus x =9, but also 10x minus x = 9x, hence 9x = 9 and since 9 != 0, x = 1.)

She shared it with her class and her teacher, leading to disbelief. One boy said he would ask his father, a software engineer, who, predictably, also refused to believe it, refused to accept the proof and couldn’t prove it wrong.

Your article will help me explain limits to my kids as a mathematical way of expressing “if it walks like a duck, talks like a duck and flies like a duck, it is a duck” or “if you can’t tell the difference, they are the same”.

The other important point you allude to is that in order to talk about “difference” you need a metric! Data “scientists” often use the naive ds² = dp² metric on probability space, seeing the unit interval as just a subset of R, so the sequence (0.9, 0.99, 0.999 …) looks like it is converging. However, in log-odds space ( x = log(p/(1-p)) , or pulling back the Euclidean dx² metric), the sequence looks like (0.954, … n-10(-n) …) and the limit of the term differences is 1.

Of course, the metric depends on what math, physics or business you are trying to do!