I thought of this after reading the first example in the comm sidebar.

In elementary Microwave Math (the subset most people learn during the normal operation of a consumer microwave), there are two places, the seconds place and the minutes place. The seconds place is constrained to [00 - 99] inclusive for one hundred total possible values in that place. The minutes place can be constrained to the same set of symbols, in which case Microwave Math is simply a base one hundred numeral system operating in a base sixty place value system, leading to the mildly humorous situation of having two ways to represent the same numerical value, e.g. 01:20 = 00:80. Some microwaves may have an hours place, or different constraints on the possible values of the minutes place, for which we’ll need…

Advanced Microwave Math! This introduces the concept of nested place value systems. Most of us are so used to place value numbering systems that we hardly notice how often we use them, but most numbering systems follow an implicit rule that the number of symbols is the same as the value of moving up a “place”. This makes sense for counting because you don’t need to move up a place until you run out of symbols, so you may as well make the value of the next place the next number you need to represent. Numeral systems don’t have to follow this rule, and Advanced Microwave Math breaks it.

The simplest case is where the minutes place is bounded to the set of all non-negative integers. In this value system there are two places, each with their own rules governing which symbols are allowed and what values they can represent. the seconds place is constrained in value to 00 - 99 (decimal, or DEC), and has a place value of one. The minutes place might be constrained to [00 - 99DEC], [000 - 999DEC], or it might be that the minutes place can contain any non-negative integer.

After that, we come to the hours place, which functions more or less the same way as the minutes place, in that it can have various constraints on what values can be used, but it still has the same place value relationship to the minutes place of sixty that minutes has to seconds. This changes with the introduction of the days place, which has a value of 24DECxhours instead of 60DEC.

Expanding this system into weeks and months and years introduces the idea that, though the system is generally presented one with positional notation (the value of place n is some [usually fixed] multiple of the value of place n+1). This isn’t necessary for Microwave Math, if each place can be defined by an arbitrary multiple of the of a base value e.g. the years place could be defined as 31557600DEC seconds (the “Julian Year”). The only requirement is that instead of position dictating the multiplier, each place must have a unique symbol denoting which multiplier is being used by that place. By convention they are arranged from largest multiplier to least, but 3 years, 6 months, and 12 seconds can just as unambiguously be written as 12 seconds, 3 years, and 6 months and refer to the same amount of microwave time (c.f. the American middle-endian date representation, a similar rule-breaking place value system that, if we insist upon using it, could really benefit from some non-positional place value indicator).

The value multiplier for a place doesn’t have to be an integer either. The introduction of leaps (day and second) and other vagaries of calculating means that we might prefer to use a “mean” value where a year might be some non-integer multiple of seconds, depending on which period of earth’s history one is in. There’s no reason the multiplier has to be an integer, or non-negative, or real, or rational, or continuous or differentiable or have any particular reference to any other place. In addition, each place has its own rules about what values can be in it, and those rules may mean that each place can have infinitely many symbols representing infinitely many values.

The inner place value systems can themselves be a simple positional place value system like decimal, or they can themselves use Microwave Math, meaning that place value systems in Microwave Math can nest infinitely. I’m not sure what kind of number that is but Microwave Math has some crazy implications to it.