Baby Sister Numbers

November 3rd, 2021
kids, math
A few days ago, Lily (7y) told me about some Nora-inspired numbers:
  • The largest number is Noranoo.

  • If you try and make any larger number, you still get Noranoo. For example, Noranoo + 1 = Noranoo, and Noranoo * 2 = Noranoo.

  • Otherwise, it behaves normally. You can have Noranoo - 1, dubbed "Norklet". This means Noranoo - 1 + 1 = Noranoo, while Noranoo + 1 - 1 = Norklet. This didn't bother her.

  • Noranoo * -1 is Norahats. It is the smallest number, and like Noranoo any attempt to go lower keeps you at Norahats.

  • These are very large numbers: much bigger than a googol.

This is a kind of saturation arithmetic, more of a computersy approach than a mathy one, since you give up associativity, distributivity, the successor function being an injection, and all that.

On the other hand, it's slightly more elegant than a typical computational implementation of saturation, because it is symmetric around zero. Normally, you are using some number of bits, which gives you 2^N distinct values, and so an even number of integers. Typically we set the minimum integer to be one larger, in absolute value, than the maximum one. In this case, though, there are an odd number of integers. I asked whether perhaps Norahats * -1 * -1 * -1 could be Norklet and not Noranoo, but Lily insisted that Noranoo and Norahats were equal in magnitude.

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