# 42

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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This is delightful. You should absolutely bring this up when she's older.

This is great! It reminds me a bit of ordinal arithmetic, in which addition is non-commutative. The ordinal numbers begin with all infinitely many natural numbers, followed by the first infinite ordinal, . The next ordinal is , which is greater than . But  is just

Subtraction isn't canonically defined for the ordinals, so  isn't a thing, but there's an extension of the ordinal numbers called the surreal numbers where it does exist. Sadly addition is defined differently on the surreals, and here it is commutative.  does exist though, and as with Norahats  does equal .

The surreals also contain the infinitesimal number , which is greater than zero but less than any real number. it's defined as the number between  on the left and all members of the infinite sequence  on the right. Not exactly Norklet (), but not too far away:  :)

(h/t Alex_Altair, whose recent venture into this area caused me to have any information whatsoever about it in my head)

ω-1 does exist as a surreal and is way better direct analog for Norklet

Yeah, agreed :) I mentioned  existing as a surreal in the original comment, though more in passing than epsilon. I guess the name Norklet more than anything made me think to mention epsilon--it has a kinda infinitesimal ring to it. But agreed that  is a way better analog.

One big difference, I think, is that you can't get to ω by (finite) counting, but you can get to Noranoo?

That was mainly the motivation for my counting question but the answer isconsistent with Noranoo invoking transfinite induction to get counted.

I think the pattern of -"Is it possible to count to Noranoo?" -yes and -"Can I count to Noranoo?" -no would isolate only transfinite induction.

Another way this could be asked as a dad joke is if ever one is tired and being asked about it claim that "I spent the night counting to Noranoo". If this story is incredile then it is recognised as a supertask. If there is a reaction like "wow you count fast" that would point to it being some very high finite integer.

I mean, she may think that it is such a large number that it is unrealistic that I could count that high overnight? Or even in my lifetime?

For example, if you claimed to have counted to a googol overnight I wouldn't believe you, but it's still finite.

I guess I am trying to fish for a scneario that would prompt a resonpce that would clearly support that. Another approach that would more strongly differentiate against googol-likeness would be to start counting and then increasingly blur the words together and then slow down "... norklet, noranoo" the thinking going that even if your mouth was perfectly dexterous the counting might detect "cheating detection" as it migth not be respectful of the vastness of the number.

But to be frank it was more that I thought I kinda understood the difference but I find myself struggling to figure out what would be a fair operationalization, suggesting I don't understand it.

Is Noranoo a prime?

Is Norklet a prime?

Is the product of all primes below Noranoo, plus one, a prime?

What is the sum of all positive integers below Norklet?

I asked Lily, and she told me that Noranoo is even, so I guess it isn't prime. Norklet is odd, but she doesn't know what prime means so I don't have a good way to find out.

The product of all primes below Noranoo is (not asking Lily) Noranoo. (Because otherwise it would be greater than Noranoo, and so clamps to Noranoo)

The sum of all positive integers below Norklet is Noranoo, since again it clamps.

Noranoo is obviously even in the sense that Noranoo = 2 * Noranoo, although this doesn't imply that Norklet is odd

Lily is operationalizing "even" as "you can divide it into two piles that are the same size".

Since you can do this for Noranoo, you can't for Norklet.

Well, if we consider two piles of size Noranoo then the total size is still Noranoo, isn't it?

You can't have two piles each of size Noranoo; that's too big

(This was initially my interpretation, but after Lily woke up I asked her, and that was also her response)

Does Noranoo have a square root? Is the square root an integer?

Noranoo has many square roots since, for example, Noranoo * Noranoo = Noranoo.

True, I guess I meant a square root that doesn't overflow, though I'm not sure how to define that in an elegant way.

I don't know how to find out, since Lily doesn't understand square roots yet.

Maybe you could introduce it via areas? If you drew squares of different sizes and talked about the relationship between area and side lengths.

She doesn't have area yet either, sadly

Is Norklet a regular integer?  I'd think it has to be, unless there's a distinct counting system for distance from Noranoo (and Norahats).  Which just means that this is regular numbers, but with an overflow policy of staying at the endpoint.

There are very few limitations posed. I strikes to me taht with properties given this could be near analogous with 1,2,3,4,5,...,ω-5,ω-4,ω-3,ω-2,ω-1,ω or how negative temperature works.

Is Norklet a regular integer?

Yes

regular numbers, but with an overflow policy of staying at the endpoint.

Yes

Can you get to Noranoo by +1 from 0?

Yes. If you start at zero and add 1 Noranoo times, you'll get Noranoo.

If you count up "one, two, three..." at what point are you supposed to say Noranoo?

Right after norklet