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Achieving every integer through mathematical functions

I was offered a problem in which you were given four 9's like this:

9 9 9 9

And by adding mathematical functions (+, -, /, *, floor/ceiling, log, ln, trig, factorial) you could make it sum to the numbers 1-100.

Which made me think: Given a positive integer \(n\), how many integers can you achieve from using some sequence of these mathematical functions -- can you get all of them? Only positive integers? I have a feeling that you can get at least every positive integer, but I don't have the evidence to prove it.

What are your thoughts?

Note by Michael Tong
3 years, 5 months ago

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So, what exactly are we allowed to use? Just one \(n\), or \(n\) 9's, or something else? Also, do we have to use all the digits we are given? Jimmy Kariznov · 3 years, 5 months ago

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@Jimmy Kariznov Well the introductory problem was just stating motivation for the question. I mean for any positive integer \(n\), greater than \(1\) I guess (since factorial is the only way to grow). You're allowed to use I guess the functions I stated (since I'm sure there are some crazy functions out there that can easily allow you to get any number). Michael Tong · 3 years, 5 months ago

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@Michael Tong Define \(\spadesuit(n) = n+1\). Then, \(\spadesuit(1) = 2\), \(\spadesuit(\spadesuit(1)) = 3\), \(\spadesuit(\spadesuit(\spadesuit(1))) = 4\), etc.

So, you don't really need a "crazy" function in order to make every positive integer. This is why providing a list of well defined functions is necessary for this to be a well posed question.

Regardless, what digits and how many of them are we allowed to use? Without any restrictions, I can "make" any number by starting with that number (no functions needed). Jimmy Kariznov · 3 years, 5 months ago

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@Jimmy Kariznov I said to look at the ones I said in the problem... did you miss that part of my post?

And \(n\) is an arbitrary integer, not every integer. I think that should be pretty obvious, to be frank. Michael Tong · 3 years, 5 months ago

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@Michael Tong As for functions, I did not miss that part of your post. I was merely demonstrating that we don't need a "crazy" function to make every positive integer.

My question was what digits are we allowed to use. The example you have has four 9's. Is that what we are starting with here? Jimmy Kariznov · 3 years, 5 months ago

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