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I hope you welcome non-mathematicians here. I am trying to find out how to write the following idea as a math equation, currently it is in computer code but I want to be able to discuss this with some of the math guys at my work and at least start the conversation in their terminology. I will show what I think I know and I will hope to learn from everyone here. Thank you in advance.

I will start by defining the variables involved...

  1. Let b be an integer greater than or equal to 2. I think I write this as: $$\{b\in\Bbb Z\,:\,b\ge 2\}$$

  2. Let c be an integer in the range 1 to (b-1). I think I write this as: $$\{c\in\Bbb Z\,:\,0<c<b\}$$

  3. Finally there is a function of c that will resolve as a real number. I think I write this as: $$ f(c) = \mathbb{R} $$

What I need to do is take a sequence of expressions and multiply all the members of that sequence to produce a single real number. The basic form of each expression can be simplified to: $${f(c) \over b}$$ and then for each value of b an equation using a sequence of expressions is created as follows.

For b = 2 the equation is simply: $$A = \left\{ {f(1) \over b} \right\}$$

For b = 3 the equation is: $$A = \left\{ {f(1) \over b} \times {f(2) \over b} \right\}$$

and so on.

In general the equation would look something like...

For b = n:

$$A = \left\{ {f(1) \over b} \times {f(2) \over b} \times \ldots \times {f(n-1) \over b}\right\}$$

but because of the first case (b = 2) I am not sure how to properly write this.

So, how did I do as far as figuring this out on my own? What did I get wrong and how do I fix it?

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  1. Let b be an integer greater than or equal to 2. I think I write this as: $$\{b\in\Bbb Z\,:\,b\ge 2\}$$

This is the set of all such numbers. You could just write: $$ b \in \mathbb{Z}, b \ge 2. $$ or more like your version $$ b \in \{ b' \in \mathbb{Z} \mid b' \ge 2 \} $$

  1. Let c be an integer in the range 1 to (b-1). I think I write this as: $$\{c\in\Bbb Z\,:\,0<c<b\}$$

Same issue. Just write $$ c \in \{1, 2, \dotsc, b-1 \} $$ or if the dots are not rigorous enough $$ c \in [1, b-1] \cap \mathbb{Z} $$ or $$ c \in \{ c' \in\mathbb{Z} \mid 0 < c'< b \} $$

  1. Finally there is a function of c that will resolve as a real number. I think I write this as: $$ f(c) = \mathbb{R} $$

If you want to give a rough description of $f$, provide its domain and codomain sets: $$ f : A \to \mathbb{R} $$ or $$ f \in \mathbb{R}^A $$ and decide how precise you want to describe your domain set $A$, is it just $\mathbb{N}$ or $\{ c \in \mathbb{Z} \mid 0 < c < b \}$?

If you want only to describe the codomain set of $f$ then $f(c) \in \mathbb{R}$ might be enough.

I am not sure about your sequence, it might be $$ s_b = \prod_{k=2}^b \frac{f(k-1)}{b} $$ then \begin{align} s_2 &= \frac{f(1)}{2} \\ s_3 &= \frac{f(1)}{3}\frac{f(2)}{3} \\ & \vdots \\ s_n &= \frac{f(1)}{n}\frac{f(2)}{n} \dotsb \frac{f(n-1)}{n} \end{align}

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  • $\begingroup$ Wonderful. I will need to do some reading to understand the product notation but it seems similar to sum notation so I think I will grasp it with a little effort. Everything else I am familiar with (though a little wobbly on) from my college algebra classes. On the matter of $f(c)$ my intent here was just to describe the codomain as $\Bbb R$ for simplification of this question, with more formal descriptions being used with different functions of $c$. Thank you mvw. I will go do that reading and come back in a few days to either ask a little more or else accept the answer as given. $\endgroup$ May 20 '18 at 13:42
  • $\begingroup$ There are different ways to express different functions depending on some parameter $c$. E.g. they might form a family of curves. One way is e.g. $f_c(x) = x^2 + c$ another $f(x;c) = x^2 + c$ or simply $f(x,c) = x^2 + 2$. But maybe different functions are very different. It depends a bit on the concrete example. $\endgroup$
    – mvw
    May 20 '18 at 15:21
  • $\begingroup$ I understand. This is part of what I want to talk to the guys at work about, figuring out the best $f(c)$ to get the desired results for the software routine I am developing. Sometimes they can be a bit snobbish because I am not a math guy so I try to show them I am making an effort to work in their world whenever I need their help. It is my way of showing respect for their expertise. $\endgroup$ May 20 '18 at 16:39
  • $\begingroup$ Computer programs can be described at various levels, from real code to pseudo code to a very mathematical modeling in the realm of theoretical computer science. The basic motivation should be to find a common ground such that everyone involved gets it. Hard to give you a good recommendation without knowing more background. You might overdo it the way we discussed it here. :-) $\endgroup$
    – mvw
    May 20 '18 at 16:54
  • $\begingroup$ I appreciate what you are saying mvw, but it would be tricky to go into more detail without exposing some proprietary algorithms involved. That is why I dramatically simplified the sequence expression to a bare bones function and quotient. Frankly I would love to just lay it all out here and get the help of all the SE people both CS and Math. Sure would make my life a lot easier if I could. I do appreciate the help you have given and the candid advice as well. $\endgroup$ May 20 '18 at 20:28

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