eLisp: how to write a correct remove-number-grouping function?

Question

In EmacsWiki, sub page "Add Commas To Numbers", it says:

Here is how to transform a number such as 12345 into the string “12,345”.

(defun add-number-grouping (number &optional separator)
  "Add commas to NUMBER and return it as a string.
Optional SEPARATOR is the string to use to separate groups.
It defaults to a comma."
  (let ((num (number-to-string number))
      (op (or separator ",")))
      (while (string-match "\\(.*[0-9]\\)\\([0-9][0-9][0-9].*\\)" num)
        (setq num (concat
             (match-string 1 num) op
                  (match-string 2 num))))
                  num))

Note that you can also specify a different separator:

(add-number-grouping 12345 "'")
=> "12'345"

That's nice, and it works!  We've done some minor code adjustments, simplifying it, so that it only takes a dot . as a separator, not a comma or possibly anything else.

|----------+------------+---|
|        a |          b | c |
|        4 |         12 |   |
|----------+------------+---|
|        2 |          4 |   |
|       79 |          1 |   |
|       33 |         99 |   |
|        2 |         55 |   |
|        c |          n |   |
|        r |          9 |   |
|        9 |          8 |   |
|----------+------------+---|
| 84729823 | 84.729.823 |   |
|     3248 |      3.248 |   |
|       22 |         22 |   |
|    34458 |     34.458 |   |
|  1298869 |  1.298.869 |   |
|----------+------------+---|
|        7 |          8 |   |
|        4 |          9 |   |
|       31 |        779 |   |
|----------+------------+---|
#+TBLFM: @III$2..@IIII$2='(add-number-grouping $1);N

Even in Orgmode spreadsheets with wanting it to operate just on certain column ranges.

But the thing is this: Where turning 12.521.876 into 12521876 is needed, need for the reverse, turning 12521876 back into 12.521.876 again, is not far away.

With my bloody-elisp-beginner skills, it didn't take me long to figure out the reverse the manual way.

|----+------------+----------|
|  a |          b |        c |
|  4 |         12 |          |
|----+------------+----------|
|  2 |          4 |          |
| 79 |          1 |          |
| 33 |         99 |          |
|  2 |         55 |          |
|  c |          n |          |
|  r |          9 |          |
|  9 |          8 |          |
|----+------------+----------|
|    | 84.729.823 | 84729823 |
|    |      3.248 |     3248 |
|    |         22 |       22 |
|    |     34.458 |    34458 |
|    |  1.298.869 |  1298869 |
|----+------------+----------|
|  7 |          8 |          |
|  4 |          9 |          |
| 31 |        779 |          |
|----+------------+----------|
#+TBLFM: @III$3..@IIII$3='(replace-regexp-in-string "\\.\\([0-9][0-9][0-9]\\)" "\\1" $2)

And now the problem: I want to place add-number-grouping into my Emacs initial config file, so that this function is always at hand and doesn't need to be coded again manually before invocation; and just like that I want if for the reverse, remove-number-grouping.

But I just fail miserably in doing the deceptively simple step to take what I already have – that last Elisp formula – and turning it into a proper function!  And it's probably really, because I'm totally new to Elisp and don't actually know what I'm doing there.  I'm coming from bash, feel comfy with it, and accompanied by tiny peek at Python, looking at Elisp code I'm seeing traces of familiar basic concepts like arguments, variables, loops, but the Elisp syntax just breaks me – it's so cryptic and the error messages don't really help.

1st attempt – minor changes on add-number-grouping code as thought fit:

(defun remove-number-grouping (string)
      (let ((str "string"))
          (replace-regexp-in-string "\\.\\([0-9][0-9][0-9]\\)" "\\1" str)
                      str))

Hitting RET echo area in Emacs is saying remove-number-grouping.  Great!

Now invoking it,…

M-; in Emacs spawns eval function in the bottom line for evaluating, using functions

prompted for input, I type in (remove-n TAB, and it auto-completes! That's a good sign, it knows what we're up to!

(remove-number-grouping

and then continuing to

(remove-number-grouping 12.553)

hit RET

what we wish we got

"12553"

what we actually got

remove-number-grouping: Wrong type argument: sequencep, 12.553

Attempt #2: input:

Eval: (defun remove-number-grouping (string)
          (replace-regexp-in-string "\\.\\([0-9][0-9][0-9]\\)" "\\1" string))

M-;
(remove-number-grouping 33.552)
RET

output:

remove-number-grouping: Wrong type argument: sequencep, 33.552

another error message often appearing in my attempts to make it work:

Symbol's value as variable is void: remove-number-grouping

That popped up often, as well:

eval: Symbol's value as variable is void: 12\.424\.989

Done so for hours. What's the solution? What's the problem, that Elisp is not happy when I'm giving it the function remove-number-grouping and a value to work with, like 56.6454.32.34242?