Kinds of division
An unusual form of "division" is possible within this closed set of Fibonacci numbers.
It is not only a surprise but a great joy to discover that a set of numbers can accomplish a feat beyond the range of the
more common sets,
like the integers for example. It is true that addition, subtraction, and
multiplication of integers will always produce an integer result, but the same cannot be said for
division. Since we will be dealing here with a form of division
of Fibonacci numbers, it seems preferable to start by defining what we mean by "division".
Normal division In dealing with integers, we understand the process of division to mean  "division into equal parts". Dividing 21 by 3 means making 3 piles of 7 each ; the value of each pile (the quotient) remaining constant. When the original larger value (21) is produced by the process of multiplication (3x7), this multiplication is an abbreviation for repeatedly adding a constant increment (7), and the multiplication merely records the number of times (3) that the basic value (7) is added to itself.
Fibonacci division In dealing with Fibonacci numbers, we will, with the same values, have a different kind of "division", this time F_{n}21 divided by F_{n}3, in which we find the values 8, 5, 8, quotients of different sizes, which we could express as 1 pile of 5 plus 2 piles of 8, all Fibonacci numbers, both piles and values.
