MathNovatory/Pure Maths/Fibonacci Numbers/Division

Dividing in a different numbering system

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".

fib01

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.

fib02

Fibonacci division
In dealing with Fibonacci numbers, we will, with the same values, have a different kind of "division",
this time Fn21 divided by Fn3, 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.

More on Fibonacci Division
For a closer look at Fibonacci Division, have a look at these sections describing the operation and conditions. Also try the Fibonacci Divider a unique Fibonacci division calculator.