RosettaCodeData/Task/Zeckendorf-arithmetic/00DESCRIPTION

This task is a ''total immersion'' zeckendorf task; using decimal numbers will attract serious disapprobation.

The task is to implement addition, subtraction, multiplication, and division using [[Zeckendorf number representation]]. [[Zeckendorf number representation#Using_a_C.2B.2B11_User_Defined_Literal|Optionally]] provide decrement, increment and comparitive operation functions.

;Addition
Like binary 1 + 1 = 10, note carry 1 left. There the similarity ends. 10 + 10 = 101, note carry 1 left and 1 right. 100 + 100 = 1001, note carry 1 left and 2 right, this is the general case.

Occurrences of 11 must be changed to 100. Occurrences of 111 may be changed from the right by replacing 11 with 100, or from the left converting 111 to 100 + 100;

;Subtraction
10 - 1 = 1. The general rule is borrow 1 right carry 1 left. eg:
<pre>
  abcde
  10100 -
   1000
  _____
    100  borrow 1 from a leaves 100
  + 100  add the carry
  _____
   1001
</pre>
A larger example:
<pre>
  abcdef
  100100 -
    1000
  ______
  1*0100 borrow 1 from b
   + 100 add the carry
  ______
  1*1001

Sadly we borrowed 1 from b which didn't have it to lend. So now b borrows from a:

    1001
  + 1000 add the carry
    ____
   10100
</pre>

;Multiplication
Here you teach your computer its zeckendorf tables. eg. 101 * 1001:
<pre>
  a = 1 * 101 = 101
  b = 10 * 101 = a + a = 10000
  c = 100 * 101 = b + a = 10101
  d = 1000 * 101 = c + b = 101010

  1001 = d + a therefore 101 * 1001 =

  101010
   + 101
  ______
 1000100
</pre>

;Division
Lets try 1000101 divided by 101, so we can use the same table used for multiplication.
<pre>
  1000101 -
   101010 subtract d (1000 * 101)
  _______
     1000 -
      101 b and c are too large to subtract, so subtract a
     ____
        1 so 1000101 divided by 101 is d + a (1001) remainder 1
</pre>

[http://arxiv.org/pdf/1207.4497.pdf Efficient algorithms for Zeckendorf arithmetic] is interesting. The sections on addition and subtraction are particularly relevant for this task.