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Fibonacci and ascending continued fractions

Ascending continued fractions occur in that part of Liber abbaci where he explains the division of numbers written using Hindu-Arabic numerals. Fibonacci wrote the mixed fractions in the form , that is the reverse order as we do it today. Then he introduces a new symbol resembling the fractional notation, where over and under the fractional line are groups of numbers having the same number of terms at the beginning of Chapter 5. He explains it as follows [1] , p.49-50:  

Also if under a fraction line several numbers are put, and above each of them other numbers are written, then the number which will be put over at the head of the fraction line on the right part of it will denote the number of parts determined by the number placed under it. ... If under a certain fraction line one puts and , and over the is , and over the is , as here is displayed, , four sevenths plus one half of one seventh are denoted. ... Also under another fraction line are , and ; and over is , and over the is , and over the is , as is here displayed , the seven that is over the at the head of the fraction line represents seven tenths, and the that is over the denotes five sixths of one tenth, and the which is over the denotes one half of one sixths of one tenth, and thus singly, one at a time they are understood; ...

Here the symbol has in our modern notation the meaning

Similarly,

Before we explain how Fibonacci divides let us show how he multiplies mixed fractions.  For instance, he writes [1] , p 79:

Again if you will wish to multiply by , then the number is written down as is shown here; you multiply the by the , and you add that is over the ; there will be thirds. Also you multiply the by part of the rule, that is , and you add ; there will be sevenths that you multiply by the thirds; there will be twenty-firsts. This you divide by the and by that are under the fraction lines; you put them under the fraction line thus: ; the entire quotient will be , ...

Actually and here .

From reasons of  the “energy savings” at division Fibonacci starts with a decomposition of the divider into smaller factors (not necessarily primes) and the division is carried out in a succession of division by these smaller factors. He precomputed a table of such decompositions, which he gives in the form [1] , p.65:

etc. From other decompositions given here let us mention the following ones

How Fibonacci found this notation? In the chapter where he explains the division he gives several examples, For instance he explains the division of by [1] , p.70, as follows:

Since one will to divide by , he notes the rule for finding in numbers the factor , and he finds the rule for , that is . He divides the by ; the quotient is , and there remains which he puts over the in the fraction, and he divides the by , namely by that which precedes the in the fraction; the quotient is , and there remains ; this he puts over , and he divides again by the , that which is at the end of the fraction; the quotient is , and three remains ; the he puts over the , and the he puts before the fraction, and thus one has the sought division , as is shown here. ...

We can formalize Fibonacci’s procedure to get the following scheme: Let is the quotient and , , the remainder after the division of by , that is . Let (what Fibonacci always tacitly assumes) and divide by the product .  Then