Editing Liber Abaci (section)

==Fibonacci's notation for fractions==
In reading {{Lang|la|Liber Abaci}}, it is helpful to understand Fibonacci's notation for rational numbers, a notation that is intermediate in form between the [[Egyptian fraction]]s commonly used until that time and the [[vulgar fraction]]s still in use today.{{r|moyon-spiesser}}

Fibonacci's notation differs from modern fraction notation in three key ways:

# Modern notation generally writes a fraction to the right of the whole number to which it is added, for instance <math>2\,\tfrac13</math> for 7/3. Fibonacci instead would write the same fraction to the left, i.e., <math>\tfrac13\,2</math>.
# Fibonacci used a ''composite fraction'' notation in which a sequence of numerators and denominators shared the same fraction bar; each such term represented an additional fraction of the given numerator divided by the product of all the denominators below and to the right of it. That is, <math>\tfrac{b\,\,a}{d\,\,c} = \tfrac{a}{c} + \tfrac{b}{cd}</math>, and <math>\tfrac{c\,\,b\,\,a}{f\,\,e\,\,d} = \tfrac{a}{d} + \tfrac{b}{de} + \tfrac{c}{def}</math>.  The notation was read from right to left. For example, 29/30 could be written as <math>\tfrac{1\,\,2\,\,4}{2\,\,3\,\,5}</math>, representing the value <math>\tfrac45+\tfrac2{3\times5}+\tfrac1{2\times3\times5}</math>. This can be viewed as a form of [[mixed radix]] notation and was very convenient for dealing with traditional systems of weights, measures, and currency. For instance, for units of length, a [[foot (length)|foot]] is 1/3 of a [[yard]], and an [[inch]] is 1/12 of a foot, so a quantity of 5 yards, 2 feet, and <math> 7 \tfrac34</math> inches could be represented as a composite fraction: <math>\tfrac{3\ \,7\,\,2}{4\,\,12\,\,3}\,5</math> yards. However, typical notations for traditional measures, while similarly based on mixed radixes, do not write out the denominators explicitly; the explicit denominators in Fibonacci's notation allow him to use different radixes for different problems when convenient. Sigler also points out an instance where Fibonacci uses composite fractions in which all denominators are 10, prefiguring modern decimal notation for fractions.{{sfn|Sigler|2002|p=7}}
# Fibonacci sometimes wrote several fractions next to each other, representing a sum of the given fractions. For instance, 1/3+1/4 = 7/12, so a notation like <math>\tfrac14\,\tfrac13\,2</math> would represent the number that would now more commonly be written as the mixed number <math> 2\,\tfrac{7}{12}</math>, or simply the improper fraction <math>\tfrac{31}{12}</math>. Notation of this form can be distinguished from sequences of numerators and denominators sharing a fraction bar by the visible break in the bar. If all numerators are 1 in a fraction written in this form, and all denominators are different from each other, the result is an Egyptian fraction representation of the number. This notation was also sometimes combined with the composite fraction notation: two composite fractions written next to each other would represent the sum of the fractions.

The complexity of this notation allows numbers to be written in many different ways, and Fibonacci described several methods for converting from one style of representation to another. In particular, chapter II.7 contains a list of methods for converting an improper fraction to an Egyptian fraction, including the [[greedy algorithm for Egyptian fractions]], also known as the Fibonacci–Sylvester expansion.