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\textbf{PROGRAMME LEONARDO TOI}\\
\textbf{PROJECT PIT.AGOR\`A} \\
\textbf{Cup code : G12f10000140006}\\

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\textbf{Mathematics Workshops}

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{\LARGE
\textbf{Numbers and Counting with the Hieroglyphs of the Ancient Egyptians}
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%\textbf{Notes by R. Petti}

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The Garden of Archimedes.

\emph{A Museum for Mathematics}




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\title{Numbers and Counting with the Hieroglyphs of the Ancient Egyptians
}
\date{}
\maketitle



\section*{Introduction}
\begin{itshape}
The way we count is certainly one of the most powerful and fully developed systems ever invented. But it is also one of the most complex and difficult to learn. Other strategies, preliminary or alternative, other viewpoints, more primitive but in some cases no less effective, give us a better understanding of some aspects of counting, and help us to overcome some difficulties, to grasp the potentials of the way we count, and also provide us with some details of the fascinating history of counting. 

With this in mind, Garden of Archimedes laboratories were set up, dedicated to numbering systems and to some of these ancient ways of counting, and designed for all types of schools and grades of students.

The purpose of this booklet, devoted to the numbering system of the Ancient Egyptians, is to provide teachers who want to carry out this activity with some theory and many practical tips about how to carry out these workshops.  


    
    
\end{itshape}


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\section*{Historical Remarks}
\addcontentsline{toc}{section}{Historical Remarks}

The civilization of Ancient Egypt has left us some of the oldest traces of mathematics in written form. The first representations were found on buildings and in inscriptions dating back to the beginning of the 3rd millennium b. C. They are inscriptions using the hieroglyphic system, the Ancient Egyptian system of wall writing. In the hieroglyphic system there are seven distinctive signs for numbers:\\

\includegraphics[scale=0.15]{../materiali/immagini/egizi/1.eps}
\includegraphics[scale=0.15]{../materiali/immagini/egizi/10.eps}
\includegraphics[scale=0.2]{../materiali/immagini/egizi/100.eps}
\includegraphics[scale=0.2]{../materiali/immagini/egizi/1000.eps}
\includegraphics[scale=0.2]{../materiali/immagini/egizi/10000.eps}
\includegraphics[scale=0.2]{../materiali/immagini/egizi/100000.eps}
\includegraphics[scale=0.2]{../materiali/immagini/egizi/1000000.eps}
%\\includegraphics[scale=0.7]{../materiali/immagini/egizi/1000000p.eps}

The symbols correspond to increasing powers of ten, respectively 1, 10, 100, 1000, 10000, 100000, 1000000. The other numbers are represented by writing these symbols the number of times required to express the required value. For example the number 1203 was written with one lotus flower, two ropes and three sticks: 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1000p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}.

The origin of the symbols is unknown. Some are fairly easily identifiable and related to significant objects but in other cases the explanations are less certain. The forms are a stick or short rope, a stick or rope in an upside down "U" shape, a rope wound into a coil, a lotus flower in bloom, a finger, a tadpole in the process of metamorphosis, and a man kneeling with arms raised to the sky. Ropes were commonly used by the Ancient Egyptians to measure fields, and the principal unit of measurement was the khet, equal to 100 cubits. Lotus flowers grew in their thousands along the banks of the Nile, where tadpoles about to turn into frogs were also a common sight. The finger could be an allusion to counting on the fingers and the symbol for a million, the highest value in the Ancient Egyptian numbering system, might represent a man kneeling before immensity. Some theories attribute the origins of the symbols to phonetic properties. 
In the most ancient representations, the symbols do not have an established order. Gradually, over time, they took on recurrent positions. In this way symbols up to four in number were placed in line:
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}. If there are five identical symbols, they are written three above and two below:
\begin{tabular}{c}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
  \end{tabular}.
Six symbols are written on two lines, three above and three below. Seven and eight symbols are written respectively four above three below, and four above four below. Nine symbols usually occupied three lines:   \begin{tabular}{c}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
  \end{tabular}.
The writing could be read from right to left or left to right, and sometimes the symbols were back to front.


Hieroglyphs were used for decorative and commemorative purposes, had a function of solemnity and were used essentially for inscriptions on buildings. For counting, censuses, inventories, testaments, economic, administrative, legal, literary, religious, mathematical, astronomical documents and the like, the scribes used cursive writing known as the hieratic system.
This was a quicker form of writing suitable for papyrus; the symbols comprised one or two rapid strokes, a form of abbreviation, schematization and extreme simplification of the hieroglyphic signs, which they often only distantly resemble.
In hieratic writing, the signs for numbers are also schematized versions of the hieroglyphic symbols and various groupings of symbols. In the first documents of the third millennium b.C., these symbols were still recognizable, but subsequent transformations led to new forms apparently unrelated to the original signs. At the end of this formal process, with the frequent use of abbreviations or  repeated symbols, the hieratic system uses symbols not only for the fundamental values 1, 10, 100, 1000, etc. but also for all composite numbers: 2, 3, 4, 5, 6, 7, 8, 9, 20, 30, ...., 90, 200, 300, ..., 900, 2000, 3000, ...., 9000.\\

\includegraphics[scale=0.4]{../materiali/immagini/egizi/ieratico2.eps}\\

The mathematical texts before the Greek period, i.e. between 2000 and 1600 b.C., are in hieratic script and comprise the most important source of knowledge about Ancient Egyptian mathematics. The sources include the so-called Rhind Papyrus, the Moscow Papyrus and the Leather Roll, which are elementary mathematical texts of a practical kind, containing calculation tables and problems of various kinds. Ancient Egyptian arithmetic had a very interesting way to represent fractions, which generally had a unit numerator, i.e. in the form of 1/n. One of the practical problems faced by ancient Egyptian mathematicians, was how to express general fractions as a sum of unitary fractions. The technique of multiplication by  repeatedly doubling the number  was particularly important. Division, the opposite procedure, was also carried out using this technique.


\section*{The Arithmetic of Hieroglyphs}
\addcontentsline{toc}{section}{The Arithmetic of Hieroglyphs}


\subsection*{Representation and Changes}
\addcontentsline{toc}{subsection}{Representation and Changes}

The hierogliphic system is a decimal system. Each symbol corresponds to ten symbols of a lower order. In other words, each of the seven symbols used correspond to an increasing power of 10:


\begin{center}
    \begin{tabular}{cccc}
\includegraphics[scale=0.15]{../materiali/immagini/egizi/1.eps}&
\includegraphics[scale=0.15]{../materiali/immagini/egizi/10.eps}&
\includegraphics[scale=0.2]{../materiali/immagini/egizi/100.eps}&
\includegraphics[scale=0.2]{../materiali/immagini/egizi/1000.eps}
\\
$1$&$10^1$&$10^2$&$10^3$
\\
1&10&100&1000
\\
  \end{tabular}.

  \begin{tabular}{ccccccc}
\includegraphics[scale=0.2]{../materiali/immagini/egizi/10000.eps}&
\includegraphics[scale=0.2]{../materiali/immagini/egizi/100000.eps}&
\includegraphics[scale=0.2]{../materiali/immagini/egizi/1000000.eps}\\
$10^4$&$10^5$&$10^6$\\
10000&100000&1000000\\
  \end{tabular}.
\end{center}
Representation is based on a strictly additive principle. Each symbol always has the same value and the overall value of an expression with several symbols is given by adding the values of the symbols. Thus 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps} 
represents the number two hundred, whilst
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps},  which can be written either as 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps} represents the number one hundred and two.
The order makes no difference.  The symbols
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10000p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}  
represent the number ten thousand and twenty and 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100000p.eps}%
\begin{tabular}{c}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\end{tabular}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}
the number one hundred thousand six hundred and two.

Twelve vertical strokes 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps},
represent the number twelve, but the same number  can be written more simply using one heel bone and two vertical strokes: 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}.
In general, every time a symbol is repeated ten times it is better to replace it with the symbol of one order above. This is a sort of change-over, i.e. one group or symbols is replaced by a single symbol of a different shape and of the same value. In this way, the number of symbols required to represent a number can be reduced and the number is easier to understand at a glance.

In some cases, in some calculation procedures, the change-over will need to be in the opposite manner, i.e. instead of one symbol, ten symbols from the lower order are required.


\subsection*{Additions}
\addcontentsline{toc}{subsection}{Addizioni}

As in all additive systems of representation, additions are immediate. To add one or more numbers, it is sufficient to put all the symbols of each addend together. If we need to add two numbers, we should copy out the symbols of a given type which occur in the first number next to the symbols of the same type which occur in the second number, and so on for each symbol .
Adding is automatic, simply by copying out the symbols. The symbols together will give the result.
For example, if I want to add 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps} 
to
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps},
I copy the coiled ropes in the first and second numbers, obtaining
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps},
next to which I copy out the heel bones in the first and second number and then the vertical strokes in both numbers. This gives the result without having to make any kind of calculation:
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}.
In our symbols, we have added 132 to 213 obtaining 345.



If, in the result, there are ten symbols of the same type, in order to use as few symbols as possible, we change ten symbols into one of a higher order. For example,if we sum of  
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\begin{tabular}{c}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\end{tabular}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}
and
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps},
at the end of the copying procedure we have
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}, in which there are eleven heel bones. So I can rewrite the number in a more compact form using one coiled rope instead of ten heel bones. Therefore, the result is: 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}. 

\subsection*{Subtractions}
\addcontentsline{toc}{subsection}{Subtractions}

Subtractions in hieroglyphics are carried out as follows: the numbers are written one after the other. In the first number (the minuend, the number from which the second number is to be subtracted), you cross out, one after the other, each symbol that also appears in the second number, i.e. the subtrahend. You then write off again all the symbols that remain in the second number and this gives the result of the subtraction.

For example, if we take  
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}
and then subtract from it 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps},
we write out the two numbers. In the first number we then cross out as many strokes as we find in the second number, i.e. one of them:
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}. We then go on to cross out all the heel bones that we find in the second number, i.e. two of them:
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}. Now we have crossed out all the symbols in the second number. We then write out the symbols that remain:
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}. This is the result.
In this way we have subtracted 21 from 32, obtaining 11.

Sometimes the first number does not contain all symbols that are in the second number. For example in the subtraction 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1000p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}
minus 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps},
we can cross out two coiled ropes, a vertical stroke and three heel bones:
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1000p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}. To cross out in the first number everything that appears in the second number, we need to cross out a heel bone in the left-hand number, but there is no heel bone. So we have to replace the remaining coiled rope with ten heel bones,
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1000p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/100Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1Np.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps},  and then cross one out, giving the result of:
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1000p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}.
which means we have subtracted 241 from 1332 and obtained 1091.



\subsection*{Multiplications}
\addcontentsline{toc}{subsection}{Multiplications}

In Ancient Egyptian arithmetic there are no multiplication tables to remember, that is to say a table of the product of a set of numbers from 1 to 9 . It is based on repeatedly doubling the number to be multiplied. Doubling and halving were very important in ancient calculations. Multiplying by two was also very easy because you just had to copy out each symbol of the number to be doubled. Sometimes there are change-overs, where one symbol replaces ten symbols of one order down, in order to reduce the total number of symbols required. These change-overs can be done either at the end of the computation  or as you go along, every time you find ten symbols of the same kind. In general, with the first and simplest examples, the former is more useful, in order not to interfere with the overall calculation. Where the number of symbols appears to be building up, it is better to replace ten symbols with a single symbol to make the calculation easier to follow.
Let’s look at how two numbers can be multiplied using the doubling technique. The calculation is done in two separate columns, both used to double numbers. In the left-hand column we begin with one and double it repeatedly, two, four, eight and so on. In the right-hand column we start with the second number to be multiplied. For example if we want to multiply 6 by 12, in the left-hand column we start with \includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps} and we stop when we have the sufficient number of strokes that form the first multiplication number 6 . In this example, three calculation lines are sufficient. Six is the sum of two and four, from the second and third lines respectively:\\
\begin{tabular}{r|l|l|}
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}
&
$\qquad$
\\
$\longrightarrow$
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\\
$\longrightarrow$
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\end{tabular}\\
In the right-hand column we start with the second number to be multiplied, 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps},  and double it repeatedly: \\
\begin{tabular}{|l|l|}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\end{tabular}\\
We can now look for the result. This is done by putting together the symbols in the right-hand column that are on the same lines that formed the first number  in the left-hand column, i.e. in this example the second and third lines. \\
\begin{tabular}{r|l|l|l}
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}
&
$\,\,$
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\\
%\cline{3-3}
$\rightarrow$
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\fbox{
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
}
&
$\longleftarrow$
\\
$\rightarrow$
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\fbox{
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
}
&
$\longleftarrow$\\
%\cline{3-3}
&&&
\end{tabular}\\
Therefore, we obtain: 
 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps} which, after changing ten strokes with one heel bone, can be written more simply as:
\begin{tabular}{c}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\end{tabular}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}.
Translating, we have multiplied 6 by 12, obtaining 72.

It should be noted that multiplying 6 by 12 is not the same as multiplying 12 by 6 although the results must be the same. In the second case, in the left-hand column, the number 1 must be doubled until at least 12 strokes are obtained. Therefore, the left-hand column will have 4 lines, respectively with one, two, four and eight strokes. The last two lines give twelve strokes. The right-hand line will also occupy four lines, doubling 6 repeatedly. We then write out the symbols corresponding to the third and fourth lines,
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
 and
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}. If we have replaced ten strokes each time they occur with a heel bone, we obtain a total of 

\begin{tabular}{c}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\end{tabular}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}.


 \subsection*{Divisions}
 \addcontentsline{toc}{subsection}{Divisions}

Using the parallel doubling technique again, we can also divide. This time we start out in the right-hand column, doubling the divider, stopping before the number to be divided is reached. In the left-hand column, we once again double 1 repeatedly until we obtain the same number of lines as in the right-hand column. 
Now we need to locate the lines that give us the result. To do this, we look at the second column and find the lines that taken together give us the number to be divided. We then look at the corresponding lines on the left-hand column and add up the symbols, which give the result. 

For example if we want to divide 65 by 13, on the right we double 13  three times, because the fourth line would be more than 65.\\

\begin{tabular}{|r|l|}
$\qquad$
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\

&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\end{tabular}\\
On the left, we double three times the unit.\\

\begin{tabular}{|c|l|}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\end{tabular}\\
On the right we find the lines whose sum gives 65. They are the first and third line, 13 and 52 respectively, which added together make 65. We can now get the result, adding the  first and third line in the first column. \\
\begin{tabular}{l|c|l|r}
%\cline{2-2}
$\longrightarrow$
&
\fbox{
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
}
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
$\leftarrow$
\\
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\\
%\cline{2-2}
$\longrightarrow$
&
\fbox{
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
}
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
$\leftarrow$
\\
%\cline{2-2}
\end{tabular}\\
 Putting together their symbols
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}
   and 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps},
   we obtain 
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}.
So we have divided 65 by 13 and obtained the result of 5. 

Obviously, it is not always possible to obtain the dividend perfectly in the right-hand column. This occurs only in case of exact divisions. In the cases in which the number cannot be obtained perfectly, we use the lines that come as close as possible to the dividend, whilst remaining short of it. What is missing will be the remainder from the division. The quotient is the sum of the symbols in the left-hand column which correspond to the lines in the right-hand column which have been chosen. 

Say, for example, we want to divide 
\begin{tabular}{c}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\end{tabular}%
\begin{tabular}{c}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\end{tabular}%
 by
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}.
We begin doubling the fourteen sticks in the right-hand column: 14, 28, 56 and we stop at 56 because the next time we double we go over the dividend. On the left we double the units as many times as the lines on the right:\\
\begin{tabular}{|c|c|}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
$\,$
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\begin{tabular}{c}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\end{tabular}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\begin{tabular}{c}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\end{tabular}%
\end{tabular}\\
Now in the right-hand column we try to add up to obtain the dividend, choosing the lines that come closest to the number. Adding the third and second line we obtain eighty-four strokes , which is the number closest to eighty-six. The two strokes  left over give us the remainder from the division. The quotient is obtained by reading the symbols in the left-hand column in the second and third lines, so we get:\\
\begin{tabular}{l|c|l|r}
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
$\,$
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
&
\\
$\longrightarrow$
&
\fbox{
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
}
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\begin{tabular}{c}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\end{tabular}%
&
$\leftarrow$
\\
$\longrightarrow$
&
\fbox{
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
}
&
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/10p.eps}%
\begin{tabular}{c}
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\\
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\end{tabular}%
&
$\leftarrow$
\\
\end{tabular}\\
Therefore the result is:        
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
        with a remainder of:
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}%
\includegraphics[scale=0.5]{../materiali/immagini/egizi/1p.eps}.



 \subsection*{Fractions}
 \addcontentsline{toc}{subsection}{Fractions}

The fractions used by the Ancient Egyptians were all in the form of 1/n, except 2/3. To represent fractions a special notation was used:  a mouth placed above the number which shows the subdivisions. Thus 1/3 was shown as three strokes with a mouth above them, one twelfth was shown as a heel bone and two vertical strokes with a mouth above them, one twentieth was shown by two heel bones with a mouth above them. 1/2 and 2/3 had two special symbols. In hieratic script, the mouth was replaced by a dot.

All fractions were shown as sums of unit fractions with different denominators. Therefore $\frac{2}{5}$  was not shown as
$\frac{1}{5}+\frac{1}{5}$, but as $\frac{1}{3}+\frac{1}{15}$.

Of course a fraction could be broken down in more than one way. For example  $\frac{7}{24}$ could be written as $\frac{1}{6}+\frac{1}{8}$ or as $\frac{1}{4}+\frac{1}{24}$. 
The Rhind Mathematical Papyrus contains many examples of how to break down a fraction in the form  $\frac{2}{n}$,  but it is not clear whether there was a precise method for doing so.


The use of Ancient Egyptian notation was continued by the Greeks and until the Middle Ages. 
There are traces of it in  \emph{Liber abaci} by Leonardo Pisano.
Although ordinary fractions, with any numerator, were already in use, here some exercises are given to change some fractions into the form of 1/n with different denominators, as used in Ancient Egypt.

Apart from special cases in which the numerator is a divider of the denominator, and the fraction therefore can be reduced to the form 1/n, the first and simplest method refers to a case in which the numerator of the fraction can be expressed as the sum of the dividers of the denominator. If we consider the fraction $\frac{7}{24}$, we can see that 7 can also be expressed as 6 + 1, or as 4 + 3, all divisors of 24. This gives us respectively $\frac{7}{24}=\frac{6}{24}+\frac{1}{24}=\frac{1}{4}+\frac{1}{24}$ and $\frac{7}{24}=\frac{4}{24}+\frac{3}{24}=\frac{1}{4}+\frac{1}{8}$.
 
In subsequent methods the attempt is made to reduce to the above situation.
For example, if the numerator $N$ of the given fraction $\frac{N}{D}$ is a divisor of $D+1$ and $k$ is an integer such that $kN=D+1$, the fraction  $\frac{kN}{kD}$,  equivalent to the given fraction,  is equal to  $\frac{D+1}{kD}$; this breaks down into $\frac{1}{k}+\frac{1}{kD}$.  Hence also $\frac{N}{D}$ breaks down into $\frac{1}{k}+\frac{1}{kD}$. 
 Variants of this method enable other cases to be covered: such cases include when  $N$ is the sum of the divisors of $D+1$, or when either $N-1$ or $N-2$ divide $D+1$.


Leonardo Fibonacci illustrates a procedure that can always be applied, even for fractions not included above. As the first part of breaking down the number we take the largest unitary fraction (i.e. with the smallest denominator) contained in the given fraction; the rest is handled using one of the methods above or taking once again the largest unitary fraction contained. Hence, if we want to break down  $\frac{4}{13}$, we divide 13 by 4; the quotient 3 tells us that $\frac{1}{4}$ is the largest unitary fraction smaller than  $\frac{4}{13}$; so we can write  $\frac{4}{13} = \frac{1}{4}+\frac{3}{52}$. The last of these fractions can be broken down using the same method:  $\frac{3}{52}=\frac{2+1}{52}=\frac{1}{26}+\frac{1}{52}$, obtaining  $\frac{4}{13}=\frac{1}{4}+\frac{1}{26}+\frac{1}{52}$. If we had used the same procedure again as for the maximum fraction, the breakdown would have been different; by dividing 52 by 3 we obtain 18 as the denominator; then we calculate $\frac{3}{52}-\frac{1}{18}$  obtaining $\frac{1}{468}$; so we have  $\frac{4}{13}=\frac{1}{4}+\frac{1}{18}+\frac{1}{468}$.
Since we can see that the numerators of the remainders decrease, this procedure always reaches an end. However, it can lead to rather lengthy decomposition and with large denominators, and more compact decompositions can be achieved.


Fractions are also involved in division, when we want to get a result without a remainder. To carry out these calculations, some experience and skill is required. In the customary procedure, when we get to the end, we should continue the column of doubling the divisor, now adding its sub-multiples. These should be found so as to form what is missing to complete the number to be divided. In the first column, for every new line, fractions appear which correspond to each sub-multiple considered, for example a half, a third, a tenth .... Combining the doubled numbers with sub-multiples, we try to make up the number to be divided. The result is obtained in the left-hand column, adding the numbers and fractions corresponding to those chosen in the second column.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{large}
%\end{document}

\clearpage
\onecolumn

\section*{Workshop Guidelines}
\addcontentsline{toc}{section}{Workshop Guidelines}


\textbf{Other materials:} stamps of hieroglyphs, presentations, the book ``Ahmose and the 999,999 lapis lazuli''.  \\

The presentations are contained in a CD-Rom which is included in the "Ancient Egyptians" folder''. They are slides in PowerPoint to be projected during activities. They contain images and brief comments as well as explanations on how to set up the exercises to be carried out.

The presentations are ordered by increasing complexity.
Level 0, pre-calculation, is designed for the very young, five years of age. Levels 1 and 2 are designed for the first and second class of Primary School. Level 3 is designed for Lower Secondary School and level 4 for Upper Secondary School.
References to age are general recommendations: each teacher can decide whether to use materials from a different level, according to the class, or whether to deal with a few topics. 

Some presentations include worksheets  to be printed out and used during the workshopp.

For writing activities (particularly for the very young and medium-aged), the stamps can be used. For older children, the simplified hieratic symbols could be used.

The story ``Ahmose and the 999,999 lapis lazuli'', whose characters appear in the presentations at level 0, 1 and 2, can be used and adapted for other levels and is a good thread going through the activities of the youngest and Primary School children.




\subsection*{Level 0}
%\addcontentsline{toc}{subsection}{Livello 0}

Recommended Age: 5 Years of Age.\\


\emph{Before Starting the Workshop}\\

The activity aims to exercise with counting and to make the transition from counting on fingers to written representation. Counting up to ten with the fingers is part of the activity, which further reinforces this ability. The numbers up to ten are used, naming them and representing them with the fingers of both hands. 
Following the initial part of the story of "Ahmose and the 999,999 lapis lazuli'' the written representation of 1 to 9 is introduced by repeating the sticks. This is very easy to understand. The written stick is half way between the representation of an object in the physical world and the abstract symbolism of our system. 

After working with the first few strokes, we can go on to numbers higher than ten. Using the experience of Ahmose we can experiment with the technique of '''bunching'' ten units at  a time. At the same time, a new symbol is introduced, the heel bone, which represents the ten strokes bunched together.
At this level we do not ask the children to know the names of the numbers from eleven onwards. The idea is to get them to experiment with the idea of "bunching" ten things together  at a time. In this way, in a group of identical objects, they are asked to pick out bunches of ten and to "organize" larger sums, learning how to recognize them and compare them.

Once they have learned how to bunch the first group of ten, you can go on to show them how these bunches can also be bunched into ten, so we start with small bunches in packets, then move onto packets making larger packages and finally to packages grouped together to make bags, all based on counting to ten. When the next symbols are introduced (snail shell, lotus flower, ...) they can be named (hundred, thousand, ...) indicating that they are bigger and bigger numbers without saying anything more precise than that.

The complete version of the story is designed for older children. The teacher can decide how much of the story to tell.\\


\emph{Contents of the Presentation of  the Workshop}\\

The first slides present Ahmose's need to make a written representation of quantity, and introduce the first symbol, the stroke. It is then shown how we can count up to ten by repeating the stroke. Here, it is best to compare counting strokes with counting fingers. A worksheet can be printed (slide 14, shown not filled in at the end of the presentation) in which the children are asked to repeat the same number of strokes as the number of fingers of the hand or hands being shown. 
This is followed by exercises on how to identify some numbers.

Ten is then discussed. the heel bone is introduced.  The new symbol is associated with the strokes to show numbers higher than ten . Two hands together form a heel bone, the symbol for ten. Ten strokes become one heel bone. Then the new symbols are shown to see if the children have understood it, using the hands to show the symbol; for each heel bone, both hands of a series of children will be needed.

Then, the teacher shows the converse, how the fingers of the hand can be translated into symbols. The children work together, trying to write down what they are being shown with the fingers. The bunching of ten lapis lazuli can be done using pebbles or small objects in the classroom.

Then the teacher goes on to the next examples of bunching. Referring back to the story, the next symbols are shown, ten by ten, demonstrating how different bunches of ten produce larger and larger bunches, and the new symbols are associated with each bunch. The symbols can be explained in terms of their symbolic meaning, associated  with their larger and larger quantity. So we go from the  short straight stick to the  longer heel bone, then the coiled rope, which is even longer, then the lotus flower in bloom which would have been visible in their thousands on the banks of the Nile; next is the bent finger which can be associated with the sky and its tens of thousands of stars, and then the tadpole at the moment it became a frog, which would have swarmed over the banks of the Nile, and finally we get to the kneeling man, the largest number. Numerical symbols  are shown and identified in bas-reliefs, and can be copied using stamps. Another worksheet can be used to fill in symbols and strengthen the idea of the increasing complexity the symbols represent.

The last slides show groups of symbols and children are asked which numbers are the highest. The children can be helped reminding them of the packets, packages and bags to which the symbols refer. 

\subsection*{Level 1}
%\addcontentsline{toc}{subsection}{Livello 1}


%egizi_e1e2 più nuovo    dentro Giusti nuove 3,4MB 30 aprile

Recommended Age: 6-8 Years of Age.\\

\emph{Before Starting the Workshop}\\

The system of numerical representation used by the Ancient Egyptians was a very intuitive one.
By use of the system, a quantity can be first determined and then represented symbolically. The use of decimal orders, associated with the "bunching'' of tens (possibly related to the history of  Ahmose), enables an appreciation of the basic concept of numbering: by counting to ten I can express any number. Following this principle, with Ancient Egyptian symbols, it is also possible to work with  large numbers, which are more difficult in our own numbering system. 
The principle of addition used in the Ancient Egyptian numbering system makes additions and subtractions more manifest in the procedure. \\



\emph{Contents of the Presentation of the Workshop}\\


The first slides illustrate the functioning of the system of representation based on the repetition of strokes. Numbers up to five are shown and then numbers up to nine. An analogy is drawn with the fingers of the hand. A form (shown not filled in at the end of the presentation) can be printed and handed out  in order to aid understanding on this point.  Subsequently exercises on the recognition of some numbers are recommended.

Then, what happens after ten is discussed, introducing the symbol of the heel bone.
The new symbol is used in combination with the vertical stroke to indicate numbers greater than ten.

Subsequently, the other symbols are introduced. Each new symbol is introduced when the previous symbol is repeated ten times. Using all the symbols  large numbers can be represented, way beyond the common experience of children of this age. But in fact the Ancient Egyptian system is based on the ability to count to ten and to order seven symbols; for this reason, without translating the numbers into our own system, the children will be able to work with even large numbers, without difficulty. The names of the corresponding numbers (hundred, thousand, ten thousand, hundred thousand, one million) can be called "increasingly large numbers" without reference to a specific meaning.

Numerical symbols are then shown in some bas-reliefs and need to be identified. They can be copied using the stamps. 

Then these numbers are used for the purposes of calculation.
How to do additions is explained, simply by putting all the symbols of the numbers together. 

The last two additions involve changing-over. Eight strokes are added to two strokes and someone in the class may suggest that these ten strokes be replaced with one heel bone. This also applies to eight vertical strokes plus four strokes.

The last slides are about subtraction, using the method of crossing out the symbols to be subtracted. None of the examples require changing-over  and so are easy to do even where the numbers are very large.


\subsection*{Level 2}
%\addcontentsline{toc}{subsection}{Livello 2}

Recommended Age: 8-10 Years of Age.\\

\emph{Before Starting the Workshop}\\

As mentioned for the first level, the hieroglyphic system makes manifest some features of arithmetic that are hidden in our system of representing numbers. In particular, in this level the work is directed  repeatedly to the carry-over after the number ten is reached, requiring different symbols for the "different orders". The add-on / cross-out form of adding and subtracting makes these calculations very easy to carry out and understand. The procedure for multiplication using repeated doubling rather than multiplication tables – as presented using some examples - will raise some curiosity and interest and will demonstrate how different systems can be used to solve the same problem. Multiplication is followed by division. The system used in Ancient Egypt shows that division is the "inverse" of multiplication: the same procedure is used but the order of the steps is changed. Here, the teacher might stop merely at saying that quotient is not always exact. However this case will be  dealt with at the next level. \\



\emph{Contents of the Presentation of the Workshop}\\

The first slides show how the hieroglyphic system for writing numbers works, by  writing  the numbers from one to twenty. After this, all seven symbols are introduced. Reading exercises follow, using some hieroglyphic numbers. Then, the children are asked to do the opposite, i.e. to write down some numbers in hieroglyphic script.
Then how to do the addition is explained, grouping together the symbols of the numbers. In the first examples, there is no changing-over. The next examples include one changing-over in order to reach the final result.

The next slides show how to do subtractions, by crossing out symbols. The first example is very easy. The second example is one in which a symbol has to be changed in order to cross out the appropriate symbols, replacing a coiled rope with ten heel bones. These examples are followed by exercises.

Multiplication is then explained using the doubling technique. The example is followed by some exercises.

Finally, division is explained. The first example and the exercises that follow are exact divisions, in which the number being divided can always be found. At the end there is one example where the division is not exact and there is a remainder.



\subsection*{Level 3}
%\addcontentsline{toc}{subsection}{Livello 3}


Recommended Age: 10 – 14 Years of Age.\\


\emph{Before Starting the Workshop}\\

Compared to the previous level, representations and examples are more complex. For division, examples are given to show how to continue a division beyond the remainder. To do this, unitary fractions  are introduced, together with their hieroglyphic representations; some information is given on how to break down any fraction into unitary fractions, a topic that is developed further at the next level.\\

\emph{Contents of the Presentation of the Workshop}\\

The first slides illustrate the functioning of the hieroglyphic numerical writing system using the examples of the first numbers, using only two symbols. Then all the other symbols are presented and exercises are carried out to read and interpret the symbols.
 
Additions are then explained, copying out the symbols and, if necessary, making a change-over. The examples are followed by some exercises on additions. 

This is followed by subtraction, showing the technique of crossing out symbols and then moving on to examples where one symbol needs to be replaced with ten symbols of equivalent value. This is followed by examples with and without change-overs.

The next slides illustrate multiplication and the technique of doubling numbers. This is followed by exercises.

Division is first handled giving examples where the answer is a whole number. This is followed by exercises. Then an example is given where the division is not exact, explaining how to find the quotient and the remainder. This is followed by exercises.

The last slides introduce fractions, showing how fractions were illustrated with hieroglyphs. Some simple examples of generic fractions as the sum of unitary fractions are given. 

Finally, it is shown how a division not giving a whole number can be carried out using fractions. In this case we have a quotient with a whole number and a fraction. This is followed by some exercises.



\subsection*{Level 4}
%\addcontentsline{toc}{subsection}{Livello 4}


Recommended Age: 14 and Above\\



\emph{Before Starting the Workshop}\\

Compared to the previous level, here additions and subtractions require change-overs since the beginning. At this level, it could be interesting to interpret the procedure for multiplication by means of  binary numbers. This could be done by beginning to ask why the procedure works. The first step is to think about the composition of the first factor by repeated doubling, or in other words the possibility of breaking down all numbers to the sum of various powers of two, which is the equivalent of writing a number in base  two. The second part of the procedure concerns the doubling of the second factor: this is nothing other than the application of the distributive properties of the product with respect to the sum. Having broken down the first factor into a sum, we sum  the partial products of the second factor by these addends. The fact that the base  two was chosen to represent numbers reduces the products to repeated doubling. 

Another interesting topic is the representation of fractional quantities. Comparison with the Ancient Egyptian method, using only unitary fractions, shows the advantages of extending the positional decimal representation system to fractions. Representation with decimal points makes the writing unambiguous and makes it easy to compare numbers.\\



\emph{Contents of the Presentation of the Workshop }\\

The first slides illustrate the functioning of the hieroglyphic numerical  system by writing some  small  numbers using only two symbols. Then all the other symbols are presented and exercises are carried out in reading and interpreting the symbols. 

Additions are then explained, copying out the symbols and, if necessary, making some change-overs  to obtain the final result. The examples are followed by some exercises on additions requiring change-overs. 

This is followed by subtraction, showing the technique of crossing out symbols and then moving on to examples where one symbol needs to be replaced with ten symbols of equivalent value. This is followed by examples.

The next slides illustrate multiplication and the technique of repeated doubling. We suggest seeing what happens when the two factors are exchanged. The next slide helps us understand why the procedure works. This is followed by exercises.

Division is first handled giving examples where the answer is a whole number. This is followed by exercises. Then an example is given where there is a remainder. This is followed by exercises.

The next slides introduce fractions, showing how fractions were illustrated with hieroglyphs. Some simple examples are given to show how to write generic fractions as the sum of unitary fractions using hieroglyphs. 
More complex examples are then given of numbers broken down using the method described in Liber abaci by Leonardo Pisano (Fibonacci). 

Finally, it is shown how a division not giving a whole number  can be carried out using fractions. Examples are given of divisions giving an answer with a fraction. This is followed by some exercises.






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