You contracted for a job to be paid and were to be paid $9,100 when the job was completed. The person wrote you a check made out for $1,900. Would you overlook the mistake since the person’s only mistake was to transpose two digits? If not, why would you want the person to write a different check?

        You would probably not accept the check because the interchanging of the two digits would be a huge change in the amount of money paid. In our numeral system (the Hindu-Arabic numeration system), the position of the numerals is important. The Hindu-Arabic numeration system is a place-value system, which means that the position of the numerals affects the value of the number it represents. (Remember that a numeral is the symbol and that a number is the value a numeral represents.) In the situation above, the 9 in the $9,100 represents nine thousand dollars and the 1 represents one hundred dollars.  In $1,900, the 9 represents nine hundred dollars and the 1 represents one thousand dollars.

        Notice the close relationship the place-value system has to sets, cardinality, and the set operation of union. The amount of money represented by $9,100 may be thought of as the cardinality of the union of a set containing nine thousand dollars with a set containing one hundred dollars. Similarly, the amount of money represented by $1,900 may be thought of as the cardinality of the union of a set containing one thousand dollars with a set containing nine hundred dollars.

        The Hindu’s developed the system before the 9^th century. The Persian mathematician al-Khwarizmi wrote a book on the system in about 825 A.D. after which the system was adopted by the Arabs. Though the system was first introduced into Europe in the 10^th century, it was not widely used in Europe until after the invention of the printing press in the 15^th century.

        In this session, we will consider the structure of the Hindu-Arabic numeration system and how it relates to the standard algorithm for addition