MTH 231 Numeration Systems Past and Present

Overview In Chapter 3 we consider how numbers have been represented historically, with an emphasis on decimal (base-ten) systems. A potentially good time to discuss these number systems would be while studying the cultures in which the number systems were used. As we translate into our numeration system—and ways to model it—the idea of exchange (“borrowing” or “carrying”) becomes critical. One important development for teachers is making distinction between a number, a numeral, and a digit.

Digits, Numerals, and Numbers A digit is a single symbol used in combination with other digits to make a numeral (e.g., 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 are the digits in our Indo-Arabic numeration system). A numeral is a symbol or name that stands for a number (e.g., 12 and twelve stand for the same number). A number is a count or measurement that is really an idea of size of quantity (e.g., the length of a ruler in inches or the number of doughnuts in a dozen). Numbers can be represented in ways other than using numerals (e.g., holding up fingers or clapping your hands).

Tally Marks The earliest means of recording numbers was simply making one mark, or tally, for each item being counted. Eventually, the tallies were grouped by fives for easier counting.

The Egyptians (c B.C.) The Egyptian system of recording numbers was based on hieroglyphics, or pictorial representations. This system was based on the number 10 (Why?), and the symbols were written vertically.

The Romans (c. 500 B.C.) The Roman system is still used today in watches and clocks, cornerstones, buildings, and (up until this year) Super Bowls.

The Indo-Arabic System Jointly named for the East Indian scholars who invented it c. 800 B.C. and the Arabs who transmitted it to the Western (read, European) world. A base-ten system with 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), the position of each digit indicates its value relative to a power of 10:

Expanded Notation

Modeling Using manipulatives to model our Indo-Arabic number system leads the way to performing operations of whole numbers. The following slides show images of commonly-used manipulatives.

Sticks In Bundles

Unifix Cubes

Units, Strips, and Mats

Base-ten Blocks

Homework Adjustment Problem 1: omit (d), (e), and (f) Problem 2: omit (c), (d), and (e) Problem 10: omit completely