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. 3400 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

0123 109103108104

19103100810004

93080040004839

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