mth 231 numeration systems past and present. overview in chapter 3 we consider how numbers have been...
TRANSCRIPT
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