CHAPTER 5 SEC 1 The Evolution of Numeration Systems

Early Societies  If you were an sheepherder, back in the very old days, how would you count something?  Tied knots in a vine, cut notches in a tally stick, or keep pebbles in a bag to keep track of them.

 These counting methods eventually led to the invention of the abstract concept of number.

 What is a number?  A number tells us how many objects we are counting.  What is a numeral?  Symbol which represents a number.

Modern Day  What is our current number system called?  Hindu-Arabic

3 Early Societies  The Egyptian  The Roman  The Chinese

Egyptian  The Egyptian hieroglyphic system is more than 5,500 years old. This system is an example of a simple grouping system.

 One must remember that order does not matter.  Convert to Hindu-Arabic.  100,122

 Convert 1,235,642 into Egyptian notation.

Addition and Subtraction  Write the solution to this problem.

Opinion  Is the Egyptian counting is it simple, hard, or tedious?

Historical  1798, the French Emperor Napoleon sailed with a large army to conquer Egypt and disrupt the lucrative trade routes to India. Although he was defeated, his defeat turned out to be a scientific triumph for Europe.  My question is why?

 Napoleon had taken some scholars to Egypt and they brought back a wealth of information of this ancient civilization. But the material was written in hieroglyphics called demotic script, which no one was able to translate it.  The good thing is that Napoleon brought back with him a key to help solve this puzzle.  Called the Rosetta Stone.

 A French Mathematician Jean-Baptiste Fourier showed some hieroglyphics to an 11-yr old boy named Jean Francois Champollion. Fourier stated that no one could read these hieroglyphics, the boy replied, “I will do it when I am older.” From that day on he dedicated himself to it.  Do you think he did it?

Answer.  The legend says that Champollion finally solved the mystery of the hieroglyphics, he exclaimed, “I’ve got it,” and fainted.

The Romans  The roman numeration system, which was developed between 500 BC and 100 AD, has several improvements over the Egyptian system.

Evaluating Roman Numerals  In Roman notation, we add the values of the numerals from left to right, provided we never have a numeral with a smaller value than the numeral to its right.  Example  DCLXXVIII  =678

 Notice that the values of the numerals either stay the same or decrease, but never increase.  However, if the value of a numeral is ever less than the value of the numeral to its right, then the value of the left numeral is subtracted from the value of the numeral to its right.

 Example,  IV represents 5-1= 4  IX represents 10-1= 9  XL represents = 40  CM represents = 900

Restrictions  There are 2 restrictions on this subtraction principle.  1. We can only subtract the numerals I, X, C, and M. For example we cannot use VL to represent 45.  2. We can only subtract numerals from the next two higher numerals. For instance, we can only subtract I from V and X; therefore, we cannot use IC to represent 99.

 1. Convert MCMXLIII to Hindu-Arabic.  M = 1000; CM = = 900; XL = = 40; III = 3  1943  2. Write 496 in Roman numerals.  CDXCVI

Other advantages  The use of multiplication principle.  The bar above a symbol means to multiply by 1000  A symbol between two vertical lines means multiply by 100.

Chinese

 This system is a multiplicative system. Originated during the Han Dynasty, which extended from 206 BC to 220 AD.

Convert.

Convert in to Chinese numerals  54,921  945,687