EDMT 3420/MATH3821/MATH7821 Fall 2012 Week 10 : Number Systems

1. EDMT 3420/MATH3821/MATH7821Fall 2012Week 10 : Number Systems • Introduction • Number systems: • Familiar systems: Decimal, binary, Base 3; Base 4; Base 5. • Unfamiliar Systems: Aztec, Maya; Inca(quipu)

2. Numeration systems • What? Why? How? • Are they culturaly specific?

3. 1 2 7 4 9

4. 8 2 7 6 4 9 5

5. Definitions • Decimal number systemDefinition: A positional system of numeration that uses decimal digits and a base of 10Context: The decimal number system is the most common numeral system used around the world. • Expanded notationDefinition: A numeral expressed as a sum of the products of each digit and its place valueContext: Expanded notation is used when converting between the Maya number system and • GlyphDefinition: A symbolic figure or a character usually incised or carved in reliefContext: The Maya used glyphs to represent days and months in their calendar. • Place valueDefinition: The value of a digit as determined by its position in a numberContext: In the Maya number , the top dot is in the 202 (four hundreds) place, the shell is in the 201 (twenties) place, and the bottom dot is in the 200 (ones). • Vigesimal number systemDefinition: Any positional system of numeration that uses a base of 20Context: The Maya number system is a vigesimal number system.

6. I. 1. Decimal system • Positional : Right to Left Columns • Base 10: Powers of 10 • Ten symbols: 0 1 2 3 4 5 6 7 8 9 Groups: • How would you read and represent: 245; 1,350; 10,500; 125,400 • QUESTION: How would you count nine objects using 0 1 2 ?

7. 2. Binary Number System • Positional: Right to Left Columns • Base 2: powers of 2 • Two symbols: on/off switches 0 1 • One switch(one 1 or one 0) is called a bit. Eight bits (eight ones and zeros) is called a byte. The byte is basic unit in binary code.01100101=? 10101110=?01101000=? • Practice writing numbers using different bases: 3,4,5 etc.

8. II. 1. Number Systems: Maya Origin: • Geographically, the Mayan civilization covered what is now Belize, Guatemala, Honduras, El Salvador, and part of Mexico • The art of the Maya has been called the richest of the New World because of the great complexity of patterns and variety of media expressions. The ancient Maya also mastered architecture, and a glyph system of writing on stone, ceramics, and paper • The Mayans were advanced and made many innovations in math and astronomy, which they used to calculate an accurate calendar • Mayan were best known for their cooking, their pyramids , and their metal tools

9. Maya Number Systems • Origin:

10. Maya Number System • Positional: Bottom-Up • Base 20 (vigesimal): Powers of 20 • Three symbols: . One Five Examples: Activity: Practice writing the world records using the Mayan System.

12. Maya Number System Perform the following operations using the Maya system: • 203 + 413 • 4567 + 5678 • 656-200 • 97459- 52963

13. Maya Arithmetic: Addition

14. Maya Arithmetic: Subtraction

15. 2. Aztec Number system Origin: • The Aztecs were the Native American people who dominated northern Mexico at the time of the Spanish conquest in the early 16th century. • A nomadic culture, the Aztecs eventually settled on several small islands where they founded the town of Tenochtitlan, modern-day Mexico City. • They built their city, Tenochtitlan in 1325 in the marshes of Lake Texcoco, and quickly adopted much of the culture and language of their new neighbors. • Fearless warriors and pragmatic builders, the Aztecs created an empire during the 15th century that was surpassed in size in the Americas only by that of the Incas in Peru. • The Aztecs adopted a writing system that had been used for many centuries before and shared by many of the other Nahuatl-speaking nations of Central Mexico. • Aztec writing had three primary functions, namely to mark calendrical dates, to record accounting mathematical calculations, and to write names of people and places.

16. Number systems

17. 3. Quipu Number system • Origin: • The Inca Empire flourished on the western coast of South America from 1438 AD where they set out their base in Cuzco. • And during the next 50 years, the Cuzco dynasty, known as the Incas, have control of an empire stretching from Quito in modern Ecuador to the Maule River in Chile – a distance of nearly 2500 miles. • The Inca established a totalitarian state that enabled the tribal ruler and a small minority of nobles to dominate the population. • Four hundred years ago the wealth possessed by the Incas was discovered, then systematically plundered by Spanish. The treasure they carried home altered the whole European economic system. • Cuzco was the center of the Inca Empire, with its advanced hydraulic engineering, agricultural techniques, architecture, textiles, ceramics and ironworks. • Incas had a complex system of government and taxes and QUIPUS were used to record data. ( Initially deciphered in 1923)

18. Quipu: An Inca Counting System • Base 10 • NO written symbols: Strings of knots • Positional :The closer to the large cord a knot was placed, the greater its value. • Three types of knots:

19. A number is represented as a sequence of knot clusters in base 10. • Powers of ten are shown by position along the string, and this position is aligned between successive strands. • Digits in positions for 10 and higher powers are represented by clusters of simple knots (e.g. 40 is four simple knots in a row in the "tens" position). • Digits in the "ones" position are represented by long knots (e.g. 4 is a knot with 4 turns). Because of the way the knots are tied, the digit 1 cannot be shown this way and is represented in this position by a figure-of-eight knot. • Zero is represented by the absence of a knot in the appropriate position.