Number Representation and Calculation

1. Number Representation and Calculation Chapter 4

2. OVERVIEW Babylonian Mayan Egyptian numeration system – a system to represent numbers Our system is called the Hindu-Arabic system. Computers use only 0’s and 1’s; it is the binary system. We will also study: • Roman • Chinese • Ionic Greek

3. Our Hindu-Arabic System and Early Positional Systems Section 4.1

4. Why/how were numbers developed? number – an abstract idea to answer “How many?” numeral – symbol to represent a number • Basically understood concepts of less/more • Needed a system of counting • Developed tally method • Acquired vocal sounds (words) • Wrote symbols (numerals) for words system of numeration – set of basic numerals and the rules for combining those numerals to represent various numbers

5. Example tally method Roman Hindu-Arabic | | | | | | | | ∕ 9 VII Mayan Chinese Ionic Greek q 九

6. World Geography North America Europe Asia Africa Australia South America Antarctica How many continents? 7

7. Our Numeration System Invented in India about 5th century AD

8. World Geography India

9. Our Numeration System Brought to Europe by Arabs about 10th century AD (during the Middle Ages) • Invented in India about 5th century AD

10. World Geography Europe India

11. Our Numeration System Brought to Europe by Arabs about 10th century AD (during the Middle Ages) • Invented in India about 5th century AD • Called Hindu-Arabic numerals • Was significant because: • It includes zero. • It uses only ten symbols (digits). • It is a positional system. Europeans called them Arabic numerals and Arabs called them Hindu numerals.

12. Our Numeration System In India the Brahmi numerals were used: No zero symbol at this point…Brahmi numbers had symbols for ten, hundred, thousand, etc. Last numerals were probably arbitrary, but could have been hurried tally marks. Probably symbol of the 4 directions as in other languages Probably tally marks, but written downward like Chinese writing By the time the numerals were brought to Europe, they looked like this:

13. 46 is different than 64 Positional Systems A positional system is a numeration system in which the position a digit occupies determines its value. It is also called place-value notation. The positions are generally based on powers. The use of positions probably developed from use of abacus in nearby China. Example3,257 = 3,000 200 50 + 7 3,257 Thousands place Ones place Tens place Hundreds place

14. Positional Systems A positional system is a numeration system in which the position a digit occupies determines its value. It is also called place-value notation. The positions are generally based on powers. Need to review powers, also called EXPONENTS.

15. Exponents Exponential notation: 52 = 5  5 = 25 2 of base number are multiplied together Base Exponent • am an = am+n • am an = am–n • (am)n = amn • Laws of Exponents: • a1 = a • a0= 1

16. Our Numeration System: Hindu-Arabic • Uses a base of 10 • Probably because we have 10 fingers 3,257 = 3,000 + 200 + 50 + 7 100 = 1 101= 10 102= 100 103= 1,000 104= 10,000 = 31000 + 2100 + 510 + 71 Textbook uses ( ). This is fine but not necessary. = 3103 + 2102 + 5101 + 7100 This is expanded form. • Back to the Example of 3,257

17. Example: Write in expanded form. 53,129 = 5104 + 3103 + 1102 + 2101 + 9100 704 = 7102 + 0101 + 4100 = 7102 + 4100 610,037 = 6105 + 1104 + 3101 + 7100

18. Example: Write as a Hindu-Arabic numeral. 3104 + 2103 + 6102 + 8101 + 5100 = 32,685 7105 + 2104 + 1102 + 9100 7105 + 2104+ 0103+ 1102+ 0101+ 9100 = 720,109 9106 + 1103 + 7102 = 9,001,700

19. Positional Systems Positional Systems: Babylonian System Our Hindu-Arabic system is a positional system based on the number 10 and using 10 symbols. • City of Babylon was 55 miles south of present-day Baghdad (the capital of Iraq). • The Babylonian system was a positional system based on the number 60 and using 2 symbols. Base 60 is called sexigesimal. Base 10 is called decimal.

20. World Geography Iraq

21. Positional Systems: Babylonian System • The Babylonian system was a positional system based on the number 60 and using 2 symbols. • City of Babylon was 55 miles south of present-day Baghdad (the capital of Iraq). • Civilization existed from 2000 BC to 600 BC. • Writing was called cuneiform (“wedge shape”). It was drawn into wet clay tablets, using a stylus to make two wedge shapes, then heated and dried. • Mathematical tables, including Pythagorean triples, have been found on Babylonian tablets.

22. Positional Systems: Babylonian System Example Write in Babylonian numerals. 31 23 46 OR

23. Positional Systems: Babylonian System Example Write in Hindu-Arabic. 50 14 38

24. Positional Systems: Babylonian System 600 = 1 601= 60 602= 3,600 603= 216,000 • Use powers of 60 • Spaces separate place values Example Write in Hindu-Arabic. = 7,882 2602 + 11601 + 22600 2 11 22 Babylonians didn’t have a symbol for zero, so used  for missing value. 23600+ 1160 + 221 7200 + 660 + 22

25. Positional Systems: Babylonian System 600 = 1 601= 60 602= 3,600 603= 216,000 • Use powers of 60 • Spaces separate place values Example Write in Hindu-Arabic. = 79,883 22 11 23 22602 + 11601 + 23600 223600 + 1160 + 231 79,200 + 660 + 23

26. Positional Systems: Babylonian System 600 = 1 601= 60 602= 3,600 603= 216,000 • Use powers of 60 • Spaces separate place values Example Write in Hindu-Arabic. = 5,220,662 24 10 1 2 24603+ 10602 + 11601 + 2600 24216,000 + 103,600 + 1160 + 21 5,184,000 + 36,000 + 660 + 2

27. Positional Systems: Positional Systems: Mayan System • The Mayans were a tribe of Central American Indians, that lived in the Yucatan Peninsula of Mexico, Belize, Guatemala, and part of Honduras

28. World Geography Yucatan Peninsula (Maya)

29. Positional Systems: Mayan System • The Mayans were a tribe of Central American Indians, that lived in the Yucatan Peninsula of Mexico, Belize, Guatemala, and part of Honduras • Their civilization lived from 300 – 1000 AD and was much more advanced than Europe at the time. • Mayans are famous for their architecture, knowledge of math and astronomy, and excellence in the arts

30. Positional Systems: Mayan System • Mayans were the 1st to have a symbol for zero! • The Mayan system used base 20, sort of. Base 20 is called vigesimal. Perhaps due to warm climate and no need for shoes, base 20 arose from total fingers and toes. • Mayan numerals were written vertically, with the bottom being the ones place.

31. Positional Systems: Mayan System Place values in Mayan: 204  18 = 2,880,000 203  18 = 144,000 Mayan calendar had 18 months of 20 days each, with separate 5-day calendar of danger and bad luck. 202  18 = 7,200 20  18 = 360 201 = 20 = 1 200

32. Positional Systems: Mayan System Example Write in Hindu-Arabic.  202  18 = 14  7,200 = 100,800 14  20  18 = 0  360 = 0 0  201 = 7  20 = 140 7 = 12  1 = 12 12  200 + 12 100,952

33. Positional Systems: Mayan System Example Write in Hindu-Arabic.  202  18 = 11  7,200 = 79,200 11  20  18 = 3  360 = 1,080 3  201 = 0  20 = 0 0 + 13 = 13  1 = 13 13  200 80,293

34. Homework From the Cow book 4.1 pg168 # 1 – 49 EOO, 61, 62 NOTE: EOO means “every other odd” NOTE: Tables for Babylonian numerals and Mayan numerals are on pages 166 and 167 of the textbook respectively. These will not have to be memorized for the test. Mayan Numerals