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Numerical Systems

Numerical Systems. Miss Chrishele Hruska Pre-Calculus, Grade 11 April 19, 2009 EDLT 302-Electronic Literacy Dr. David D. Carbonara Spring 2009. Start. Student Objectives.

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Numerical Systems

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  1. Numerical Systems Miss ChrisheleHruska Pre-Calculus, Grade 11 April 19, 2009 EDLT 302-Electronic Literacy Dr. David D. Carbonara Spring 2009 Start

  2. Student Objectives • The students will be able to write numbers in simple, multiplicative, and positional number systems after completing the Interactive PowerPoint to 100% correctness. • The students will be ale to write numbers in base 20 and base 60 after completing the Interactive PowerPoint to 100% correctness. • The students will be able to identify Egyptian Heiroglyphics, Mayan numerals, Babylonian Cuneiform, Ancient Chinese- Japanese numerals, and Attick Greek after completing the Interactive PowerPoint to 100% correctness. Directions

  3. Directions Dear Student, Complete this Interactive PowerPoint to learn about different numerical systems. You will have three days to complete all of the slides. Don’t worry about getting a question wrong! You can always go back to review the previous slides and try again. Remember to record your answers to the questions on your worksheet that you will hand in. If you have any questions while you are completing the PowerPoint, please contact me before you move on. Good luck! Get started!

  4. Numerical Systems • Numerical systems are based on grouping systems that rely on certain bases. • Represent a useful set of numbers • Give every number represented a unique representation • Reflect the algebraic and arithmetic structure of the numbers • Our numerical system is a base 10 system. • Can you think of different ways that we use a base 10 system everyday? Next

  5. A base-5 system has been used in many cultures for counting. It originated from counting the number of fingers on a human hand. • A base-8 system was devised by the Yuki tribe of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. Continue

  6. There are many other bases that have been used in ancient times and in present day. • Base 5- South American Tribes • Base 20- Mayan • Base 60- Babylonian Cuneiform • Can you think of things that are base 12 that we see everyday? Continue

  7. From what did some early numeral systems originate? The Sun Fingers Calendars Successive Kings

  8. Incorrect Answer! Here’s a hint! Think of base 5 and base 10 systems. Go back and review Try Again!

  9. Correct Answer! • Great job! Keep going! Next

  10. Base 10 Systems • Along with our modern number system that we use everyday, many other civilizations also use base 10 systems like the Egyptians, Ancient Chinese-Japanese, Attic Greek, and Romans. • But how did our system come to be how it is today? Let’s Find Out!

  11. Hindu-Arabic Number System • As early as 250 B.C., the Hindus of India invented a new number system. • The oldest preserved examples are found on stone columns erected in India by King Asoka. • It is likely that traders and travelers of the Mediterranean coast introduced the new number system to the Arabs. • In 711 A.D., the Arabs invaded Spain and the number system emerged into Europe. Next

  12. The Arabs invaded Spain in 711 A.D., bringing their number system along with them. http://www.acs.ucalgary.ca/~vandersp/Courses/maps/fullmap2.jpg Continue

  13. The early engravings do not have a zero or the positional notation. • The Hindu-Arabic number system became popular because it was easier to write out calculations. • Later, when zero and the positional system were developed, the Hindu-Arabic number system became superior than any of the other number systems being used at the time. • Aryabhatta of Kusumapura who lived during the 5th century developed the place value notation and Brahmagupta later introduced the symbol zero in the 6th century. Let’s see what you’ve learned!

  14. Why is our modern number system called the Hindu-Arabic number system? The Hindus and the Arabs developed the number system at the same exact time. The Hindus invented the number system, and then Arabs continued to spread it around Europe. The Hindus and the Arabs were at war with each other over the number system. The Arabs invented the number system and the Hindus stole it from them.

  15. Incorrect Answer! • Sorry, you didn’t get the question right. Here’s a hint: Remember who brought the number system to Spain during the conquest of 711 A.D. Go back and review Try Again!

  16. Correct! • Great Job! You’re ready to learn more about different number systems! Next

  17. Simple Number Systems • Many numeral systems are simple. • Simple means that there is a symbol for the base number, b, and also a symbol for b2, b3, b4, etc. • A number is expressed by using these symbols additively, repeating the symbol a certain number of times. • Examples of simple number systems are: Egyptian Hieroglyphics, Attic Greek, and Roman Numerals. Learn about Egyptian!

  18. Egyptian Hieroglyphics • This system was used in Egypt until the first century B.C. • Hieroglyphics are based on a scale of 10 and consecutive bases of 10. • There are symbols for 1, 10, 102,103,104, 105, and 106. • Multiples of these values were expressed by repeating the symbol as many times as needed Learn more!

  19. Egyptian Hieroglyphics were used in Egypt throughout Persian rule in the 6th and 5th centuries B.C. and even after Alexander the Great’s conquest during the Macedonian and Roman periods. Learn more! http://www.iziko.org.za/sh/resources/egypt/images/map_e1_l.gif

  20. Egyptian Hieroglyphics Symbols • Here are the symbols that the Egyptians used for numbers. Now some examples!

  21. Examples of numbers written in Egyptian Hieroglyphics • 13 • 457 More…

  22. More examples! • 6, 123 • 10, 268 You Try!

  23. A few things to remember… • Whenever there are more than five of the same symbol, stack the symbols to save room • Examples: You Try!

  24. Write 342 in Egyptian Hieroglyphics. A B C D

  25. Incorrect Answer. Here’s a hint! Remember to write in descending order! Go back and review Try Again!

  26. Correct Answer! • Great Job! You’re ready to move on to Attic Greek! Move on

  27. Attic Greek • Attic Greek was developed sometime before the third century B.C. • Like Egyptian Hieroglyphics, there are symbols for 1, 10, 100, 1000, and 10,000. But, there is also a symbol for 5. Learn more!

  28. Attic Greek was used until the 4th century BC, when it was replaced by Koine Greek, known as “the Common Dialect”. http://www.thucydides.netfirms.com/thucydides/greece_ancient_sm.gif Next

  29. Special use of 5 ( Γ ) • Another special feature is that when there are more than 5 of the same symbol, Greeks used Γ with the symbol and wrote the remaining symbols. • Examples • 8 is written as • 700 is written as Continue

  30. Attic Green Symbols and Examples • 34 ΔΔΔ|||| • 617 HΔΓ|| • 2341 XXHHHΔ Δ Δ Δ| • 10,135 MHΔΔΔΓ http://www.jesus8880.com/chapters/gematria/images/Attic-Numerals.gif You Try!

  31. What is HHHΔΔΓ|| in our modern numeral system? 50,327 53,257 503, 212 5, 327

  32. Incorrect Answer! Here’s a hint! Remember that when there are more than 5 of a symbol, the Γ is used to hang one, and then the rest of the symbols are written. Don’t forget that a Γis also used for the number 5. Go Back and Review Try Again!

  33. Correct Answer! • Fantastic work! You’re doing great! Move on to Roman Numerals

  34. Roman Numerals • The last type of simple grouping system is Roman Numerals. • Roman Numerals were the standard numbering system in Ancient Rome and Europe until around 900 AD, when the Hindu-Arabic system emerged. • There is no symbol for 0 in Roman Numerals. Learn more!

  35. Although the Roman numerals are now written with letters of the Roman alphabet, they were originally independent symbols. Next

  36. Roman Numerals Learn the subtraction rule!

  37. Subtraction Rule • In modern times, the subtractive principle has become very common when writing Roman Numerals. • I can precede only V or X • Examples • 4 is written as IV • 9 is written as IX • X can precede only L or C • Examples • 40 is written as XL • 90 is written as XC • C can precede only D or M • Examples • 400 is written as CD • 900 is written as CM Examples!

  38. Examples of Roman Numerals • 33 • XXXIII • 54 • LIV • 147 • CXLVII • 999 • CMXCIX More

  39. More Examples • 1042 • MXLII • 2741 • MMDCCXLI • 3001 • MMMI • 5618 • MMMMMDCXVIII Let’s see what you know!

  40. You Try! What is 798 in Roman Numerals? DCCXCVIII CCCCCCCXCVIII DCCHCIIIIIIII CCMXCVIII

  41. Incorrect Answer! • Don’t give up! Try again! Here’s a hint! Don’t forget to use the subtraction rules! Go back and Review Try Again

  42. Correct Answer! • You are ready to move on to Multiplicative Number Systems! Keep going!

  43. Multiplicative Number Systems • In a multiplicative system, there are only symbols for 1-9, 10, 102, 103, etc. • We need to first write the number in expanded form. • Examples • 54= 5 x 10 + 4 • 613= 6 x 102 + 1 x 10 + 3 • So, we don’t need to have a number for 40. Instead, we can write it as 4 x 10. • One example of a multiplicative system is that of the Ancient Chinese-Japanese. Start http://bbs.chinadaily.com.cn/attachments/month_0901/chinese-paper-cutting-40120141324284_qOeMAVE7WlRD.jpg

  44. Ancient Chinese-Japanese Number System • The traditional Chinese-Japanese number system has characters for the numerals 0 through 9, 10, 100, and 1000. • Numbers are written in expanded form from top to bottom instead of left to right. • Since this system has a symbol for 0, it is used as a place holder. Learn more!

  45. In 1899 a major discovery was made at the archaeological site at the village of Xiao dun .Thousands of bones and tortoise shells were discovered there which had been inscribed with ancient Chinese numerals. Archaeologists think that they date back to the Late Shang dynasty from the 14th century BC. Continue http://www.earthquest.co.uk/china/chinamap.jp http://blog.asiahotels.com/wp-content/uploads/2008/05/japan_map.jpgg

  46. Ancient Chinese-Japanese Numerals Next

  47. 6 1000 8 100 10 3 Writing Numbers in Ancient Chinese-Japanese • First, write the number in expanded form. • Then, fill in the symbols and remember to write the number vertically. 6813 = 6 x 1000 + 8 x 100 + 1 x 10 + 3 Examples

  48. Ancient Chinese-Japanese Examples • 8,612 =8 x 1000 + 6 x 100 + 1 x 10 + 2 • 354 =3 x 100 + 5 x 10 + 4 You Try!

  49. You try! What is in our modern number system? 912 9, 120 9, 121 9.12

  50. Incorrect Answer! Here’s a hint! In the number 405, the tenths digit is a zero, instead of writing the symbol for zero, it is omitted! Go back and review Try Again

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