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Section 4.1 Additive, Multiplicative, and Ciphered Systems of Numeration

Section 4.1 Additive, Multiplicative, and Ciphered Systems of Numeration. What You Will Learn. Additive, multiplicative, and ciphered systems of numeration. Systems of Numeration. A number is a quantity. It answers the question “How many?”

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Section 4.1 Additive, Multiplicative, and Ciphered Systems of Numeration

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  1. Section 4.1Additive, Multiplicative, and Ciphered Systems of Numeration

  2. What You Will Learn • Additive, multiplicative, and ciphered systems of numeration

  3. Systems of Numeration • A numberis a quantity. It answers the question “How many?” • A numeral is a symbol such as , 10 or used to represent the number (amount).

  4. Systems of Numeration • A system of numeration consists of a set of numerals and a scheme or rule for combining the numerals to represent numbers.

  5. Types Of Numeration Systems • Four types of systems used by different cultures will be discussed. They are: • Additive (or repetitive) • Multiplicative • Ciphered • Place-value

  6. Additive Systems • An additive system is one in which the number represented by a set of numerals is simply the sum of the values of the numerals. • It is one of the oldest and most primitive types of systems. • Examples: Egyptian hieroglyphics and Roman numerals.

  7. Egyptian Hieroglyphics

  8. Example 1: From Egyptian to Hindu-Arabic Numerals • Write the following numeral as a Hindu-Arabic numeral. Solution 10,000 + 10,000 + 10,000 + 100 + 100 + 100 + 10 + 1 + 1 + 1+1 = 30,314

  9. Try This P. 176 # 11

  10. Example 2: From Hindu-Arabic to Egyptian Numerals • Write 43,628 as an Egyptian numeral. Solution 43,628 = 40,000 + 3000 + 600 + 20 + 8

  11. Try This Write 1,234,527 as an Egyptian numeral

  12. Roman Numerals

  13. Roman Numerals • Two advantages over Egyptian system: • Uses the subtraction principle as well as addition principle • DC = 500 + 100 = 600 • CD = 500 – 100 = 400 • Uses the multiplication principle for numerals greater than 1000

  14. Example 4: From Roman to Hindu-Arabic Numerals • Write CMLXIV as a Hindu-Arabic numeral. • Solution • It’s an additive system so, = CM + L + X + IV = (1000 – 100) + 50 + 10 + (5 – 1) = 900 + 50 + 10 + 4 = 964

  15. Try THis Write MCMLXXI as a Hindu-Arabic numeral

  16. Example 5: Writing a Roman Numeral • Write 439 as a Roman numeral. • Solution • 439 = 400 + 30 + 9 • = (500 – 100) + 10 + 10 + 10 + (10 – 1) • = CDXXXIX

  17. Try This Write 3794 as a Roman Numeral

  18. Multiplicative Systems • Multiplicative systems are more similar to the Hindu-Arabic system which we use today. • Chinese numerals

  19. Chinese Numerals • Written vertically • Top numeral from 1 - 9 inclusive • Multiply it by the power of 10 below it. 20 = 400 =

  20. Example 7: A Traditional Chinese Numeral • Write 538 as a Chinese numeral. Solution:

  21. Try This Write 3,271 as a Chinese Numeral

  22. Ciphered Systems • In this system, there are numerals for numbers up to and including the base and for multiples of the base. • The number (amount) represented by a specific set of numerals is the sum of the values of the numerals.

  23. Ciphered Systems • Ciphered numeration systems require the memorization of many different symbols but have the advantage that numbers can be written in a compact form.

  24. Examples of Ciphered Systems • We discuss in detail the Ionic Greek system • developed about 3000 B.C. • used letters of Greek alphabet as numerals • Base is 10 • An iota, ι, placed to the left and above a numeral represents the numeral multiplied by 1000

  25. Examples of Ciphered Systems • Hebrew • Coptic • Hindu • Brahmin • Syrian • Egyptian Hieratic • early Arabic

  26. Ionic Greek System * Ancient Greek letters not used in the classic or modern Greek language.

  27. Ionic Greek System * Ancient Greek letters not used in the classic or modern Greek language.

  28. Example 9 Write as a Hindu-Arabic numeral. Solution The sum is 839.

  29. Example 10 Write 1654 as an Ionic Greek numeral. Solution 1654 = 1000 + 600 + 50 + 4 = (1 × 1000) + 600 + 50 + 4

  30. Homework P. 176 # 9 – 75 (x 3)

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