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Section 2.4

Section 2.4. Numeration. Mathematics for Elementary School Teachers - 4th Edition. O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK. A symbol is different from what it represents. The word symbol for cat is different than the actual cat. Numeration Systems.

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Section 2.4

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  1. Section 2.4 Numeration Mathematics for Elementary School Teachers - 4th Edition O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

  2. A symbol is different from what it represents The word symbol for cat is different than the actual cat

  3. Numeration Systems Just as the written symbol 2 is not itself a number. The written symbol, 2, that represents a number is called a numeral. Here is another familiar numeral (or name) for the number two

  4. Babylonian Numeration System Egyptian Numeration System Mayan Numeration System Roman Numeration System Hindu-Arabic Numeration System Definition of Numeration System An accepted collection of properties and symbols that enables people to systematically write numerals to represent numbers. (p. 106, text)

  5. Hindu-Arabic Numeration System • Developed by Indian and Arabic cultures • It is our most familiar example of a numeration system • Group by tens: base ten system • 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 • Place value - Yes! The value of the digit is • determined by its position in a numeral • Uses a zero in its numeration system

  6. Definition of Place Value In a numeration system with place value, the position of a symbol in a numeral determines that symbol’s value in that particular numeral. For example, in the Hindu-Arabic numeral 220, the first 2 represents two hundred and the second 2 represents twenty.

  7. Models of Base-Ten Place Value Base-Ten Blocks - proportional model for place value Thousands cube, Hundreds square, Tens stick, Ones cube or block, flat, long, unit text, p. 110 2,345

  8. Models of Base-Ten Place Value Colored-chip model: nonproportional model for place value One Hundred Ten One One Thousand 3,462 chips from text, p. 110

  9. Expressing Numerals with Different Bases: Show why the quantity of tiles shown can be expressed as (a) 27 in base ten and (b)102 in base five, written 102five we can group these tiles into two groups of ten with 7 tiles left over (a) form groups of 10 27 (b) form groups of 5 102five we can group these tiles into groups of 5 and have enough of these groups of 5 to make one larger group of 5 fives, with 2 tiles left over. No group of 5 is left over, so we need to use a 0 in that position in the numeral: 102five

  10. 1324five = (1×53) + (3×52) + (2×51) + (4×50) = 1(125) + 3(25) + 2(5) + 4(1) = 125 + 75 + 10 + 4 = 214ten Expressing Numerals with Different Bases: Find the base-ten representation for 1324five Find the base-ten representation for 344six Find the base-ten representation for 110011two

  11. 60 = 1 256 - 216 61 = 6 40 62 = 36 -36 63 = 216 4 64 = 1296 Expressing Numerals with Different Bases: Find the representation of the number 256 in base six 1(63) + 1(62) + 0(61) + 4(60) 1(216) + 1(36) + 0(6) + 4(1) = 1104six

  12. 50 = 1 30 - 25 51 = 5 5 52 = 25 - 5 53 = 125 0 Expressing Numerals with Different Bases: Change 42seven to base five First change to base 10 42seven = 4(71) + 2(70) = 30ten Then change to base five 30ten = 1(52) + 1(51) + 0(50) = 110five

  13. Expanded Notation: This is a way of writing numbers to show place value, by multiplying each digit in the numeral by its matching place value. Example (using base 10): 1324 = (1×103) + (3×102) + (2×101) + (4×100) or 1324 = (1×1000) + (3×100) + (2×10) + (4×1)

  14. reed One heel bone Ten coiled rope One Hundred lotus flower One Thousand bent finger Ten Thousand burbot fish One Hundred Thousand kneeling figure or astonished man One Million Egyptian Numeration System Developed: 3400 B.C.E Group by tens New symbols would be needed as system grows No place value No use of zero

  15. one ten Babylonian Numeration System Developed between 3000 and 2000 B.C.E There are two symbols in the Babylonian Numeration System Base 60 Zero came later Place value Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the Babylonian system: 42(601) + 34(600) = 2520 + 34 = 2,554

  16. Ⅽ (one) ⅼ Ⅿ Ⅹ Ⅽ ⅼ Ⅹ (five) Ⅴ Ⅹ (ten) Ⅼ (fifty) Ⅽ (one hundred) Ⅾ (five hundred) Ⅿ (one thousand) Roman Numeration System Developed between 500 B.C.E and 100 C.E. • Group partially by fives • Would need to add new symbols • Position indicates when to add or subtract • No use of zero Write the Hindu-Arabic numerals for the numbers represented by the Roman Numerals: 900 + 90 + 9 = 999

  17. Symbols = 0 = 1 = 5 Mayan Numeration System Developed between 300 C.E and 900 C.E • Base - mostly by 20 • Number of symbols: 3 • Place value - vertical • Use of Zero Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the Mayan system: 8(20 ×18) = 2880 6(201) = 120 2880 + 120 + 0 = 3000 0(200) = 0

  18. Summary of Numeration System Characteristics

  19. The End Section 2.4 Linda Roper

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