Ch2 Notes (F09)

Ch2 Notes (F09) - Section 2-1 Numeration Systems Imagine...

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1 Section 2-1 – Numeration Systems Imagine that we are in the past—before we had counting systems. How would you keep track of the number of chickens on your farm?? Having a counting system requires that we recognize a one-to-one correspondence between two equivalent sets: the set of objects we are counting and the set of numbers we are using to count them. Numbers, Numerals, and Numeration Systems: Number – The amount being counted or measured Numeral – The symbol used to designate the amount (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) Numeration System – A collection of properties and symbols agreed upon to represent numbers systematically. The table on pg. 62 has the symbols used for different numeration systems (Babylonian, Egyptian, Mayan, Greek, Roman, Hindu, Arabic, Hindu-Arabic) Hindu-Arabic System – Base 10 System: •No tally marks •All numerals are constructed from the 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 •Place value is based on powers of 10 (the base of the system) •Concept of Zero •Expanded form For example, the number 546 = 500 + 40 + 6 = (5 * 100) + (4 * 10) + (6 * 1) = (5*10 2 ) + (4 * 10 1 ) + (6 * 10 0 ) (expanded form) We can use Base-10 Blocks to represent numbers in base-10.
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2 Examples : Draw the following numbers using the base-10 blocks and then write the number in expanded notation: a) 312 b) 1,423 Example : Rewrite the following as a base-10 numeral: (7 × 10 3 ) + (4 × 10 2 ) + (0 × 10 1 ) + (6 × 10 0 ) OTHER NUMERATION SYSTEMS: Egyptian Numeration System (pg. 65-66) : ( LEARN ON YOUR OWN ANSWERS BELOW ) The Egyptian system is an example of an additive system (tallies and pictures) where the value of the number is the sum of the numerals. It dates back to 3400 BCE. They used powers of 10: Symbol Name Astonished Person Polliwog Pointing Finger Lotus Flower Scroll Heelbone Staff Value 1,000,000 100,000 10,000 1000 100 10 1 Example : Find the value for each of the following Egyptian numerals: a) b) c) Example : Write 2,341,602 using the Egyptian system
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3 Roman Numeration System (pg. 69) : ( LEARN ON YOUR OWN ANSWERS BELOW ) Symbol M D C L X V I Value 1000 500 100 50 10 5 1 Examples : What is the value of the following numbers? a) MDLVII b) DXI Examples : Express the following in Roman numerals: c) 1,109 d) 418 e) 1,109 f) 418 Answers (Egyptian) Answers (Roman) a) 526 a) 1,557 b) 200,202 b) 511 c) 1,304,010 c) MCIX d) CDXVIII e) MCIX f) CDXVIII
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4 Babylonian Numeration System : The Babylonians used powers of 60 for their place values: …… 60 3 60 2 60 1 60 0 Symbol Value 1 10 This is an example of a positional system because the value of a numeral depends on its position in the number. (Earliest occurrence of the concept of place values in recorded history) Examples : What is the value of the following numbers? a) b) c) Examples : How would the Babylonians represent the following amounts? a) 310 b) 4,281
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5 Example: CARAMEL FACTORY Suppose you own a caramel factory, but are having a lot of trouble finding good help. In fact, the people you hired are only able to count up to four. So when your caramel machine spews out the caramels, your employees are unable to count how many there are. After some deliberation, you decide that the thing to do is to package your product by fours. Any time an employee sees four caramels, he is to put them in a pack.
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Ch2 Notes (F09) - Section 2-1 Numeration Systems Imagine...

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