MA 118 – Section 3.1: Numeration Systems Reminder : If a and m are whole numbers then exponential expression amor a to the mth power is m = ⋅⋅⋅⋅⋅ defined by a aaa... aa . m times Number a is the base ; m is the exponent or power . Example : 53 = Definition: A number is an abstract idea that indicates “how many”. A numeral is a symbol used to represent a number. A System of Numeration consists of a set of basic numerals and rules for combining them to represent numbers.
1
Section 3.1: Numeration Systems
The timeline indicates recorded writing about 10,000 years ago between 4,000–3,000 B.C. The Ishango Bone provides the first evidence of human counting, dated 20,000 years ago.
I. Tally Numeration System Example: Write the first 12 counting numbers with tally marks. In a tally system, there is a one-to-one correspondence between marks and items being counted. What are the advantages and drawbacks of a Tally Numeration System?
2
Section 3.1: Numeration Systems (continued) II. Egyptian Numeration System Staff Yoke Scroll Lotus Pointing Fish Amazed (Heel Blossom Finger (Polywog) (Astonished) Bone) Person
Example : Page 159 1 b, 2b Example : Write 3248 as an Egyptian numeral.
What are the advantages and drawbacks of the Egyptian System?
3
Section 3.1: Numeration Systems (continued) III. Roman Numeral System
Roman Numerals I V X L C D M Indo-Arabic 1 5 10 50 100 500 1000 If a symbol for a lesser number is written to the left of a symbol for a greater number, then the lesser number is subtracted. Only numerals I, X, C, M can be subtracted, according to this table: IV IX XL XC CD CM 4 9 40 90 400 900 This procedure makes unnecessary the repetition of any symbol more than three times. For example, 1959 was originally written ______later it was written ______. Examples: Page 159 1 f, 3 c Example: Write 49 in Roman numerals. What are drawbacks of the Roman Numeral System? 4
Section 3.1: Numeration Systems (continued) IV. Babylonian System The Babylonians (in Mesopotamia) used cuneiform, writing in clay tablets with a wedge-shaped stylus. The base of the Babylonian System is 60. Using powers of sixty, 604321 ,60 ,60 ,60 , and 60 0 , Babylonian numerals can be written in expanded form. A space was left to indicate place value. The later Babylonian system used a place holder for zero. Symbols are combined additively. Babylonian Numerals ∨ < Our Numerals 1 10 Example: Page 159 1 i, 4 c
The major disadvantage of the Babylonian system was no zero for a place holder.
5
Section 3.1: Numeration Systems (continued) II. Hindu (Indo) – Arabic System (Ours!)
The base of the Hindu – Arabic system is 10. In the Hindu – Arabic system, we write the numeral for any number, large or small using only ten symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits (Latin for fingers). Using powers of ten, Indo-Arabic numerals can be written in expanded form.
Base 10 Blocks
Example: Write 2048 in expanded notation.
Words (know to trillions): nonillion, octillion, septillion, sextillion, quintillion, quadrillion, trillion, billion, million, thousand, hundred 6
Section 3.1: Numeration Systems (continued) Mathematical Characteristics of Number Systems • A numeration system has a base reflected by grouping by a number. All numbers are written in powers of the base. • A numeration system is a place value system if the value of the digit is determined by its position of the numeral. • A numeration system is multiplicative if each symbol in a numeral represents a different multiple of the face value of that symbol. • A numeration system is additive if the value of the set of symbols representing a number is the sum of the values of the individual symbols. • A numeration system has a zero if there is a symbol to represent the number of elements in the empty set. Summary of Numeration Systems Worksheet
7
Section 3.2 Non-Decimal (10) Number Base Systems Activity: Numbers from Rectangles Activity: Base 5 Group Activity Activity: Number Base Equivalents
8