Numeration Systems A number is a concept or

Numeration Systems A number is a concept, or an idea, used to represent some quantity. A numeral, on the other hand, is a symbol used to represent a number. A numeration system consists of a set of symbols (numerals) to represent numbers, and a set of rules for combining those symbols.

Tally System A tally system is the simplest kind of numeration system, and almost certainly the oldest. In a tally system there is only one symbol needed and a number is represented by repeating that symbol. Most often, they are used to keep track of the number of occurrences of some event. The most common symbol used in tally systems is |, which we call a stroke. Tallies are usually grouped by fives, with the fifth stroke crossing the first four, as in.

EXAMPLE 1 Using a Tally System An emergency room nurse is interested in keeping track of the number of patients she treats during her shift. There were six patients before her first break, eight more between break and lunch, and only four during a slow afternoon. If she used a tally system to keep track of the patients, write out what that may have looked like, and use it to total the number of patients. SOLUTION The nice thing about the groups of five is that it makes it easy to find the total: 3 fives with 3 ones, which is 18 patients.

Simple Grouping Systems In an additive grouping system there are symbols that represent select numbers. Often, these numbers are powers of 10. To write a number in a simple grouping system, repeat the symbol representing the appropriate value(s) until the desired quantity is reached. One such system was used in ancient Egypt.

Heiroglyphics

The Egyptian Numeration System One of the earliest formal numeration systems was developed by the Egyptians over 5000 years ago. It used a system of pictures (known as hieroglyphics) to represent certain numbers. (Please excuse the crude drawings!) Symbol Number Description 1 Symbol Number Description Vertical staff 10, 000 Pointing finger 10 Heel bone 100, 000 Burbot fish (or tadpole) 100 Scroll 1, 000 Person 1, 000 Lotus Flower

Egyptian numeration system Insert Photo Credit Here

Egyptian numeration system =1, 333, 330 Insert Photo Credit Here

EXAMPLE 2 Using the Egyptian Numeration System Find the numerical value of each Egyptian numeral. (a) (b)

EXAMPLE 2 Using the Egyptian Numeration System SOLUTION (a) This number is made up of 3 fish (which means 3 hundred thousands), 3 pointing fingers (3 ten thousands), 2 scrolls (2 hundreds), 3 heel bones (3 tens), and 6 vertical staffs (6 ones), which makes the number 3 × 100, 000 + 3 × 10, 000 + 2 × 100 + 3 × 10 + 6 × 1 = 300, 000 + 30, 000 + 200 + 30 + 6 = 330, 236. (b) The number consists of 1 million, 2 thousands, 1 hundred, 1 ten, and 1 one; the number is 1, 000 + 2, 000 + 10 + 1 = 1, 002, 111.

EXAMPLE 3 Writing Numbers in Egyptian Notation Write each number as an Egyptian numeral. (a) 237 (b) 3, 202, 419

EXAMPLE 3 Writing Numbers in Egyptian Notation SOLUTION (a) We can write 237 as 2 × 100 + 3 × 10 + 7 × 1. So we need 2 of the hundreds symbol (scroll), 3 of the tens symbol (heel bone), and 7 of the ones symbol (vertical staff). Since 137 consists of 1 hundred, 3 tens, and 7 ones, it is written:

EXAMPLE 3 Writing Numbers in Egyptian Notation SOLUTION continued (b) Since 3, 202, 419 consists of 3 millions, 2 hundred thousands, 2 thousands, 4 one hundreds, 1 ten, and 9 ones, it is written as:

Roman Numerals…. Still in Use

Roman Numerals Symbol Number I 1 V 5 X 10 L 50 C 100 D 500 M 1, 000

The Roman Numeration System The Roman numeration system is a variation on additive grouping systems. In order to keep from having to repeat any symbol more than three times, subtraction is used. For example, 8 is written as VIII (one five plus three ones) 9 is not written as VIIII; instead, we write IX. The fact that the ones symbol is written before the tens symbol tells us to subtract 1 from 10, leaving 9.

The Roman Numeration System When a letter is repeated in sequence, its numerical value is added. For example, XXX represents 10 + 10, or 30. When smaller-value letters follow larger-value letters, the numerical values of each are added. For example, LXVI represents 50 + 10 + 5 + 1, or 66.

The Roman Numeration System When a smaller-value letter precedes a larger-value letter, the smaller value is subtracted from the larger value. For example, IV represents 5 − 1, or 4, and XC represents 100 − 10, or 90. You might find it easier to think of this as “one taken away from five” and “ten taken away from 100. ”

Roman Numerals I can only precede V or X X can only precede L or C C can only precede D or M Symbol Number IV 4 IX 9 XL 40 XC 90 CD 400 CM 900 • All numbers using 4’s or 9’s will use the subtraction principle. • Never repeat any character more that 3 times straight.

EXAMPLE 5 Finding the Value of Roman Numerals Find the value of each Roman numeral. (a) LXVIII (b) XCIV (c) MCML (d) CCCXLVI (e) DCCCLV

EXAMPLE 5 Finding the Value of Roman Numerals SOLUTION (a)LXVII: L = 50, X = 10, V = 5, and III = 3; so LXVIII = 50 + 10 + 5 + 3 = 68. (b) XCIV: XC = 90 and IV = 4; so XCIV = 94. (c) MCML: M = 1, 000, CM = 900, L = 50; so MCML = 1, 950. (d) CCCXLVI: CCC = 300, XL = 40, V = 5, and I = 1; so CCCXLVI = 346. (e) DCCCLV: D = 500, CCC = 300, L = 50, V = 5; so DCCCLV = 855.

EXAMPLE 6 Writing Numbers Using Roman Numerals Write each number using Roman numerals. (a) 19 (b) 238 (c) 1, 999 (d) 840

EXAMPLE 6 Writing Numbers Using Roman Numerals (2 of 2) SOLUTION (a) 19 is written as 10 + 9 or XIX. (b) 238 is written as 200 + 30 + 8 or CCXXXVIII. (c) 1, 999 is written as 1, 000 + 90 + 9 or MCMXCIX. (d) 840 is written as 500 + 300 + 40 or DCCCXL.

Positional Systems In a positional system, instead of writing the multiplier with another number to multiply it by (like a power of 10), we just write the multiplier alone; the number to multiply it by is understood from that multiplier’s position. If this sounds a lot like the number system you’re familiar with, it should. The numeration system you grew up with is a positional system that requires 10 symbols—the digits from 0 through 9 —to represent any number, no matter how big or small. A fundamental understanding of how that system is designed requires a clear understanding of exponents, so a quick review seems like a good idea.

Exponential Expressions For any number b and natural number n, we define the exponential expression bn as bn = b ⋅ b · · · b where b appears as a factor n times. The number b is called the base, and n is called the exponent. We also define b 1 = b for any base b, and b 0 = 1 for any nonzero base b.

Hindu-Arabic Numeration System The numeration system we use today is called the Hindu-Arabic system. It is a positional system since the position of each digit indicates a specific value. The place value of each number is given as billion hundred million ten million hundred thousand ten thousand hundred ten one 109 108 107 106 105 104 103 102 101 1 The number 82, 653 means there are 8 ten thousands, 2 thousands, 6 hundreds, 5 tens, and 3 ones. We say that the place value of the 6 in this numeral is hundreds.

Example 7 Expanded Notation To clarify the place values, Hindu-Arabic numbers are sometimes written in expanded notation. An example, using the numeral 32, 569, is shown below. 32, 569 = 30, 000 + 2, 000 + 500 + 60 + 9 = 3 × 10, 000 + 2 × 1, 000 + 5 × 100 + 6 × 10 + 9 × 10 0 = 3 × 104 + 2 × 103 + 5 × 102 + 6 × 101 + 9 × 100

Base 10 System Since all of the place values in the Hindu-Arabic system correspond to powers of 10, the system is known as a base 10 system. We’ll study other base number systems later in this chapter.

EXAMPLE 8 Writing a Base 10 Number in Expanded Form Write 9, 034, 761 in expanded notation. SOLUTION 9, 034, 761 can be written as 9, 000 + 30, 000 + 4, 000 + 700 + 60 + 1 = 9 × 1, 000 + 3 × 10, 000 + 4 × 1, 000 + 7 × 100 + 6 × 10 + 1 = 9 × 106 + 3 × 104 + 4 × 103 + 7 × 102 + 6 × 101 + 1.

Plimpton 322, Babylonia

Babylonian clay tablet YBC 7289 (between 1800 and 1600 BCE) 1. 4142135623730950488. . .

Remnants of Base 60…

Babylonian Numeration System The Babylonians had a numerical system consisting of just two symbols, shown below. (These wedgeshaped symbols are known as “cuneiform. ”) The first symbol represents the number of 10 s, and the second symbol represents the number of 1 s. The ancient Babylonian system is sort of a cross between a multiplier system and a positional system.

EXAMPLE 9 Using the Babylonian Numeration System What number does this sequence of symbols represent? SOLUTION Since there are 3 tens and 6 ones, the number represents 36.

Babylonian Numeration System, continued You might think it would be cumbersome to write large numbers in this system; however, the Babylonian system was also positional in base 60. Numbers from 1 to 59 were written using the two symbols shown in Example 9, but after the number 60, a space was left between the groups of numbers. For example, the number 2, 538 was written as shown below and means that there are 42 sixties and 18 ones. The space separates the 60 s from the ones.

The Babylonian System (Base 60)

Connection: Time = Babylonian System Hours 10 10 1 11 20 Minutes Seconds 21 12 3 20 10 2 21 2 11 11 2 21 30 Total Seconds 602 601 600 Base 10 Value

Connection: Time = Babylonian System Hours 10 10 1 11 20 Minutes Seconds 21 12 3 20 10 2 21 2 11 11 2 21 30 Total Seconds 21 723 37210 36002 4862 40271 72120 1290 602 601 600 Base 10 Value 21 723 37210 36002 4862 40271 72120 1290 1 minute = 60 seconds 1 hour = 60 minutes = 3600 seconds (60· 60) 1 hour, 21 minutes, 2 seconds 1(3600 sec) + 21(60 sec)+ 2 sec (3600+1260+2)sec 4860 sec 1(3600) + 21(60) + 2 3600 + 1260 + 2 4860

EXAMPLE 10 Using the Babylonian Numeration System Write the numbers represented. (a) (b)

EXAMPLE 10 Using the Babylonian Numeration System SOLUTION (a) There are 52 sixties and 34 ones; so the number represents 52 × 60 + 34 × 1 = 3, 120 + 34 = 3, 154 (b) There are 12 3, 600’s (602), 51 60’s, and 23 1’s. 12 × 3, 600 + 51 × 60 + 23 × 1 = 43, 200 + 3, 060 + 23 = 46, 283

EXAMPLE 10 Writing a Number in the Babylonian System Write 5, 217 using the Babylonian numeration system.

EXAMPLE 10 Writing a Number in the Babylonian System SOLUTION Since the number is greater than 3, 600, it must be divided by 3, 600 to see how many 3, 600 s are contained in the number. 5, 217 ÷ 3, 600 = 1 remainder 1, 617 The remainder is then divided by 60 to see how many 60 s are in 1, 617 ÷ 60 = 26 remainder 57 That leaves 57 ones. So, the number consists of 1 × 3, 600 + 26 × 60 + 57 × 1 = 3, 600 + 1, 560 + 57 = 5, 217. It can be written as:

Mayan Numeration System The three symbols in the Mayan system are a dot (representing one), a horizontal line (representing five), and the symbol (representing zero). The numbers are written vertically from bottom to top. 3 is written as 12 is written as (two fives and two ones)

Mayan Number System Place Values (start from the bottom) 18· 20· 20 18· 20 20 1

Mayan Numeration System For numbers greater than 19 the system becomes positional, with the top grouping describing the number of twenties, and the bottom one describing the number of ones. For example, 154 is written as below. 20 s 1 s 154 = 7 × 20 + 14 × 1 = 7 twenties + 14 ones

Mayan Numeration System The next place value after 20 is not 202 as you might expect, but 360 (which is 18 × 20). This was used because the Mayan calendar was made up of 18 months of 20 days each, with 5 “nameless days” to match the 365–day solar year. The next higher place value is 18 × 202, or 7, 200; then 18 × 203 and so on. The symbol for zero is used to indicate that there are no digits in a given place value. For example, the numeral to the right represents 18 × 360 + 0 × 20 + 11 × 1, which is 6, 491.

EXAMPLE 11 Using the Mayan Numeration System (a) The Mayan calendar lists the creation date as 3114 BCE, which was 5, 126 years before I wrote this question. Write that number of years in Mayan numerals. (b) Convert the Mayan numeral below into a Hindu. Arabic numeral.

EXAMPLE 11 Using the Mayan Numeration System SOLUTION (a) We divide 5, 126 by 360 to get 14 with a remainder of 86. So we need 14 three hundred sixties, 4 twenties, and 6 ones. The answer is shown below. (b) With four distinct groupings, the top tells us the number of 7, 200 s; the second, the number of 360 s; the third, the number of 20 s, and the last tells us the number of 1 s. In this case, we have sixteen 7, 200 s, no 360 s, nine 20 s, and no 1 s. This is 16 × 7, 200 + 9 × 20 = 115, 380.

Pa-Kua…Ancient Symbols of the Orient South Korean Flag The Pre-Heaven Eight Trigrams were supposedly developed by the Emperor Fu Hsi around the year of 2900 B. C.

Chinese Rod System ßEven place values: 100 102 104 … ßOdd place values: 101 103 105 …

The Incan Quipu