Section 4.1Early and Modern

Numeration Systems

Math in Our World

Learning Objectives Define a numeration system. Work with numbers in the Egyptian system. Work with numbers in the Chinese system. Identify place values in the Hindu-Arabic

system. Write Hindu-Arabic numbers in expanded

notation. Work with numbers in the Babylonian system. Work with Roman numerals.

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

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 ||||.

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.

EXAMPLE 1 Using a Tally System

An amateur golfer gets the opportunity to play with Tiger Woods, and, star struck, his game completely falls apart. On the very first hole, it takes him six shots to reach the green, then three more to hole out. Use a tally system to represent his total number of shots on that hole.

SOLUTIONThe total number of shots is nine, which we tally up as

Simple Grouping Systems

In a simple 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.

The Egyptian Numeration SystemOne of the earliest formal numeration systems was developed by the Egyptians sometime prior to 3000 BCE. It used a system of hieroglyphics using pictures to represent numbers.

EXAMPLE 2 Using the Egyptian Numeration System

Find the numerical value of each Egyptian numeral.

(a)(b)(c)

EXAMPLE 2 Using the Egyptian Numeration System

SOLUTION The value of any numeral is determined by counting up the

number of each symbol and multiplying the number of occurrences by the corresponding value. Then the amounts for each symbol are added.

(a)

There are 4 heel bones and 3 staffs, so to find the value…(4 x 10) + (3 x 1) = 40 + 3 = 43.

= 10 = 1

EXAMPLE 2 Using the Egyptian Numeration System

SOLUTION (b)(3 x 100,000) + (3 x 10,000) + (2 x 100) + (3 x 10) + (6 x 1)

300,000 + 30,000 + 200 + 30 + 6 = 330,236

(c)(1 x 1,000,000) + (2 x 10,000) + (2 x 1000) + (2 x 100) + (1 x 10) + (3 x 1) = 1,000,000 + 20,000 + 2000 + 200 + 10 + 3=1,022,213

EXAMPLE 3 Writing Numbers in Egyptian Notation

Write each number as an Egyptian numeral.(a) 42 (b) 3,200,419

EXAMPLE 3 Writing Numbers in Egyptian Notation

SOLUTION (a)Forty-two can be written as 4 x 10 + 2 x 1, so it consists of four tens and two ones. We would write it using four of the tens symbol (the heel bone) and two of the ones symbol (the vertical staff).

(b)Since 3,200,419 consists of 3 millions, 2 hundred thousands, 4 one hundreds, 1 ten, and 9 ones, it is written as

Positional Systems

In a positional system no multiplier is needed. The value of the symbol is understood by its position in the number. To represent a number in a positional system you simply put the numeral in an appropriate place in the number, and its value is determined by its location.

Hindu-Arabic Numeration SystemThe numeration system we use today is called the Hindu-Arabic system. It uses 10 symbols called digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.This is a positional system since the position of each digit indicates a specific value. The place value of each number is given as

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 9 Finding Place Values

In the number 153,946, what is the place value of each digit?

a) 9 b) 3 c) 5 d) 1 e) 6

SOLUTION(a) hundreds(b) thousands(c) ten thousands(d) hundred thousands(e) ones

Hindu-Arabic Numeration SystemTo 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 x 10,000 + 2 x 1,000 + 5 x 100 + 6 x 10 + 9 = 3 x 104 + 2 x 103 + 5 x 102 + 6 x 101 + 9

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.

EXAMPLE 10 Writing a base 10 Number in Expanded Form

Write 9,034,761 in expanded notation.

SOLUTION9,034,761 can be written as

9,000,000 + 30,000 + 4,000 + 700 + 60 + 1

= 9 x 1,000,000 + 3 x 10,000 + 4 x 1,000 + 7 x 100 + 6 x 10 + 1

= 9 x 106 + 3 x 104 + 4 x 103 + 7 x 102 + 6 x 101 + 1.

Babylonian Numeration SystemThe Babylonians had a numerical system consisting of two symbols. They are and . (These wedge-shaped symbols are known as “cuneiform.”) The represents the number of 10s, and represents the number of 1s.

The ancient Babylonian system is sort of a cross between a multiplier system and a positional system.

EXAMPLE 11 Using the Babylonian Numeration System

SOLUTIONSince there are 3 tens and 6 ones, the number represents 36.

Babylonian Numeration System

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 11, but after the number 60, a space was left between the groups of numbers.For example, the number 2,538 was written as

and means that there are 42 sixties and 18 ones. The space separates the 60s from the ones.

EXAMPLE 12 Using the Babylonian Numeration System

Write the numbers represented.

EXAMPLE 12 Using the Babylonian Numeration System

SOLUTION

There are 52 sixties and 34 ones; so the number represents 52 x 60 = 3,120+ 34 x 1 = 34 3,154

There are twelve 3,600s (602), fifty-one 60s and twenty-three 1s. 12 x 3,600 = 43,200 51 x 60 = 3,060+ 23 x 1 = 23 46,283

Roman Numeration SystemThe Romans used letters to represent their numbers.The Roman system is similar to a simple grouping system, but to save space, the Romans also used the concept of subtraction. For example, 8 is written as VIII, but 9 is written as IX, meaning that 1 is subtracted from 10 to get 9.

Roman Numeration SystemThere are three rules for writing numbers in Roman

numerals:1. When a letter is repeated in sequence, its numerical value is

added. For example, XXX represents 10 + 10 + 10, or 30.2. 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.

3. 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.In addition, I can only precede V or X, X can only precede L or C, and C can only precede D or M. Then 4 is written as IV, 9 is written as IX, 40 is written as XL, 90 is written XC, 400 is written as CD, and 900 is written as CM.

EXAMPLE 14 Using Roman Numerals

SOLUTION(a) L = 50, X = 10, V = 5, and III = 3; so LXVIII = 68.(b) XC = 90 and IV = 4; so XCIV = 94.(c) M = 1,000, CM = 900, L = 50; so MCML = 1,950.(d) CCC = 300, XL = 40, V = 5, and I = 1;

so CCCXLVI = 346.(e) D = 500, CCC = 300, L = 50, V = 5; so DCCCLV = 855.

Find the value of each Roman Numeral.

(a) LXVIII (b) XCIV (c) MCML

(d) CCCXLVI (e) DCCCLV

EXAMPLE 15 Writing Numbers Using Roman Numerals

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 + 900 + 90 + 9 or MCMXCIX.(d) 840 is written as 500 + 300 + 40 or DCCCXL.(e) 72 is written as 50 + 20 + 2 or LXXII.

Write each number using Roman Numerals.

(a) 19 (b) 238 (c) 1,999 (d) 840 (e) 72