Chapter 3

Whole Numbers

Section 3.1

Numeration Systems

The number system we use today to represent numbers has resulted from innovations during various times in history to be one of the most concise efficient ways to represent numbers. This section looks at the developments that have taken place in number systems throughout the years.

Tally Systems

The tally system used one character (usually a dot (●) or a stick (|) to stand for each unit represented.

Our Number 1 2 3 4 5 6 7Tally with | | || ||| |||| ||||| ||||| | ||||| ||Tally with ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

The advantage of a tally system is that is easy to understand. Some disadvantages are that it is difficult to write really big numbers (i.e. 6472) and it is hard to distinguish numbers right away:

||||||||||||||||||||||| 23 |||||||||||||||||||||||| 24

Symbol Name Value

| staff 1

heel bone 10

scroll 100

lotus flower 1,000

finger 10,000

fish 100,000

Egyptian Numeration Systems

The early Egyptians solved the problem of how to represent big numbers with a smaller number of symbols. Different symbols were assigned specific values. Writing down the number would mean to adding the values of the symbols together.

The symbols below represent the number 24,356

||||||

What number is represented by the following symbols?

||||

10,634

This advantage of this system is that it did enable people to write large numbers in a short amount of space. The problem is that new symbols were introduced for bigger numbers and numbers like 99,999 used many symbols.

Babylonian Numeration System

The Babylonians were able to make two important advancements in how numbers are expressed.

1. They used only two symbols, one to represent 1 and the other to represent 10. Later they introduced a third symbol that acted like 0.

2. They introduced the concept of place value. This has to do with where a symbol is positioned determines its value. If positioned in one place it would have a different value than in another place.

The system that was used was a base 60 system. The symbol furthest to the right represented ones. The symbols second from the right represented groups of 60. The symbols third from the right represented groups of 3600 (6060). The groups were initially separated by a space later by the symbol for 0.

Symbol Value

1

10

0

The symbols below represent the number 697.

10+1=11We have 11 groups of 60.

1160=660

30+7=37We have 37 ones.

371=37

660+37=697

What do the following represent?

30+5=35 (260) + (20+4)=144

(3060)+(10+3)

1800+13

1813

(23600)+(160)+(30+8)

7200+60+38

7298

How do you write each of the following numbers?

347

34760 = 5 remainder 47

1571

157160 = 26 remainder 11

Roman Numeration System

The Romans devised a system that used an addition/subtraction method for writing numbers. They had only 7 letters that stood for numbers given in the table below. To limit the number of symbols the Romans said that a symbol could not be used more than 3 times.

Roman Numeral I V X L C D MBase-ten Value 1 5 10 50 100 500 1000To find the value of a Roman numeral start at the left adding the numerals that are of equal or lesser value as you move to the right. If you find a numeral of smaller value than the numeral to its right subtract it from the one to the right.

Example:

MMDCCCLXVII 1000+1000+500+100+100+100+50+10+5+1+1=2867M M D C C C L X V I I

MCDXCIV 1000+(500-100)+(100-10)+(5-1)=1000+400+90+4=1494

M CD XC IV

Base-Ten Place-Value System

The sleek efficient number system we know today is called the base-ten number system or Hindu-Arabic system. It was first developed by the Hindus and Arabs. This used the best features from several of the systems we mentioned before.

1. A limited set of symbols (digits). This system uses only the 10 symbols:0,1,2,3,4,5,6,7,8,9.

2. Place Value. This system uses the meaning of the place values to be powers of 10.

For example the number 6374 can be broken down (decomposed) as follows:

6 thousands 3 hundreds 7 tens 4 ones

6000 + 300 + 70 + 4

61000 + 3100 + 710 + 4

6103 + 3102 + 7101 + 4

The last row would be called the expanded notation of the number 6374.

Visual Representations of Base-Ten Numbers

Base-Ten numbers have many different visual representations. A very common one is to use a set of Dienes blocks.

Block Name Size Value

Unit 1 by 1 1

Long 1 by 10 10

flat 10 by 10 100

Block Name Size Value

block

or

cube

10 by 10 by 10 1000

Counts vs Measures

The are two distinct ways that a number can be used. One of the is called a count the other is called a measure.

The count or cardinal number represents the number of things in a set. These are always whole numbers that can not be broken down any further. The set of whole numbers is denoted with a W.

W = {0, 1, 2, 3, 4, 5, 6, …}

Examples of counts:

The number of students in this class.

The number of nickels you have in your pocket.

The number of cars you own.

A measure or unit of measure such as pounds, inches, seconds are numbers that can be split into smaller parts.

Example of a measure:

The number of inches you are tall.