Number Systems

Tally, Babylonian, RomanAnd

Hindu-Arabic

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=1494M 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 base-ten expanded notation of the number 6374.

Write each of the numbers below in expanded notation.

a) 82,305

= 810,000 + 21,000 + 3100 + 010 + 51

= 8104 + 2103 + 3102 + 5100

b) 37.924

= 310 + 71 + 9(1/10) + 2(1/100) + 4(1/1000)

= 3101 + 7100 + 910-1 + 210-2 + 410-3

Write each of the numbers below in standard notation.

a) 6105 + 1102 + 4101 + 5100

= 600,000 + 100 + 40 + 5

= 600,145

b) 7103 + 3100 + 210-2 + 810-3

= 7000 + 3 + .02 + .008

= 7003.028

Base Symbols Place Values as Numbers Place Values as Powers

2 0,1 … , 16, 8, 4, 2, 1 … , 24, 23, 22, 21, 1

3 0,1,2 … , 81, 27, 9, 3, 1 … , 34, 33, 32, 31, 1

4 0,1,2,3 … , 256, 64, 16, 4, 1 … , 44, 43, 42, 41, 1

5 0,1,2,3,4 … , 125, 25, 5, 1 … , 53, 52, 51, 1

6 0,1,2,3,4,5 … , 216, 36, 6, 1 … , 63, 62, 61, 1

7 0,1,2,3,4,5,6 … , 343, 49, 7, 1 … , 73, 72, 71, 1

8 0,1,2,3,4,5,6,7 … , 512, 64, 8, 1 … , 83, 82, 81, 1

9 0,1,2,3,4,5,6,7,8 … , 729, 81, 9, 1 … , 93, 92, 91, 1

10 0,1,2,3,4,5,6,7,8,9 … , 1000, 100, 10, 1 … , 103, 102, 101, 1

Writing Numbers in Other Bases

A number in another base is written using only the digits for that base. The base is written as a subscripted word after it (except base 10).

For Example: 10324 is a legitimate base four number “Read 1-0-3-2 base four”

15424 is not a legitimate base four number not allowed 4 or 5

BaseFour

BaseTen

DienesBlocks

14 1 1 unit

24 2 2 units

34 3 3 units

104 41 long

114 5 1 unit1 long

124 6 2 units1 long

134 7 3 units1 long

204 82 longs

BaseFour

BaseTen

DienesBlocks

214 9 1 unit 2 longs

224 10 2 units2 longs

234 11 3 units2 longs

304 123 longs

314 13 1 unit3 longs

324 14 2 units3 longs

334 15 3 units3 longs

1004 16

1 flat

Notice that the numbers in go in order just like in base 10 but only using the symbols 0, 1, 2, 3. In base 4 numbers are grouped in blocks 1, 4, 16, ….

We can use this different number system to illustrate what it is like to try to learn to count. Give the three numbers that come before and the three numbers that come after each of the numbers below.

2367823677

23676

23675

23679

23680

23681

2135

2125

2115

2105

2145

2205

2215

Converting a number to base 10

This process is a combination of multiplication and addition. You multiply each digit by its place value and add up the results. Convert 13024 to base 10.

In expanded form this number is given by:

13024 = 1×43 + 3×42 + 0×41 + 2×40

13024

2 1 = 2

0 4 = 0

3 16 = 48

1 64 = + 64

114

13024

13014

13004

12334

13034

13104

13114

114113

112

111

115

116

117

Notice that when the numbers convert they stay in the same order.

593 = 19 r 2

193 = 6 r 1

63 = 2 r 0

23 = 0 r 2

246710 = 246 r 7

24610 = 24 r 6

2410 = 2 r 4

210 = 0 r 2

Lets convert some of these other numbers to base 10.20123 2748

2 1 = 2

1 3 = 3

0 9 = 0

2 27 = + 54

59

4 1 = 4

8 7 = 56

2 64 = + 128

188

Converting a number to a different base

To convert a number from base 10 to a different base you keep dividing by the base keeping tract of the quotients and remainders then reversing the remainders you got. The examples to the right first show how to convert a base 10 number 2467 to base 10. Then how you convert 59 to base three. (Notice 59 agrees with what we got for the base three number above.

2467

quo

tien

ts

quo

tien

ts

rem

aind

ers

rem

aind

ers

20123

20123 = 2×33 + 0×32 + 1×31 +2×30

2748 = 2×82 + 7×81 + 4×80

Base Two

The important details about base 2 are that the symbols that you use are 0 and 1. The place values in base 2 are (going from smallest to largest):

20

(1)21

(2)22

(4)23

(8)24

(16)25

(32)26

(64)27

(128)28

(256)29

(512)210

(1024)

Change the base 2 number 1100112 to a base 10 (decimal) number.

1100112

11 = 1 12 = 2 04 = 0 08 = 0116 = 16132 = 32

51

Change the base 10 (decimal) number 47 to a base 2 (binary) number.

47 2 = 23 remainder 1

23 2 = 11 remainder 1

11 2 = 5 remainder 1

5 2 = 2 remainder 1

2 2 = 1 remainder 0

1 2 = 0 remainder 1

47 = 1011112

Base 12 and 16

For bases that are larger than 10 we need to use a single symbol to stand for the "digits" in a number that represent more than 10. This is because if you use more than one symbol the place values will get off. In particular, bases 12 and 16 are sometimes very useful.

In base 12 the digit 10 is represented with a letter T and the digit 11 is represent with a letter E.

In base 16 the letters A, B, C, D, E, F represent the digits 10, 11, 12, 13, 14, 15 respectively.

Base Symbols Place Values as Numbers Place Values as Powers

12 0,1, 2, 3, 4, 5, 6, 7, 8, 9, T, E

… , 144, 12, 1 … , 122, 121, 1

16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

… , 256, 16, 1 … , 162, 161, 1

Convert T3E12 to base 10.

T3E12

E1 = 11 1 = 11312 = 3 12 = 36T144 = 10144 = 1440

1477

Write the base 16 number A2D16 in expanded form and convert it to base ten.

In expanded form A2D16 is:

A×162 + 2×161 + D×160

10×162 + 2×161 + 13×160

A2D16

D1 = 13 1 = 13216 = 2 16 = 32A256 = 10256 = 2560

2605

Converting from Base to Base

If we wish to convert from one strange base to another we do this by "going through" base ten. In other words, for example if we want to convert from base 5 to base 16, first convert base 5 to base ten then convert that base ten number to base 16.

Example, Convert 32045 to base 16.

1st convert 32045 to base 10

32045

4×1 = 40×5 = 02×25 = 503×125 = 375

429

2nd convert 429 to base 16

429 16 = 26 remainder 13 = D

26 16 = 1 remainder 10 = A

1 16 = 0 remainder 1 = 1

We get the following: 32045 = 1AD16