realistic approach of strange number system from unary to [email protected], 2....

Realistic Approach of Strange Number System from Unary to Decimal Debasis Das1 and U. A. Lanjewar2

1Assistant Professor, MCA, VMV Commerce, JMT Arts & JJP Science College, Nagpur, India 2 Professor, MCA, VMV Commerce, JMT Arts & JJP Science College, Nagpur, India

[email protected], [email protected]

Abstract

Numbers play an important role in Mathematics, also

in Computer Science. A number is a symbol or group of

symbols, or a word in a natural language that

represents a numeral, which is different from numbers

just as words differ from the things they refer to. A set

of numbers in a framework that are represented by

numerals in a consistent manner is called number

system. In computing the study of number systems is

useful to all, as a fact that various number systems are

used in computer fields. Some are familiar number

system (decimal (base 10), binary (base-2), octal (base-

8) and hexadecimal (base-16)) and others are strange

number system (SNS). Strange number system is

investigated for efficiently describing and implementing

in digital systems. In computing the study of strange

number system (SNS) will useful to all researchers.

Their awareness and detailed explanation is necessary

for understanding various digital aspects. In this paper

we have elaborate the concepts of strange number

system (SNS), needs, number representation, arithmetic

operations and inter conversion with different bases,

represented in tabulated form. This paper will also

helpful for knowledge seekers to easy understanding

and practicing of number systems as well as to

memories them.

Keywords- Strange Number System, trinary,

quaternary, quinary, senary, septenary and nonary

1. Introduction A number is a count or measurement that is really an

idea in our minds. A digit is a single symbol used to

make numbers. The concept of number is the most

basic and fundamental in the world of science and

mathematics. The first digit in any numbering system is

always a zero. For example, a base 2 (binary) number

contains 2 digits: 0 and 1, a base 3 (ternary) numbers

contains 3 digits: 0, 1 and 2, a base 4 (quaternary)

number contains 4 digits: 0 through 3 and so fourth.

Remember, a base 10 (decimal) number does not

contain the digit 10, similarly base 16 number does not

contain a digit 16.

A number system is a writing system for

expressing numbers that is a mathematical notation for

representing number of a given set, using graphemes or

symbols or words in a consistent manner. The number

sense is not the ability to count, but the ability to

recognize that something has changes in a small

collection. Data is usually combination of Numbers,

Characters and special characters.

The unary system is normally only useful for small

numbers, although it plays an important role in

theoretical computer science. Many ancient cultures

from early on calculated with numerals based on ten:

Egyptian hieroglyphs, in evidence since around 3000

BC, used a purely decimal system. Then the binary

system (base 2), was propagated in the 17th century by

Gottfried Leibniz. The first ternary machine was built

by Thomas Fowler in 1840. In 1958, a much more

complicated machine was built, called “Setun” in

Russia by Nikolai P. Brousentsov and his colleagues at

Moscow State University (Brousentsov et al, 1997). In

1973, Gideon Frieder and his colleagues at the State

University of New York designed a Base-3 machine

they called TERNAC, and went ahead to create a

software emulator for it, (Hayes, 2001) [1].

Although many students know the traditional number

system such as decimal, binary, octal and hexadecimal

and are very comfortable with performing operations

using this system, it is important for students to

understand that these four common number systems are

not the only system. By studying other number systems

such as trinary (base-3), quaternary (base-4), quinary

(base-5), senary (base-6), septenary (base-7) and

nonary (base-9), students will gain a better

understanding of how number systems work in general.

When discussing how a computer stores information,

the binary number system becomes very important

since this is the system that computers use. It is

Debasis Das et al,Int.J.Computer Techology & Applications,Vol 3 (1), 235-241

IJCTA | JAN-FEB 2012 Available [email protected]

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http://en.wikipedia.org/wiki/Word

http://en.wikipedia.org/wiki/Natural_language

http://en.wikipedia.org/wiki/Number

important that students understand that computers store

and transmit data using electrical pulses, and these

pulses can take two forms - "on" (1) or "off" (0).

This paper will stimulate the reader’s interest to the

strange number system beyond traditional number

system. In this particular paper, we are taking under the

consideration a tabulated format for few strange

number systems. It covers each system’s number

representation, their uses, their arithmetic and inter-

conversion of numbers from one system to another.

2. Need of the Strange Number System (SNS) In digital world when we deal with computer and

information technology, normally we require a working

knowledge of traditional number system, i.e. binary,

octal, decimal and hexadecimal. Apart from these basic

number systems, the strange number system also plays

a significant role in computing. The strange number

system poses some extra features which distinguish

them from the other existing number systems and make

them worth an extra look, some of these features

include:

Greater speed of arithmetic operations realization

Greater density of memorized information

Better usage of transmission paths

Decreasing of interconnections complexity and

interconnections area

Decreasing of pin number of integrated circuits

and printed boards

3. Types of Number System The number systems are basically categorized in two

types, non-positional number system and positional

number system.

3.1 Non-Positional Number System

In early days, human beings counted with fingers.

When ten fingers were not adequate, stones, pebbles, or

sticks were used to indicate values. This method of

counting uses an additive approach or the non-

positional number system. In this system, we have

symbols such as I for 1, II for 2, III for 3, IIII for 4, IIIII

for 5, etc. The unary number system is under this

category.

3.2 Positional Number System

The general idea behind positional numbering systems

is that a numeric value is represented through

increasing powers of a radix (or base). This is often

referred to as a weighted numbering system because

each position is weighted by a power of the radix.

In a positional number system, there are only a few

symbols called digits, and these symbols represent

different values depending on the position they occupy

in the number. The value of each digit in such a number

is determined by three considerations:

The digit itself,

The position of the digit in the number, and

The base of the number system.

The binary, ternary, quaternary, quinary, senary,

septenary, octal, nonary and decimal number systems

are the example of positional number systems.

4. Number Representation of Strange

Number System In general in a number system with a base or radix n,

the digits used are from 0 to n-1 and the number can be

represented as:

x = anbn + an - 1b

n - 1 + ... + a1b

1 + a0b

0, where

x = Number, b=Base, a= any digit in that base

Any real number x can be represented in a positional

number system of base "b" by the expression

x = anbn + an - 1b

n - 1 + ... + a0b

0 + a-1b

-1 + ... + a-(n-1)b

-(n-1)

+ a-nb-n

.

For example number 128 can be represented in various

number systems as follows:

Table 1 Number Representation of Strange Number System

Number

System

Bas

e Symbol Number Representation

Unary 1 | ----- -----

Ternary 3 0,1,2 (201.02)3

2×32 + 0×31 + 1×30 +

0×3-1 + 2×3-2

Quaternary 4 0,1,..,3 (302.03)4

3×42 + 0×41 + 2×40 +

0×4-1 + 3×4-2



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Quinary 5 0,1,..,4 (403.04)5 4×52 + 0×51 + 3×50 +

0×5-1 + 4×5-2

Senary 6 0,1,..,5 (504.05)6

5×62 + 0×61 + 4×60 +

0×6-1 + 5×6-2

Septenary 7 0,1,..,6 (605.06)7 6×72 + 0×71 + 5×70 +

0×7-1 + 6×7-2

Nonary 9 0,1,..,8 (807.08)9 8×92 + 0×91 + 7×90 +

0×9-1 + 8×9-2

5. Overview of Strange Number System Number systems provide the basics for all operations in

information processing systems. In this section strange

number systems and their applications in the area of

different fields have been discussed.

5.1 Unary Number System

The unary number system is the bijective base-1

numeral system. Compared to standard positional

numeral systems, the unary system is inconvenient and

is not used in practice for large calculations. It is the

simplest numeral system to represent natural numbers:

in order to represent a number N, an arbitrarily chosen

symbol representing 1 is repeated N times. For

example, using the symbol | (a tally mark), the number

6 is represented as ||||||. There is no explicit symbol

representing zero in unary as there is in other traditional

bases, so unary is a bijective numeration system with a

single digit.

Addition and subtraction are particularly simple in the

unary system, as they involve little more than string

concatenation. However, multiplication and division

are more cumbersome.

The standard method of counting on one's fingers is

effectively in a unary system. Unary is most useful in

counting or tallying ongoing results, such as score’s in

a game of sport, since no intermediate result is in a

need to be erased or discarded. The unary system was

used in ancient mathematics such as, the Moscow

Mathematical Papyrus, dating from circa 1800 BC.

The example of a unary counting system was clustered

in the Chinese, Japanese and Korean number system. In

Brazil but also France, a variation of this system is

commonly used.

5.2 Ternary Number System

The number system with base three is known as the

ternary number system. Only three symbols are used to

represent numbers in this system and these are 0, 1 and

2. It is also a positional number system that each bit

position corresponds to a power of 3. Moreover, it has

two parts the Integral part or integers and the fractional

part or fractions, set a part by radix point. For example

(1202.021)3

In ternary number system the leftmost bit is known as

most significant bit (MSB) and the right most bit is

known as least significant bit (LSB). The following

expression shows the position and the power of the

base 3:

…...333

23

13

0.3

-13

-23

-3…..

As for rational numbers, ternary number system offers

a convenient way to represent one third; but a major

drawback is that, ternary number system does not offer

a finite representation for the most basic fraction: one

half (and thus, neither for one quarter, one sixth, one

eighth, one tenth, etc.), because 2 is not a prime factor

of the base .

Perhaps the most important and immediate use of

ternary technology is in the new and emerging field of

Quantum Computing, a technology which has been

described as “a promising and flourishing research

area‟ (Khan, 2004). Ternary logic also serves as a

stepping stone on the way to Multi-Valued-Logic

(MVL) which is currently being probed for its

application in artificial intelligence.

5.3 Quaternary Number System

The number system with base four is known as the

quaternary number system. Only four symbols are used

to represent numbers in this system and these are 0, 1, 2

and 3. It is also a positional number system that each

bit position corresponds to a power of 4. Moreover, it

has two parts the Integral part or integers and the

fractional part or fractions, set a part by radix point. For

example (202.321)4

In quaternary number system the leftmost bit is known

as most significant bit (MSB) and the right most bit is



base 4 [2]:

…...434

24

14

0.4

-14

-24

-3…..

A carry-free arithmetic operation can be performed on

quaternary numbers. The implementation of quaternary

addition and multiplication results in a fix delay

independent of the number of digits. Operations on a

large number of digits such as 64, 128, or more, can be

implemented with constant delay and less complexity.



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Quaternary numbers are used in the representation of

2D Hilbert curves, while many of the Chumashan

languages originally used a base 4 counting system, in

which the names were structured according to multiples

of 4 and 16. Quaternary line codes have been used for

transmission, from the invention of the telegraph to the

2B1Q code used in modern ISDN circuits.

5.4 Quinary Number System

The number system with base five is known as the

quinary number system. Only five symbols are used to

represent numbers in this system and these are 0, 1, 2, 3

and 4. It is also a positional number system that each

bit position corresponds to a power of 5. As five is a

prime number, only the reciprocals of the powers of

five terminate, so its location between two composite

numbers (4 and 6) does not help make its radix

economy better.

Moreover, it has two parts the Integral part or integers

and the fractional part or fractions, set a part by radix

point. For example (503.341)5

In quinary number system the leftmost bit is known as




base 5:

…...535

25

15

0.5

-15

-25

-3…..

The main usage of base 5 is as a biquinary system,

which is decimal using five as a sub-base. Many

languages use quinary number systems, including

Gumatj, Nunggubuyu, Kuurn Kopan Noot and

Saraveca. Roman numerals are a biquinary system.

5.5 Senary Number System

The number system with base six is known as the

senary number system. In this system six symbols are

used to represent numbers and these are 0, 1, 2, 3, 4 and

5. It is also a positional number system that each bit

position corresponds to a power of 6. Moreover, it has

two parts the Integral part or integers and the fractional

part or fractions, set a part by radix point. For example

(423.523)6

In senary number system the leftmost bit is known as




base 6 [2]:

…...636

26

16

0.6

-16

-26

-3…..

The arithmetic operations like addition, subtraction,

multiplication and division operations of decimal

numbers can be also performed on senary numbers.

Senary may be considered useful in the study of prime

numbers since all primes, when expressed in base-six,

other than 2 and 3 have 1 or 5 as the final digit.

5.6 Septenary Number System

The number system with base seven is known as the

septenary number system. In this system seven symbols

are used to represent numbers and these are 0, 1, 2, 3, 4,

5 and 6. It is also a positional number system that each

bit position corresponds to a power of 7. Hence, any

septenary number can not have any digit greater than 6.

Moreover, it has two parts the Integral part or integers



In septenary number system the leftmost bit is known

as most significant bit (MSB) and the right most bit is



base 7:

…...737

27

17

0.7

-17

-27

-3…..

The arithmetic operations like addition, subtraction,

multiplication and division operations of decimal

numbers can be also performed on senary numbers.

5.7 Nonary Number System

The number system with base nine is known as the

nonary number system. In this system nine symbols are

used to represent numbers and these are 0, 1, 2, 3, 4, 5,

6, 7 and 8 but not the digit 9. Nonary notation can be

used as a concise representation of ternary data. It is

also a positional number system that each bit position

corresponds to a power of 9.

Moreover, it has two parts, the Integral part or integers



In nonary number system the leftmost bit is known as




base 9:

…...939

29

19

0.9

-19

-29

-3…..

Except for three, no primes in nonary end in 0, 3 or 6,

since any nonary number ending in 0, 3 or 6 is divisible

by three. A nonary number is divisible by two, four or



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eight, if the sum of its digits is also divisible by two,

four or eight respectively.

6. Operations and Conversion of Strange

Number System Decimal arithmetic is the oldest and most common

branch of mathematics, used by almost everyone, from

simple day-to-day counting to advanced science and

business calculations. Arithmetic is at the heart of the

digital computer, and the majority of arithmetic

performed by computers is binary arithmetic, that is,

arithmetic on base two numbers. The arithmetic

operations such as addition, subtraction, multiplication

and division on strange numbers also play a significant

role in computing. Table II shows some basic

arithmetic operations performed on strange numbers.

Addition is the basic operation of arithmetic. In its

simplest form, addition combines two numbers into a

single number called the sum of the numbers. The

procedure to adding more than two numbers is called

summation.

Subtraction is one of the four basic binary arithmetic

operations. It is the inverse of addition. Subtraction

finds the difference between two numbers, the minuend

minus the subtrahend. If the minuend is larger than the

subtrahend, the difference is positive; if the minuend is

smaller than the subtrahend, the difference is negative;

if they are equal, the difference is zero.

Multiplication is the one of the basic operation of

arithmetic. Multiplication also combines two numbers

into a single number called the product. The two

original numbers are called the multiplier and the

multiplicand.

Division is essentially the opposite of multiplication.

Division finds the quotient of two numbers when the

dividend divided by the divisor. Any dividend divided

by zero is undefined. For positive numbers, if the

dividend is larger than the divisor, the quotient is

greater than one; otherwise it is less than one.

Table 2 Arithmetic Operations of Strange Number System

2.1 Addition and Subtraction

Arithmetic

Number

System

Addition Subtraction

Ternary

Quaternary

Quinary

(11)3 + (02)3 = (20)3

(36)4 + (06)4 = (120)4

(42)5 + (16)5 = (113)5

(11)3 - (02)3 = (02)3

(36)4 - (06)4 = (30)4

(42)5 - (16)5 = (21)5

Senary

Septenary

Nonary

(52)6 + (24)6 = (120)6

(26)7 + (13)7 = (42)7

(38)9 + (05)9 = (44)9

(52)6 - (24)6 = (24)6

(26)7 - (13)7 = (13)7

(38)9 - (05)9 = (33)9

2.2 Multiplication and Division

Arithmetic

Number

System

Multiplication Division

Ternary

Quaternary

Quinary

Senary

Septenary

Nonary

(11)3 * (02)3 = (22)3

(36)4 * (06)4 = (1230)4

(42)5 * (16)5 = (1432)5

(52)6 * (24)6 = (2212)6

(26)7 * (13)7 = (404)7

(38)9 * (05)9 = (214)9

(11)3 / (02)3 = (02)3

(36)4 / (06)4 = (03)4

(42)5 / (16)5 = (02)5

(52)6 / (24)6 = (02)6

(26)7 / (13)7 = (02)7

(38)9 / (05)9 = (07`)9

This section describes the conversion of numbers from

one number system to another. Radix Divide and

Multiply Method is generally used for conversion.

There is a general procedure for the operation of

converting a decimal number to a number in base r. If

the number includes a radix point, it is necessary to

separate the number into an integer part and a fraction

part, since each part must be converted differently. The

conversion of a decimal integer to a number in base r is

done by dividing the number and all successive

quotients by r and accumulating the remainders. The

conversion of a decimal fraction is done by repeated

multiplication by r and the integers are accumulated

instead of remainders.

When converting numbers from one base to another,

there is five different types of conversions that the

students will need to remember:

Any Base to Decimal and vice versa

Any Base to Binary and vice versa

Mixed Conversions (e.g. base 3 to base 5)



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When performing these conversions, it becomes

important for all to indicate the base that the

number is written in, as it will become very easy to

confuse how the number is being represented.

Table III shows the inter-conversions between

various numbers.

Table 3 Base Inter Conversion of Strange Number

system

Base Inter Conversion

Step-1

Part-A Decimal to any base [binary, … , nonary]

Integer: repeated division by base (method)

Fraction: repeated multiplication by base (method)

Part-B Any base [binary, … , nonary] to decimal

Integer: sum of [(+ve weights)×(integer)]

Fraction: sum of [(-ve weights) ×(fraction)]

Step-2

Part-A Binary to any base [quaternary, octal...]

To quaternary: replace group of 2-binary bits by

quaternary digit

To octal: replace group of 3-binary bits by octal

digit

Part-B Any base [quaternary, octal...] to binary

From quaternary: replace each quaternary digit

by 2-bit binary

From octal: replace each octal digit by 3-bit binary

Step-3

(--)3,4,5,6,7,9, ... to (--)3,4,5,6,7,9, ...

Direct conversion not applicable

First (--)3,4,5,6,7,9,...to (--)10 Then (--)10 to (--)3,4,5,6,7,9, ...

7. Conclusion

Here we have elaborated the concepts of strange

number system (SNS) and proposed an easy, short

and simple approach to fulfill the needs, number

representation, arithmetic operations and inter

conversion with different bases, represented in

tabulated form used in the digital world specially

computer science and technology. As we have seen

that, not only traditional numbers are used in digital

world, but there are some strange numbers, which

are also very common and frequently used in most

of the digital technologies and devices. Due to the

benefits of strange number representation, which

include greater speed of arithmetic operations

realization, greater density of memorized

information, better usage of transmission paths and

decreasing of pin number of integrated circuits, this

paper concludes that strange number system even

though they are not yet more commercially

available, remain a viable field for research, and

have a promising future as a replacement for

traditional number system. This study will be very

helpful for researchers and knowledge seekers to

easy understanding and practicing of number

systems as well as to memories them for those who

are in the field of computer science and technology.

Acknowledgement

Handful of thanks to Dr. U A Lanjewar, Professor

VMV Commerce, JMT Arts & JJP Science

College, Wardhaman Nagar, Nagpur for his

guidance, support and valuable instructions.

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