# Advantages of Binary Number System

### I. INTRODUCTION

The binary number system, also called the base-2number system, is a method of representing numbers that counts by using combinations of only two numerals: zero (0) and one (1). Computers use the binary number system to manipulate and store all of their data including numbers, words, videos, graphics, and music.

The term bit, the smallest unit of digital technology, stands for "Binary digit." A byte is a group of eight bits. A kilobyte is 1,024 bytes or 8,192 bits.

Using binary numbers, 1 + 1 = 10 because "2" does not exist in this system. A different number system, the commonly used decimal or

### base-10

number system, counts by using 10 digits (0,1,2,3,4,5,6,7,8,9) so 1 + 1 = 2 and 7 + 7 = 14. Another number system used by computer programmers is the hexadecimal system,### base-16

, which uses 16 symbols (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F), so 1 + 1 = 2 and 7 + 7 = E. Base-10 and base-16 number systems are more compact than the binary system. Programmers use the hexadecimal number system as a convenient, more compact way to represent binary numbers because it is very easy to convert from binary to hexadecimal and vice versa. It is more difficult to convert from binary to decimal and from decimal to binary.Binary is a number system used by digital devices like computers, cd players, etc.

Binary is Base 2, unlike our counting system decimal which is Base 10 (denary).

In other words,

Binary has only 2 different numerals (0 and 1) to denote a value, unlike Decimal which has 10 numerals (0,1,2,3,4,5,6,7,8 and 9).

### Here is an example of a binary number:

10011100

As you can see it is simply a bunch of zeroes and ones, there are 8 numerals in all which make this an 8 bit binary number. Bit is short for Binary Digit, and each numeral is classed as a bit.

The bit on the far right, in this case a 0, is known as the Least significant bit (LSB). The bit on the far left, in this case a 1, is known as the Most significant bit (MSB).### notations used in digital systems:

4 bits = Nibble

8 bits = Byte

16 bits = Word

32 bits = Double word

64 bits = Quad Word (or paragraph)

When writing binary numbers you will need to signify that the number is binary (base 2), for example, let's take the value 101. As it is written, it would be hard to work out whether it is a binary or decimal (denary) value. To get around this problem it is common to denote the base to which the number belongs, by writing the base value with the number, for example:

101

2 is a binary number and 10110 is a decimal (denary) value.

Once we know the base then it is easy to work out the value, for example:

101

### 2= 1*22 + 0*21 + 1*20 = 5 (five)

101

### 10= 1*102 + 0*101 + 1*100 = 101 (one hundred and one)

One other thing about binary numbers is that it is common to signify a negative binary value by placing a 1 (one) at the left hand side (most significant bit) of the value. This is called a

### sign bit

TABLE I

REPRESENTATION OF BINARY NUMBER SYSTEM

Decimal Number System |
Binary Number System |

0 |
0000 |

1 |
0001 |

2 |
0010 |

3 |
0011 |

4 |
0100 |

5 |
0101 |

6 |
0110 |

7 |
0111 |

8 |
1000 |

9 |
1001 |

10 |
1010 |

11 |
1011 |

12 |
1100 |

13 |
1101 |

14 |
1110 |

15 |
1111 |

The term 'bit', the smallest unit of digital technology stands for binary digit. Where as a byte is a group of 8 bits. The leftmost bit in a given binary number with highest weight is called as Most Significant Bit (MSB) whereas the rightmost bit in a given number with the lowest weight is called as Least Significant Bit (LSB). We typically write binary numbers as a sequence of bits. We have defined boundaries for these bits. These boundaries are:

The Bit: The smallest 'unit' of data is defined as a single bit. With a single bit we can represent any two distinct items like true or false, on or off, male or female, right or wrong, etc.

The Nibble: A nibble is a combination of four bits. With a nibble, we can represent up to 16 distinct values.

The Byte: A byte is a combination of 8 binary bits. The number of distinct values represented by a byte 256, ranging from 00000000 to 111111111. A byte contains two nibbles. Byte is the most important data item in the microprocessor.

The Word: A word is a combination of 16 bits. Hence it consists of two bytes.

The Double Word: A double word is a combination of 32 bits or 4 bytes or 8 nibbles.

Computer systems are constructed of digital electronics. That means that their electronic circuits can exist in only one of two states: on or off. Most computer electronics use voltage levels to indicate their present state. For example, a transistor with five volts would be considered "on", while a transistor with no voltage would be considered "off." Not all computer hardware uses voltage, however. CD-ROM's, for example, use microscopic dark spots on the surface of the disk to indicate "off," while the ordinary shiny surface is considered "on." Hard disks use

### TABLE II

### BINARY NUMBER FORMATS

Name |
Size |
Example |

Bit |
1 |
1 |

Nibble |
4 |
0101 |

Byte |
8 |
0000 0101 |

Word |
16 |
000000000000 0101 |

Double Word |
32 |
000000000000 000000000000 00000101 |

II. HISTORY

A full set of eight trigrams and 64 hexagrams, analogous to the three-bit and six-bit binary numerals, was known to the ancient Chinese through the classic text I Ching. An arrangement of the hexagrams of the I Ching, ordered according to the values of the corresponding binary numbers (from 0 to 63), and a method for generating them, was developed by the Chinese scholar and philosopher Shao Yong in the 11th century. However, there is no evidence that Shao understood binary computation; the ordering is also the lexicographical order on sextuples of elements chosen from a two-element set.

The Indian writer Pingala (c. 200 BC) developed advanced mathematical concepts for describing prosody, and in doing so presented the first known description of a binary numeral system.

Similar sets of binary combinations have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been widely applied in sub-Saharan Africa.

In 1605 Francis Bacon discussed a system by which letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded as scarcely visible variations in the font in any random text. Importantly for the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of a twofold difference only; as by Bells, by Trumpets, by Lights and Torches, by the report of Muskets, and any instruments of like nature".

The modern binary number system was fully documented by Gottfried Leibniz in the 17th century in his article Explication de l'Arithmétique Binaire. Leibniz's system uses 0 and 1, like the modern binary numeral system. As a Sinophile, Leibniz was aware of the I Ching and noted with fascination how its hexagrams correspond to the binary numbers from 0 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.

In 1854, British mathematician George Boole published a landmark paper detailing an algebraic system of logic that would become known as Boolean algebra. His logical calculus was to become instrumental in the design of digital electronic circuitry.

In 1937, Claude Shannon produced his master's thesis at MIT that implemented Boolean algebra and binary arithmetic using electronic relays and switches for the first time in history. Entitled A Symbolic Analysis of Relay and Switching Circuits, Shannon's thesis essentially founded practical digital circuit design.

In November 1937, George Stibitz, then working at Bell Labs, completed a relay-based computer he dubbed the "Model K" (for "

Kitchen", where he had assembled it), which calculated using binary addition. Bell Labs thus authorized a full research programme in late 1938 with Stibitz at the helm. Their Complex Number Computer, completed January 8, 1940, was able to calculate complex numbers. In a demonstration to the American Mathematical Society conference at Dartmouth College on September 11, 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a teletype. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were John Von Neumann, John Mauchly and Norbert Wiener, who wrote about it in his memoirs.

### III. WORKING

The decimal number system that people use every day contains ten digits, 0 through 9. Start counting in decimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Since there are no more digits left. So to continue counting with only ten digits we add a second column of digits, worth ten times the value of the first column and start counting again: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 , 21, 22, 23, ... , 94, 95, 96, 97, 98, 99, Oops! Once again, there are no more digits left. The only way to continue counting is to add yet another column worth ten times as much as the one before. If we continue counting: 100, 101, 102 ... 997, 998, 999, 1000, 1001, 1002, .... we will get the picture at this point.

Another way to make this clear is to write decimal numbers in expanded notation. 365, for example, is equal to 3×100 + 6×10 + 5×1. 1032 is equal to 1×1000 + 0×100 + 3×10 + 2×1. By writing numbers in this form, the value of each column becomes clear.

The binary number system works in the exact same way as the decimal system, except that it contains only two digits, 0 and 1. As there are no more binary digits except 0 and 1,in order to keep counting, we need to add a second column worth twice the value of the column before. We continue counting again: 10, 11, time to add another column again. Counting further: 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111.... If we watch the pattern of 1's and 0's we will see that binary works the same way decimal does, but with fewer digits.

Binary uses two digits, so each column is worth twice the one before. This fact, coupled with expanded notation, can be used convert between from binary to decimal. In the binary system, the columns are worth 1, 2, 4, 8, 16, 32, 64, 128, 256, etc. To convert a number from binary to decimal, we simply write it in expanded notation. For example, the binary number 101101 can be rewritten in expanded notation as 1×32 + 0×16 + 1×8 + 1×4 + 0×2 + 1×1. By simplifying this expression, we can see that the binary number 101101 is equal to the decimal number 45.

### IV. ADVANTAGES

The advantage of the binary system is its simplicity. A computing device can be created out of anything that has a series of switches, each of which can alternate between an "on" position and an "off" position. These switches can be electronic, biological, or mechanical, as long as they can be moved on command from one position to the other. Most computers have electronic switches.

When a switch is "on" it represents the value of one, and when the switch is "off" it represents the value of zero. Digital devices perform mathematical operations by turning binary switches on and off. The faster the computer can turn the switches on and off, the faster it can perform its calculations.

There are various number systems in binary number system:

i. Unsigned integer

ii. Two's Complement Integer

iii. Unsigned Fractional

iv. Two's Complement Signed Fractional

v. Gray Code

vi. Signed Magnitude

vii. Offset Two's Complement

viii. One's Complement

ix. Floating Point

x. Block Floating Point

TABLE III

### SUMMARY OF BINARY NUMBER SYSTEM

System |
Advantages |

Unsigned Integer |
Universal Numbering System. Easy to perform arithmetic operations such as addition or subtraction. |

Two's Complement |
Stores both positive and negative numbers. Easy to perform arithmetic with regular adders. |

Unsigned Fractional |
Stores positive numbers greater than and less than 1. Operations are identical to unsigned integer operation. |

Two's Complement Signed Fractional |
Stores positive and negative number both greater than less than 1. Operations are identical to two's complement operation. |

Gray Code |
Only One bit changes between adjacent numbers which facilitates interfaces with physical systems. |

Signed Magnitude |
Useful for application that require the magnitude to be distinct from the sign. |

Offset Two's Complement |
Used by many Analog to Digital and Digital to Analog converters. Easy to perform arithmetic operations. |

One's Complement |
Easy to perform negations. |

Floating Point |
Very large dynamic range |

Block Floating Point |
Large dynamic range and requires minimal hardware. |

### V. BINARY OPERATION

Binary numbers can be manipulated with the same familiar operations used to calculate decimal numbers, but using only zeros and ones. To add two numbers, there are only four rules to remember:

Therefore, to solve the following addition problem, start in the rightmost column and add 1 + 1 = 10; write down the 0 and carry the 1. Working with each column to the left, continue adding until the problem is solved.

To convert a binary number to a decimal number, each digit is multiplied by a power of two. The products are then added together. For example, to translate the binary number 11010 to decimal, the formula would be as follows:

- COUNTING IN BINARY

## Decimal | ## Binary |

0 |
0 |

1 |
1 |

2 |
10 |

3 |
11 |

4 |
100 |

5 |
101 |

6 |
110 |

7 |
111 |

8 |
1000 |

9 |
1001 |

10 |
1010 |

Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Decimal counting uses the symbols

### 0through 9, while binary only uses the symbols

### 0 and 1.

When the symbols for the first digit are exhausted, the next-higher digit (to the left) is incremented, and counting starts over at 0. In decimal, counting proceeds like so:

000, 001, 002 ... 007, 008, 009, (rightmost digit starts over, and next digit is incremented)

010, 011, 012...

090, 091, 092 ... 097, 098, 099, (rightmost two digits start over, and next digit is incremented)

### 100, 101, 102...

After a digit reaches 9, an increment resets it to 0 but also causes an increment of the next digit to the left. In binary, counting is the same except that only the two symbols 0 and 1 are used. Thus after a digit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next digit to the left:

0000,0001, (rightmost digit starts over, and next digit is incremented)

0010, 0011, (rightmost two digits start over, and next digit is incremented)

0100, 0101, 0110, 0111, (rightmost three digits start over, and the next digit is incremented)

### 1000, 1001...

Since binary is a base-2 system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on. To determine the decimal representation of a binary number simply take the sum of each of the product binary digits with the power of 2 which they represent. For example, the binary number:

100101

is converted to decimal form by:

[(1) × 25] + [(0) × 24] + [(0) × 23] + [(1) × 22] + [(0) × 21] + [(1) × 20] =

[1× 32] + [ 0× 16] + [0× 8] + [1× 4] + [0× 2] + [1× 1] = 37To create higher numbers, additional digits are simply added to the left side of the binary representation.

### HOW BINARY WORKS

The decimal number system that people use every day contains ten digits, 0 through 9. Start counting in decimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, Oops! There are no more digits left. How do we continue counting with only ten digits? We add a second column of digits, worth ten times the value of the first column. Start counting again: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 (Note that the right column goes back to zero here.), 21, 22, 23, ... , 94, 95, 96, 97, 98, 99, Oops! Once again, there are no more digits left. The only way to continue counting is to add yet another column worth ten times as much as the one before. Continue counting: 100, 101, 102, ... 997, 998, 999, 1000, 1001, 1002, ....

Another way to make this clear is to write decimal numbers in expanded notation. 365, for example, is equal to 3×100 + 6×10 + 5×1. 1032 is equal to 1×1000 + 0×100 + 3×10 + 2×1. By writing numbers in this form, the value of each column becomes clear.

The binary number system works in the exact same way as the decimal system, except that it contains only two digits, 0 and 1. Start counting in binary: 0, 1, Oops! There are no more binary digits. In order to keep counting, we need to add a second column worth twice the value of the column before. We continue counting again: 10, 11, Oops! It is time to add another column again. Counting further: 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111.... Watch the pattern of 1's and 0's. You will see that binary works the same way decimal does, but with fewer digits.

Binary uses two digits, so each column is worth twice the one before. This fact, coupled with expanded notation, can be used convert between from binary to decimal. In the binary system, the columns are worth 1, 2, 4, 8, 16, 32, 64, 128, 256, etc. To convert a number from binary to decimal, simply write it in expanded notation. For example, the binary number 101101 can be rewritten in expanded notation as 1×32 + 0×16 + 1×8 + 1×4 + 0×2 + 1×1. By simplifying this expression, you can see that the binary number 101101 is equal to the decimal number 45.

### VIII. BITWISE OPERATION

Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated using Boolean logical operators. When a string of binary symbols is manipulated in this way, it is called a bitwise operation; the logical operators AND, OR, and XOR may be performed on corresponding bits in two binary numerals provided as input. The logical NOT operation may be performed on individual bits in a single binary numeral provided as input. Sometimes, such operations may be used as arithmetic short-cuts, and may have other computational benefits as well. For example, an arithmetic shift left of a binary number is the equivalent of multiplication by a (positive, integral) power of 2.

The binary number system is a radix-2 number system with '0' and '1' as the two independent digits. All larger binary numbers are represented in terms of '0' and '1'. The procedure for writing higher order

binary numbers after '1' is similar to the one explained in the case of the decimal number system.

For example, the first 16 numbers in the binary number system would be 0, 1, 10, 11, 100, 101, 110,111, 1000, 1001, 1010, 1011, 1100, 1101, 1110 and 1111. The next number after 1111 is 10000, which is the lowest binary number with five digits. This also proves the point made earlier that a maximum of only 16 (= 24_ numbers could be written with four digits. Starting from the binary point, the place

values of different digits in a mixed binary number are 20, 21, 22 and so on (for the integer part) and 2−1, 2−2, 2−3 and so on (for the fractional part).

### Example:-

Consider an arbitrary number system with the independent digits as 0, 1 and X. What is the radix of this number system? List the first 10 numbers in this number system.

### Solution

• The radix of the proposed number system is 3.

• The first 10 numbers in this number system would be 0, 1, X, 10, 11, 1X, X0, X1, XX and 100.

### IX. ADVANTAGES

Logic operations are the backbone of any digital computer, although solving a problem on computer could involve an arithmetic operation too. The introduction of the mathematics of logic by George Boole laid the foundation for the modern digital computer. He reduced the mathematics of logic to a binary notation of '0' and '1'. As the mathematics of logic was well established and had proved itself to be quite useful in solving all kinds of logical problem, and also as the mathematics of logic (also

known as Boolean algebra) had been reduced to a binary notation,

the binary number system had a clear edge over other number systems for use in computer systems. Digital Electronics Yet another significant advantage of this number system was that all kinds of data could be

Conveniently represented in terms of 0s and 1s. Also, basic electronic devices used for hardware implementation could be conveniently and efficiently operated in two distinctly different modes.

For example, a bipolar transistor could be operated either in cut-off or in saturation very efficiently. Lastly, the circuits required for performing arithmetic operations such as addition, subtraction, multiplication, division, etc., become a simple affair when the data involved are represented in the form of 0s and 1s.

### X. WHY DO WE NEED BINARY SYSTEM

The binary system is essential in technology. The reason is that any electronic circuit can have only two possible states, on or off. A simple example is the light in your room. The switch has only two options, on or off. Another example of a binary system would be Morse code. It also works with only two digits, a dot or a dash. Anything expressed in Morse code is done with these two digits. Electronic circuits work the same way, they are either on or off. And every sequence of these two signals has a certain meaning. Every communication that takes place inside your computer uses this binary system.

### REFRENCES

[1] Publications, Pune, 2009, pp.1.11-1.13, 2.11, 2.17.

[2] R.P.Jain, Modern Digital Electronics, 3rd ed., Tata McGraw -Hill Publishing Company Limited, New Delhi, 2008, pp. 22-23.

[3] Thomas L.Floyd and R.P.Jain, Digital Fundamentals, 8th ed., Pearson Education, 2008, pp. 18-19.

[4] J.B.Gupta, Electronics Devices and Circuits, 3rd ed., S.K.Kataria and Sons, New Delhi, 2009, pp.711-712.

[5] Gurmeet Singh Bains, Mandeep Singh Walia and Navdeep Singh Dhaliwal, 1st ed., PBS Education, Jalandhar, 2009, pp.6-8.