Binary

Number System

This is a number system in which base (or radix) is 2.

There are only two digits – 0 and 1. These two digits can mean many things like

0 representing OFF, LOW, FALSE etc. and 1 representing HIGH, ON, TRUE etc. This

is the only number system that computers can understand directly. In all digital

circuits, binary or its some other versions of binary (Like BCD, Gray, Excess –

3 etc.) are used.

Why Binary?

In all logic systems, the functions are realized with electronic

circuits and decision is taken based on voltage levels. Traditionally, humans

use decimal number system in which base is 10 (perhaps because we have 10

fingers in our hands). So, we make all calculations in that system. But implementing

such a system with 10 different levels is something which is next to impossible

(Though researches are going on in this field even today), it is far easier to

design a system with just two voltage levels (LOW / 0V and HIGH/+5 V) and to

convert all calculations to the new system.

Difficulty in representing large numbers is a serious

drawback of binary system (For e.g., even to represent 100, we need 7 bits).

Due to this, two other systems, Octal (base – 8) and Hexadecimal (base – 16)

have been developed and they act as a shorthand representations of binary. It

is also very easy to convert between Binary – octal – Hexadecimal systems.

Apart from the basic Binary code, Some other varieties

of binary are used:

1.

Gray

code

Gray code (also known as

Reflected Binary code) is a non – positional number system in which two

successive digits change by only one unit. This is used extensively in real

time systems, shaft encoders etc.

Necessity

of Gray code:

In machines with rotating

parts (Like CNC machine, machines used to dig tunnels, DVD/CD player etc.) we

use a device known as rotary encoder to know the angular position of the

rotating disc. This is accomplished by dividing the surface of the discs into

various sectors and representing each sector in binary (For e.g., we can divide

the surface of the disk into 8 parts (each sector being 450 wide)

and represent each sector by binary code 000, 001, 010 etc.) Problem with this

representation lies in the fact that all sensors have different response times.

So, during the transition from one sector to another (Like from 3 (011) to four

(100), all sensors may not change value simultaneously resulting in momentary

erroneous readings (E.g. 011 may become

100 in the following order: 011 – 111 – 110 – 100 ). This has serious repercussions in

industry especially if the data so obtained is used to control some other

process like time keeping.

To solve this problem, a

variant of binary known as Reflected Binary Code / Gray Code, in which

successive digits have change in only a single position. (E.g. 3 in this system

is represented as 010 and four as 110). This reduce the chance of error because

only one sensor has to change state when moving from one sector to another.

Grey code is not a

positional number system and is not used for calculations.

2.

Binary

Coded Decimal (BCD)

This is one of the simplest

variants of binary in which each digit of a decimal number is replaced by

corresponding 4 bit binary. This system has numbers from 0000 to 1001. The

numbers 1010 to 1111 are invalid.

E.g. 2017 in BCD = 0010

0000 0001 0111 (spaces are added for clarity)

This is also not a

positional system and arithmetic rules are different from usual binary. ICs

that convert From BCD to 7 segment displays are very common.

3.

Excess

3 (XS 3) code

Excess – 3 is also a non

– positional system in which each decimal digit (0 – 9) is represented by

equivalent 4 bit binary from 3 to 12 (i.e., 0011 – 1100) . Combinations like

0000, 0001, 0010, 1101, 1110,1111 are invalid.

It is obtained as

XS– 3 = BCD + 0011

It is easier to use XS –

3 in adder/subtractor circuits as 9’s complement of a digit in XS – 3 can be

obtained by taking one’s complement (Inverting all bits) in the XS – 3

representation.

E.g. 9’s complement of 8

= 1

In XS -3 , 8 = 1011 and 1

= 0100 ( 1’s complement of 1011)

Using this property, we

can design cheap adder/subtractor circuits.

4.

ASCII

This stands for American

Standard Code for Information Interchange. This is a 7-bit character coding standard.

In this code, characters are represented in 7-bit binary code and has a total

of 128 characters. Extended ASCII code uses 8 bits.

ASCII code for some

common characters:

48 to 57 in ASCII -> 0 (zero) to 9 (nine)

65 to 90 -> A ( capital A) to Z (capital A)

97 – 122 -> a ( small

a) to z (small z)

E.g. NITC is Represented

in ASCII as 1001110 1001001 1010100 1000011 ( i.e., 78 73 84 67)

Binary

to Grey Code/ Grey Code to binary converter with mode control

This is a device which

converts from binary number system to Grey Code or vice versa depending upon

the value of a control variable known as mode control. In this experiment, the

designed device take a 3 bit binary number as input and generate the equivalent

Gray code as output when mode control is 1.