Lesson 01: Digital Systems and Number Systems

Binary

Binary

In a computer, all the information is stored using digital signals that translate to binary numbers. A single bit can represent two possible states: on (1) or off (0).

Characteristics of the binary number system are as follows:

• Base-2
• Digits: {0, 1}
• Binary Digits are called bits
• Positional system
• Example:
  1101.1₂ = (1 x 2³) + (1 x 2²) + (0 x 2¹) + (1 x 2⁰) + (1 x 2^-1)

Octal

Octal

Characteristics of the binary number system are as follows:

• Base-8
• Digits: {0, 1, 2, 3, 4, 5, 6, 7}
• A group of 3 binary digits can be used to represent each octal digit
• Positional system
• Example:
  357₈ = (3 x 8²) + (5 x 8¹) + (7 x 8⁰)

Hexadecimal

Hexadecimal

Characteristics of the hexadecimal number system are as follows:

• Base-16
• Digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}
• Groups of 4 bits represent each base 16 digit
• Positional system
• Example:
  1F4₁₆ = (1 x 16²) + (F x 16¹) + (4 x 16⁰)

BCD

BCD

BCD (Binary Coded Decimal) is a decimal number system, where each decimal digit is represented by a 4-bit binary number.

Do not get confused, BCD is not the same as hexadecimal. The hexadecimal also uses a 4-bit binary to represent one hex digit. Each hex digit is valid up to (F)₁₆ representing binary (1111)₂. But the one decimal digit is only up to 9 binary (1001)₂. This means, that the 4-bit binary can represent 16 numbers (2⁴), in the BCD number system, the six binary codes from (1010)₂ (decimal 10) to (1111)₂ (decimal 15) are not used.

The main advantage of BCD is that it allows easy conversion between decimal and binary forms. However, the disadvantage is that the BCD code is wasteful as the states between (1010)₂ and (1111)₂ are not used.

For example, give the BCD representation for the decimal numbers 90 and 873.

The BCD representation of the number 90 is written as follows:

(90)₁₀ = (1001 0000)_BCD

For the number 873:

(873)₁₀ = (1000 0111 0011)_BCD

Gray Code

Gray Code

A Gray Code is an encoding of numbers so that adjacent numbers have one single digit differing by 1. This coding technique was invented by Frank Gray, thus it is named so. The term Gray code is often used to refer to a "reflected" code, or more specifically still, the binary reflected Gray code.

Gray Code is a non-weighted code, as it does not ascribe a specific weight to each bit position. It is not used for arithmetic calculations.

Gray code is used in Karnaugh maps and in the design of logic circuits. They also find application in rotary encoders, where the predisposition to errors increases with the number of bits that change logical states between two consecutive positions.

ASCII Code

ASCII Code

ASCII code (American Standard Code for Information Interchange) has seven bits allowing for the representation of 2⁷ = 128 symbols.

Table 1: ASCII Table

Decimals to Base-r Number

Use Radix Divide Method

The equation for decimal to base-r number:

Integral Part

1. Divide (integral part) by the base- r number:
   
2. Take the remainder as a coefficient
3. Take the quotient and repeat the division, until the quotient is equal to zero
   

Fractional Part

1. Multiply the number by the base- r number
   
2. Take the integer (either 0 or 1) as a coefficient
3. Take the resultant fraction and repeat the multiplication
   

Binary ↔ Octal

• 8 = 2³
• Each group of 3 digits in binary represents an Octal digit

Binary ↔ Hex

• 16 = 2⁴
• Each group of 4 binary bits represents a Hexadecimal digit

Octal ↔ Hex

Convert to Binary as an intermediate step

Example: (26.2)₈ to Hex

Works both ways (Octal to Hex & Hex to Octal)

Binary ↔ BCD Code

Binary-to-BCD Conversion

To convert binary to BCD, we will first convert the binary number to decimal and then the decimal number to BCD number. Similarly, to convert BCD to binary, we will first convert the BCD to decimal, and then decimal to equivalent binary.

Binary ↔ Gray Code

Binary-to-Gray Code Conversion

Binary to Gray Code Conversion

The conversion process from binary code to gray code involves the following steps:

1. MSB (most significant bit) or the leftmost bit of the given binary is the same as MSB of the gray number.
2. To get the next bit of gray code, add MSB with the next bit of binary number (move from left to right). Write the sum and discard the carry.
3. Repeat step 2 for all the bits in the sequence till LSB.

Gray to Binary Code Conversion

The steps given below are required to be followed in order to convert a given gray value into its binary equivalent:

1. MSB does not change. So, write the leftmost bit of gray code as the MSB of binary code.
2. Now, add the recently achieved binary digit with the next adjacent gray code bit. The sum must be written as the next bit of binary equivalent, while the carry must be discarded.
3. The above-discussed step must be followed for all the bits present in the sequence.

The 1-bit adder can be represented as xor logic.

   
Figure 2: Gray Code Conversion using XOR logic