IT Essentials Lab 4: Number Systems and Arithmetic Operations

Binary Arithmetic Operations

Binary numbers use only two digits: 0 and 1. Arithmetic operations are performed using similar principles to decimal arithmetic but with a base of 2.

Binary Addition

Binary addition follows these rules:

• $0+0=0$
• $0+1=1$
• $1+0=1$
• $1+1=0$ (with a carry of $1$)
• $1+1+1=1$ (with a carry of $1$)

Example:

  1011 (11 in decimal)
+ 0110 (6 in decimal)
------
 10001 (17 in decimal)

Explanation:

• Rightmost column: $1+0=1$
• Second column from right: $1+1=0$ (carry-over $1$)
• Third column from right: $0+1+1$ (carry-over) $=0$ (carry-over $1$)
• Leftmost column: $1+0+1$ (carry-over) $=10$

Binary Subtraction

Binary subtraction also has specific rules:

• $0-0=0$
• $1-0=1$
• $1-1=0$
• $0-1=1$ (borrow $1$ from the next higher bit)

Example:

  1101 (13 in decimal)
- 0110 (6 in decimal)
------
  0111 (7 in decimal)

Explanation:

• Rightmost column: $1-0=1$
• Second column from right: $0-1=1$ (borrow $1$ from the third column, third column is now $0$)
• Third column from right: $0-1=1$ (borrow $1$ from the fourth column, fourth column is now $0$)
• Leftmost column: $0-0=0$

Binary Multiplication

Binary multiplication is similar to decimal multiplication:

• $0\times 0=0$
• $0\times 1=0$
• $1\times 0=0$
• $1\times 1=1$

Example:

    101 (5 in decimal)
x   110 (6 in decimal)
--------
    000
   1010
+ 10100
--------
  11110 (30 in decimal)

Explanation:

• First step is to mulitply $0\times 101=000$
• Next, multiply $1\times 101=1010$ (In the second digit one zero, on the third we add two zeros and so on)
• Next, mulitply $1\times 101=10100$
• Finally we will add the results $000+1010+10100=11110$

ASCII Code

ASCII (American Standard Code for Information Interchange) is a character encoding standard for electronic communication. Each character (letters, numbers, symbols) is assigned a unique 7-bit binary code (ranging from 0 to 127).

Character ranges:

• Digits: ASCII values from 48 -> 0 to 57 -> 9.
• Uppercase Letters: ASCII values from 65 -> A to 90 -> Z.
• Lowercase Letters: ASCII values from 97 -> a to 122 -> z.

Example:

• A $=65$ (decimal) = $01000001$ (binary)
• a $=97$ (decimal) = $01100001$ (binary)
• 0 $=48$ (decimal) = $00110000$ (binary)

Binary-Coded Decimal (BCD)

BCD represents each decimal digit (0-9) with a 4-bit binary code.

Example:

The decimal number 29 is represented in BCD as:

• $2$ = $0010$
• $9$ = $1001$

Therefore, $29$ in BCD is $00101001$.

BCD is useful for applications where decimal representation is important, such as digital displays.

Decimal	BCD (4-bit Binary)
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111
8	1000
9	1001

Practice Your Learning

1. Numbering Systems Conversion

Convert the decimal number 45 into binary, octal, and hexadecimal.

Solution:

• Binary: $45\to 101101$
• Octal: $45\to 55$
• Hexadecimal: $45\to 2D$

2. Binary Addition

Add the binary numbers 1011 and 1101.

Solution:

        1011
      + 1101
      ------
      11000 (Binary)

Decimal equivalent: $11+13=24$

3. Binary Subtraction

Subtract the binary number 1101 from 10101.

Solution:

       10101
    -   1101
    --------
       1000 (Binary)

Decimal equivalent: $21-13=8$

4. Binary Multiplication

Multiply the binary numbers 101 and 11.

Solution:

        101
      ×  11
    --------
        101
    + 1010
    --------
       1111 (Binary)

Decimal equivalent: $5\times 3=15$

5. ASCII Code

What is the ASCII code for the character "A"

Solution:

• ASCII code for A is $65$ (decimal) or $01000001$ (binary).

Find the character corresponding to the ASCII code 97.

Solution:

• ASCII code 97 corresponds to the character a.

6. Binary-Coded Decimal (BCD)

Convert the decimal number 59 into BCD.

Solution:

• Decimal 59 in BCD: $01011001$

7. BCD to Decimal

Convert the BCD 0110 0101 into decimal.

Solution:

• BCD $01100101$ corresponds to decimal $65$.