Number Systems & Conversion: Binary, Octal, Hex, Decimal & Operations

TAKEAWAYS:

• Computers use binary (base-2) numbers internally, but humans use decimal (base-10); conversions between them are fundamental.
• Octal (base-8) and hexadecimal (base-16) are shortcuts for writing long binary numbers compactly.
• Conversion rules (division-remainder, grouping, and positional value) simplify switching between bases.
• Binary arithmetic (addition, subtraction, multiplication) follows simple rules but requires careful handling of carries.
• NEB exam tests conversions, binary operations, and real-world applications (e.g., IP addresses in binary).

What is a Number System?

A number system is a way to represent numbers using symbols and rules. Different systems use different bases (the number of digits/symbols used). For example:

• Decimal (base-10): Uses digits 0–9 (e.g., 256).
• Binary (base-2): Uses digits 0–1 (e.g., 1010).
• Octal (base-8): Uses digits 0–7 (e.g., 372).
• Hexadecimal (base-16): Uses digits 0–9 and letters A–F (e.g., 1A3).

Why do we need different systems? Computers use binary because it’s simple (only two states: on/off, 1/0). Humans use decimal because it’s natural (10 fingers). Octal and hexadecimal are used to simplify binary for humans.

Binary Number System (Base-2)

Binary is the language of computers. Each digit is called a bit (binary digit). The value of each bit depends on its position (right to left, starting at 0):

graph LR
    B7["2^7 (128)"] --> B6["2^6 (64)"]
    B6 --> B5["2^5 (32)"]
    B5 --> B4["2^4 (16)"]
    B4 --> B3["2^3 (8)"]
    B3 --> B2["2^2 (4)"]
    B2 --> B1["2^1 (2)"]
    B1 --> B0["2^0 (1)"]

Example: Convert 1011 (binary) to decimal.

• Break it down: $1\times 2^3=8$ $0\times 2^2=0$ $1\times 2^1=2$ $1\times 2^0=1$
• Total: $8+0+2+1=11$ (decimal).

Trace: Convert 11010 to decimal.

1. Write powers of 2: $16,8,4,2,1$.
2. Multiply each bit by its power: $1\times 16=16$ $1\times 8=8$ $0\times 4=0$ $1\times 2=2$ $0\times 1=0$
3. Answer: $16+8+0+2+0=26$.

Decimal to Binary Conversion

Use the division-remainder method:

1. Divide the decimal number by 2.
2. Write the remainder (0 or 1).
3. Repeat until the quotient is 0.
4. Read remainders from bottom to top.

Example: Convert 23 (decimal) to binary.

23 ÷ 2 = 11 (remainder 1)
11 ÷ 2 = 5  (remainder 1)
5  ÷ 2 = 2  (remainder 1)
2  ÷ 2 = 1  (remainder 0)
1  ÷ 2 = 0  (remainder 1)

Read remainders upward: 10111.

Trace: Convert 45 to binary.

45 ÷ 2 = 22 (1)
22 ÷ 2 = 11 (0)
11 ÷ 2 = 5  (1)
5  ÷ 2 = 2  (1)
2  ÷ 2 = 1  (0)
1  ÷ 2 = 0  (1)

Answer: 101101.

Octal (Base-8) Number System

Octal uses digits 0–7. It’s useful because 3 binary digits (bits) = 1 octal digit. This makes binary numbers shorter and easier to read.

Example: Convert 372 (octal) to decimal.

• Break it down: $3\times 8^2=3\times 64=192$ $7\times 8^1=7\times 8=56$ $2\times 8^0=2\times 1=2$
• Total: $192+56+2=250$ (decimal).

Trace: Convert 506 (octal) to decimal.

1. $5\times 64=320$
2. $0\times 8=0$
3. $6\times 1=6$ Answer: $320+0+6=326$.

Hexadecimal (Base-16) Number System

Hexadecimal uses digits 0–9 and letters A–F (where A=10, B=11, ..., F=15). It’s even more compact than octal: 4 binary digits = 1 hex digit.

Example: Convert 1A3 (hex) to decimal.

• Break it down: $1\times 16^2=256$ $A(10)\times 16^1=160$ $3\times 16^0=3$
• Total: $256+160+3=419$ (decimal).

Trace: Convert FF (hex) to decimal.

1. $F(15)\times 16^1=240$
2. $F(15)\times 16^0=15$ Answer: $240+15=255$.

Conversion Between Binary, Octal, and Hexadecimal

Binary to Octal

Group binary digits into sets of 3 (from right to left). If needed, pad with leading zeros. Convert each group to octal.

Example: Convert 11010110 (binary) to octal.

1. Group: 001 101 011 0 (pad with one zero).
2. Convert each group: 001 = 1 101 = 5 011 = 3 000 = 0
3. Answer: 1530 (octal).

Trace: Convert 1011001 to octal.

1. Group: 001 011 001 (pad with two zeros).
2. Convert: 001 = 1 011 = 3 001 = 1 Answer: 131 (octal).

Binary to Hexadecimal

Group binary digits into sets of 4 (from right to left). Pad with leading zeros if needed. Convert each group to hex.

Example: Convert 11010110 (binary) to hex.

1. Group: 1101 0110.
2. Convert each group: 1101 = D 0110 = 6
3. Answer: D6 (hex).

Trace: Convert 101100101 to hex.

1. Group: 0001 0110 0101 (pad with three zeros).
2. Convert: 0001 = 1 0110 = 6 0101 = 5 Answer: 165 (hex).

Binary Arithmetic

sequenceDiagram
    participant A as Binary Addition
    participant B as 1011
    participant C as 1101
    participant D as Result
    A->>B: Addend 1
    A->>C: Addend 2
    loop Carry
        A->>D: 1 (carry)
    end
    A->>D: 11000
    note right of D: 1011 + 1101 = 11000 (27 in decimal)

Binary Addition

Add binary digits like decimal, but remember:

• $0+0=0$
• $0+1=1$
• $1+1=10$ (write 0, carry 1)
• $1+1+1=11$ (write 1, carry 1)

Example: Add 1011 + 1101.

  1011
+ 1101
-------
 11000

Trace: Add 1101 + 0110.

  1101
+ 0110
-------
 10011

Binary Subtraction

Subtract binary digits, borrowing as needed:

• $0-0=0$
• $1-0=1$
• $1-1=0$
• $0-1=1$ (borrow 1 from the next left digit)

Example: Subtract 1011 - 0110.

  1011
- 0110
-------
  0101

Trace: Subtract 1101 - 0011.

  1101
- 0011
-------
  1010

Binary Multiplication

Multiply like decimal, but use binary rules:

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

Example: Multiply 101 × 11.

    101
   ×  11
   -----
    101   (101 × 1)
   101    (101 × 1, shifted left)
   -----
  10011

Trace: Multiply 110 × 101.

    110
   × 101
   -----
    110   (110 × 1)
   000    (110 × 0, shifted left)
  110     (110 × 1, shifted left twice)
  -----
 100110

Comparison Table: Number Systems

Applications of Number Systems

1. Binary in Computers:

   • Computers store data as binary (e.g., 10101010 for text or images).
   • Example: IP addresses like 192.168.1.1 are binary at the hardware level.

2. Hexadecimal in Programming:

   • Used to represent colors (e.g., #FF0000 = red).
   • Memory addresses in programming (e.g., 0x0040).

3. Octal in Legacy Systems:

   • Older computers (e.g., PDP-8) used octal for simplicity.

Common Mistakes to Avoid

1. Forgetting to pad zeros when converting binary to octal/hex.

   • Wrong: 10110 → 101 10 (missing a zero).
   • Correct: 010 110 → 26 (octal).

2. Misaligning bits when grouping for octal/hex conversion.

   • Always start grouping from the right.

3. Incorrect carry handling in binary addition/subtraction.

   • Practice with small numbers first.

NEB Board-Style Questions

Short Answer (2 marks)

1. Convert 101101 (binary) to decimal. Answer: $32+8+4+1=45$.

2. Write 47 (decimal) in binary. Answer: 101111.

3. What is the hexadecimal equivalent of 11011010 (binary)? Answer: DA.

Long Answer (5 marks)

4. Convert 3A7 (hex) to binary and then to octal.

   • Step 1: Hex to binary: 3 = 0011, A = 1010, 7 = 0111 → 001110100111.
   • Step 2: Binary to octal: Group: 001 110 100 111 → 1647 (octal).

5. Perform the following binary operations: a) 1101 + 1011 Answer: 11000. b) 10110 - 1101 Answer: 01001. c) 1101 × 101 Answer: 1000111.

Exam Tip

1. Memorize conversion shortcuts:

   • Binary to octal: Group 3 bits.
   • Binary to hex: Group 4 bits.

2. Practice binary arithmetic:

   • NEB often tests addition/subtraction. Write steps clearly.

3. Watch units:

   • $2^4=16$ (hex), $8^2=64$ (octal). Know powers of 2, 8, and 16.

4. Real-world connections:

   • IP addresses (e.g., 192.168.1.1) are binary at the core.
   • Colors in web design use hex (e.g., #00FF00 = green).

5. Time management:

   • If stuck on a conversion, move to the next question and return later.

Summary Checklist

Before the exam, ensure you can: ✅ Convert decimal ↔ binary using division-remainder. ✅ Convert binary ↔ octal/hex using grouping. ✅ Perform binary addition, subtraction, and multiplication. ✅ Recognize applications (IP addresses, colors, memory). ✅ Avoid common mistakes (padding, carry handling).