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Number Base Conversion — Detailed Notes (Computer Science)

1. Introduction

In computer science, numbers are often represented in different bases.
The most common ones are:

Base Name Digits Typical Use
2 Binary 0–1 Machine operations, logic circuits
8 Octal 0–7 Legacy systems, UNIX permissions
10 Decimal 0–9 Human-readable numbers
16 Hexadecimal 0–9, A–F Memory addresses, color codes, assembly

Understanding base conversion is essential for programming, digital logic, and low-level systems.

2. Converting Between Bases

2.1 Decimal → Binary / Octal / Hexadecimal

Method A: Repeated Division (for integer part)

  1. Divide the number by the target base.

  2. Record the remainder.

  3. Use the quotient for the next division.

  4. Stop when the quotient is 0.

  5. The result is the remainders read in reverse order.

Example: Convert 45₁₀ → Binary

45 ÷ 2 = 22 R 1 22 ÷ 2 = 11 R 0 11 ÷ 2 = 5 R 1 5 ÷ 2 = 2 R 1 2 ÷ 2 = 1 R 0 1 ÷ 2 = 0 R 1

Binary = 101101₂

Method B: Multiply-by-base (for fractional part)

  1. Multiply fraction by the target base.

  2. The integer part becomes the next digit.

  3. Use the fractional part as next input.

  4. Continue until fraction becomes 0 (or precision limit reached).

Example: Convert 0.625₁₀ → Binary

0.625 × 2 = 1.25 → 1 0.25 × 2 = 0.5 → 0 0.5 × 2 = 1.0 → 1

Binary fraction = 0.101₂

2.2 Binary ↔ Octal / Hexadecimal (Fast Grouping Method)

Grouping rules

  • Binary → Octal: group 3 bits from right to left.

  • Binary → Hexadecimal: group 4 bits.

Add leading zeros if necessary.

Example: Binary → Hex

Convert 1101011011₂ to hex:

Group into 4 bits:

0011 0101 1011

Convert each group:

Binary Hex
0011 3
0101 5
1011 B

Hexadecimal result = 35B₁₆

Example: Binary → Octal

1101011011₂ grouped as 3 bits:

1 101 011 011 → 001 101 011 011

Binary Oct
001 1
101 5
011 3
011 3

Octal = 1533₈

2.3 Hexadecimal / Octal → Binary

Simply convert each digit individually:

Hex → Binary Table

Hex Binary
0 0000
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111
8 1000
9 1001
A 1010
B 1011
C 1100
D 1101
E 1110
F 1111

Octal → Binary Table

Oct Binary
0 000
1 001
2 010
3 011
4 100
5 101
6 110
7 111

2.4 Binary / Octal / Hexadecimal → Decimal

Positional Value Expansion

For a number with digits dndn−1...d1d0.d−1d−2...d_n d_{n-1} ... d_1 d_0 . d_{-1} d_{-2} ...dn​dn−1​...d1​d0​.d−1​d−2​...

Decimal value =

∑i=kmdi⋅(base)i\sum_{i=k}^{m} d_i \cdot (\text{base})^ii=k∑m​di​⋅(base)i

Example: Convert 101101₂ → Decimal

1×2⁵ + 0×2⁴ + 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 32 + 0 + 8 + 4 + 0 + 1 = 45

Example: Convert A3₁₆ → Decimal

A = 10 A3₁₆ = 10×16¹ + 3×16⁰ = 160 + 3 = 163

3. Converting Between Non-Decimal Bases (Binary ↔ Octal ↔ Hex)

Rule

Always go through binary when converting between oct and hex.

Example: Convert 57₈ → Hex

  1. 5 → 101
    7 → 111

→ 101111₂

  1. Group into 4 bits: 0010 1111

→ 2F₁₆

4. Two’s Complement for Negative Numbers

Steps to get two’s complement

  1. Write the positive number in binary.

  2. Invert all bits (0 ↔ 1).

  3. Add 1.

Example: Represent −5 in 8-bit two’s complement

  1. +5 = 0000 0101

  2. Invert → 1111 1010

  3. Add 1 → 1111 1011₂

  4. by this way a signed integer can be calculated by the same way as unsigned integers.

5. Common Pitfalls

1. Forgetting to pad zeros

Binary → hex requires 4-bit groups.
Binary → oct requires 3-bit groups.

2. Mixing up fractional conversion

Integer part uses division, fractional uses multiplication.

3. Truncation errors

Some decimals (e.g., 0.1) cannot be represented exactly in binary.

6. Quick Reference Table

Powers of 2 and Conversions

Power Value Approx Decimal
2¹⁰ 1024 ≈ 10³
2⁸ 256 —
2⁴ 16 —
2¹⁶ 65536 —
2³² 4,294,967,296 —

Hex ↔ Binary Shortcuts

Hex Binary
1 0001
2 0010
4 0100
8 1000

How C Determines the Base of Numeric Literals

In C, the base (radix) of an integer literal is determined solely by its textual form. The compiler does not guess based on context or variable type.

1. Decimal literals (base-10)

Rule:

  • Must not start with 0.

  • Digits: 0–9.

Examples:

123 42 2025

2. Octal literals (base-8)

Rule:

  • Start with a leading 0.

  • Digits must be 0–7.

Examples:

0755 012 0007

Invalid:

09 // 9 is not allowed in octal

3. Hexadecimal literals (base-16)

Rule:

  • Start with 0x or 0X.

  • Digits: 0–9, a–f, A–F.

Examples:

0xFF 0X1a3 0x7B

4. Binary literals (base-2)

Binary literals are not supported in older C standards (C89/C99/C11).
They are supported starting from C23, and also by many compilers as extensions (e.g., GCC).

Rule (C23 or compiler extension):

  • Start with 0b or 0B.

  • Digits: 0 or 1.

Examples:

0b1010 0B11001