1.2 Number System Conversions

Understanding how to convert numbers between decimal, binary, octal, and hexadecimal systems is a fundamental skill in computer science. Since computers operate primarily in binary, but humans are accustomed to decimal, these conversions are crucial for interpreting data, debugging programs, and working with hardware. This section will guide you through the various methods of converting numbers from one base to another.

1.2.1 Decimal to Other Number Systems

When converting a decimal number to another base (binary, octal, or hexadecimal), the standard method is successive division for integer parts and successive multiplication for fractional parts.

Method (Integers – Successive Division)

1.     Divide the decimal number by the target base.

2.     Record the remainder (this becomes a digit in the new base).

3.     Divide the quotient by the target base again.

4.     Repeat steps 2 and 3 until the quotient is 0.

5.     Read the remainders from bottom to top (last remainder = Most Significant Digit, first remainder = Least Significant Digit).

Example 1: Convert Decimal 25 to Binary

Target base = 2

Division	Quotient	Remainder
25 ÷ 2	12	1
12 ÷ 2	6	0
6 ÷ 2	3	0
3 ÷ 2	1	1
1 ÷ 2	0	1

Reading from bottom to top: 11001₂
So, 25₁₀ = 11001₂

Example 2: Convert Decimal 179 to Hexadecimal

Target base = 16 (A=10, B=11, …, F=15)

Division	Quotient	Remainder	Hexadecimal
179 ÷ 16	11	3	3
11 ÷ 16	0	11	B

Reading from bottom to top: B3₁₆
So, 179₁₀ = B3₁₆

1.2.2 Binary to Other Number Systems

Binary can be converted to decimal using positional notation or to octal/hexadecimal using bit grouping.

Method 1: Positional Notation (Binary → Decimal)

1.     Start from the rightmost digit.

2.     Multiply each digit by 2ⁿ, where n is its position (starting from 0).

3.     Sum all the products.

Method 2: Grouping Bits (Binary → Octal/Hexadecimal)

·         Binary to Octal: Group digits in sets of 3 (from right to left). Pad with leading zeros if needed. Convert each group to octal.

·         Binary to Hexadecimal: Group digits in sets of 4 (from right to left). Pad with leading zeros if needed. Convert each group to hexadecimal.

Example 1: Convert Binary 10110 to Decimal

Binary: 10110₂

·         0 × 2⁰ = 0

·         1 × 2¹ = 2

·         1 × 2² = 4

·         0 × 2³ = 0

·         1 × 2⁴ = 16

Sum: 0 + 2 + 4 + 0 + 16 = 22
So, 10110₂ = 22₁₀

Example 2: Convert Binary 1101101110 to Hexadecimal

Binary: 1101101110₂

Group into fours: 0011 0110 1110

·         0011 → 3

·         0110 → 6

·         1110 → E

Combined: 36E₁₆
So, 1101101110₂ = 36E₁₆

1.2.3 Octal to Other Number Systems

Octal is easily converted to decimal using positional notation and to binary using direct digit mapping.

Method 1: Positional Notation (Octal → Decimal)

1.     Start from the rightmost digit.

2.     Multiply each digit by 8ⁿ, where n is its position (starting from 0).

3.     Sum all the products.

Method 2: Direct Conversion (Octal → Binary)

1.     Convert each octal digit into its 3-bit binary equivalent.

2.     Combine the groups.

Example 1: Convert Octal 73 to Decimal

·         3 × 8⁰ = 3

·         7 × 8¹ = 56

Sum: 3 + 56 = 59
So, 73₈ = 59₁₀

Example 2: Convert Octal 504 to Binary

·         5 → 101

·         0 → 000

·         4 → 100

Combined: 101000100₂
So, 504₈ = 101000100₂

1.2.4 Hexadecimal to Other Number Systems

Hexadecimal is commonly converted to decimal via positional notation and to binary via direct digit mapping (4 bits per digit).

Method 1: Positional Notation (Hex → Decimal)

1.     Start from the rightmost digit.

2.     Multiply each digit (A=10, B=11, etc.) by 16ⁿ, where n starts at 0.

3.     Sum all the products.

Method 2: Direct Conversion (Hex → Binary)

1.     Convert each hexadecimal digit into its 4-bit binary equivalent.

2.     Combine the groups.

Example 1: Convert Hexadecimal A5 to Decimal

·         5 × 16⁰ = 5

·         A × 16¹ = 10 × 16 = 160

Sum: 5 + 160 = 165
So, A5₁₆ = 165₁₀

Example 2: Convert Hexadecimal 1F8 to Binary

·         1 → 0001

·         F → 1111

·         8 → 1000

Combined: 000111111000₂
So, 1F8₁₆ = 111111000₂