Number System Conversions

Being able to fluently convert between binary, decimal, octal, and hexadecimal is an essential skill in digital electronics. This guide covers all conversion paths with detailed methods.

Conversion Map

1. Decimal to Binary

Method: Repeated Division by 2

Convert 156₁₀ to binary:

Fractional Part: Repeated Multiplication by 2

Convert 0.3125₁₀ to binary:

2. Binary to Decimal

Method: Positional Weight Addition

Convert 11010110₂ to decimal:

3. Decimal to Octal

Method: Repeated Division by 8

Convert 500₁₀ to octal:

4. Decimal to Hexadecimal

Method: Repeated Division by 16

Convert 1000₁₀ to hex:

5. Binary to Octal (Direct)

Group binary bits in 3s from right:

Convert 110101011₂ to octal:

6. Binary to Hexadecimal (Direct)

Group binary bits in 4s from right:

Convert 110101011110₂ to hex:

7. Octal to Hexadecimal

Convert through binary:

Example: Convert 372₈ to hexadecimal

8. Hexadecimal to Octal

Example: Convert 2AC₁₆ to octal

Quick Conversion Table

Conversion with Fractional Numbers

Example: Convert 25.75₁₀ to all bases

Common Mistakes to Avoid

1. Grouping direction: Always group from the decimal point outward
2. Padding: Add leading/trailing zeros when groups are incomplete
3. Hex letters: Remember A=10, B=11, C=12, D=13, E=14, F=15
4. Octal digits: Never use 8 or 9 in octal (max digit is 7)
5. Verification: Always verify by converting back to decimal

Interview Questions

Q1: What is the fastest way to convert between hex and octal? Convert through binary as an intermediate. Hex→Binary (4-bit groups)→Octal (3-bit groups) or vice versa. This avoids decimal arithmetic entirely and is a mechanical process of expanding and regrouping bits.

Q2: Convert 0.1₁₀ to binary. What do you observe? 0.1×2=0.2→0; 0.2×2=0.4→0; 0.4×2=0.8→0; 0.8×2=1.6→1; 0.6×2=1.2→1; 0.2×2=0.4→0; (repeats). Result: 0.000110011001100... It's a non-terminating repeating binary fraction. This is why floating-point representation of 0.1 introduces rounding errors in computers.

Q3: How many hex digits are needed to represent an n-bit binary number? Ceil(n/4) hex digits. For example, a 32-bit number needs exactly 8 hex digits, a 10-bit number needs 3 hex digits (with the leading digit using only 2 of its 4 bits).

Q4: Convert 777₈ to hexadecimal. 777₈ → 111 111 111₂ → 0001 1111 1111₂ → 1FF₁₆. Alternatively, 777₈ = 7×64+7×8+7 = 511₁₀ = 1×256+15×16+15 = 1FF₁₆.

Q5: Why can some decimal fractions not be represented exactly in binary? A decimal fraction terminates in binary only if its denominator (in lowest terms) has only 2 as a prime factor. For example, 0.5=1/2 terminates, 0.25=1/4 terminates, but 0.1=1/10 has 5 as a factor in the denominator, so it cannot be represented exactly in binary.