Decimal ToolsFree Interactive Tool

convert decimal fraction to binary

Convert continuous decimal fractions (e.g. 0.625) into binary fractions using repeated multiplication by 2. Designed for software developers, electrical engineers, students, and computer architecture researchers requiring deterministic client-side accuracy.

Engine: Client-Side Verified (0ms Latency)

Decimal Fraction

Enter a decimal fractional value between 0 and 1.

Calculated Output

Primary Representation

Calculating...

Step-by-Step Mathematical Proof

Active Derivation

Ready. Enter input above to generate derivation.

Concept

What Is convert decimal fraction to binary?

Converting a decimal fraction to binary is the mathematical technique of representing non-integer quantities in base 2. In base 10, fractional positions represent tenths, hundredths, and thousandths (10^-1, 10^-2, 10^-3). In base 2, fractional positions represent halves, quarters, eighths, and sixteenths (2^-1 = 0.5, 2^-2 = 0.25, 2^-3 = 0.125, 2^-4 = 0.0625).

Methodology

How Does It Work?

Multiply the fractional part by 2 repeatedly. The integer part of the product (which will be 0 or 1) becomes the next binary bit to the right of the radix point. Discard the integer part and repeat with the remaining fraction until the fraction reaches 0 or begins repeating infinitely.

Formula & Rules

Mathematical Algorithm

F_{k+1} = \text{frac}(F_k \times 2), \quad b_{-k} = \lfloor F_k \times 2 \rfloor

Worked Problem

Step-by-Step Example

Convert 0.625_{10} to binary: Step 1: 0.625 × 2 = 1.25 -> record 1, remainder 0.25 Step 2: 0.25 × 2 = 0.50 -> record 0, remainder 0.50 Step 3: 0.50 × 2 = 1.00 -> record 1, remainder 0.00 Fraction reached 0.00! Reading bits from top to bottom yields: 0.101_2.

Important Rules & Edge Cases

Read the generated bits in normal top-to-bottom order (unlike integer division which is read bottom-to-top).
If the fraction does not terminate after repeated multiplications, it is an infinite repeating binary fraction.
Input must be a valid real number with a decimal point.

Practical Applications in Engineering

Synthesizing floating-point numbers in compiler lexers and parsers.
Configuring digital-to-analog converters (DACs) and pulse-width modulation (PWM) duty cycles.
Understanding floating-point arithmetic precision limits in financial algorithms.

Common Mistakes to Avoid

Caution: Reading the binary bits bottom-to-top instead of top-to-bottom.
Caution: Expecting numbers like 0.1 or 0.2 to terminate in binary (they produce infinite periodic sequences in base 2).

FAQ

Frequently Asked Questions

Why does 0.1 decimal repeat infinitely in binary?

A fraction terminates in base B only if all prime factors of its reduced denominator divide base B. Decimal base 10 has prime factors 2 and 5. Binary base 2 only has prime factor 2. Because 5 does not divide 2, any fraction with a factor of 5 in its denominator (like 1/10 = 0.1) produces an infinite repeating binary fraction (0.00011001100...).

What is 0.75 in binary?

0.75 in binary is 0.11: (1 × 0.5) + (1 × 0.25) = 0.75.