Fraction Decimal to Binary Converter: Fixed-Point & IEEE 754 Tables

If you are searching for a fraction decimal to binary converter, the exact binary representation of the common fraction 5/8 (decimal 0.625) is 0.101. Unlike whole numbers, fractional conversion relies on successive multiplication by two rather than division. The core formula is: Fraction × 2 = Integer + New Fraction. Substituting our anchor value: 0.625 × 2 = 1.25 (record 1), 0.25 × 2 = 0.5 (record 0), and 0.5 × 2 = 1.0 (record 1), yielding 0.101. However, in embedded systems and power electronics, the assumption that fixes the answer is your bit-depth and per-unit (PU) scaling base. A pure mathematical conversion is only half the battle; mapping that fraction to a microcontroller register requires strict adherence to fixed-point or IEEE 754 floating-point standards.

The Core Algorithm and Neighboring Values

The manual algorithm for converting a fractional decimal to binary is straightforward: multiply the fractional part by 2. If the result is 1 or greater, record a 1 and subtract 1 from the result. If the result is less than 1, record a 0. Repeat this process until the fractional part reaches zero or you hit your memory limit.

When designing digital filters or PWM duty cycles, you rarely work with isolated numbers. Below is a table of neighboring values within a ±20% range of our anchor (0.500 to 0.750), showing how the binary bitstream shifts as the decimal value increments by 1/16ths.

Hardware Scaling: How System Voltage Shifts the Binary Register

In power electronics DSP programming (such as using a TI C2000 microcontroller for motor drives), you aren't just converting pure math; you are mapping physical electrical values to binary comparator registers. The assumption that fixes the answer here is your ADC reference voltage and per-unit (PU) base scaling.

How the answer shifts for 120V vs 230V vs 3-phase systems: If you are programming a digital comparator threshold for a 0.5 (50%) fault limit, the binary register value changes drastically based on the system topology:

• 120V Single-Phase: With a base peak voltage of ~170V, a 50% threshold is 85V. Assuming a 12-bit ADC (0-4095 range) scaled to 200V max, the binary register is 011010110000.
• 230V Single-Phase: The base peak shifts to ~325V. A 50% threshold is now 162.5V. The binary register shifts to 110011011000 to maintain the same physical percentage.
• 400V 3-Phase: The DC link bus sits around 565V peak. To prevent integer overflow while maintaining precision across the wider voltage swing, engineers abandon 12-bit fixed-point and shift to 32-bit IEEE 754 floating-point registers, where 0.5 is universally stored as 0x3F000000 regardless of the physical voltage base.

For a deep dive into how DSPs handle these shifts, refer to the IEEE 754 Standard for Floating-Point Arithmetic, which governs how these physical scalings are normalized in silicon.

When Fractional Conversion Becomes Meaningless

A fraction decimal to binary conversion is mathematically meaningless when dealing with denominators that are not powers of two (e.g., 1/3, 1/5, 1/10) unless a precision limit is explicitly defined.

Take the decimal 0.1 (1/10). In binary, this evaluates to an infinitely repeating sequence: 0.00011001100110011... If you ask a converter for 'the' binary of 0.1 without specifying a bit-depth, the question is flawed. In finite memory, you must truncate or round.

For embedded systems, always use fixed-point math (e.g., Q15 format) or integer scaling (measuring in millivolts instead of volts) when dealing with repeating binary fractions to avoid cumulative rounding errors in PID control loops. The All About Circuits digital logic textbook provides excellent foundational reading on why base-2 struggles with base-10 human conventions.