Binary ToolsFree Interactive Tool

Binary Floating-Point Converter

Dissect real decimal numbers into binary floating-point scientific notation (significand and exponent). Designed for software developers, electrical engineers, students, and computer architecture researchers requiring deterministic client-side accuracy.

Engine: Client-Side Verified (0ms Latency)

Real Number

Enter any real decimal number with optional decimal point.

Calculated Output

Primary Representation

Calculating...

Step-by-Step Mathematical Proof

Active Derivation

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Concept

What Is Binary Floating-Point Converter?

Binary floating-point representation expresses real numbers in normalized binary scientific notation: (-1)^sign × 1.fraction × 2^exponent.

Methodology

How Does It Work?

Convert the integer part to binary. Convert the fractional part by repeated multiplication by 2. Normalize the combined binary string so exactly one leading 1 appears before the binary point, and adjust the power of 2 accordingly.

Formula & Rules

Mathematical Algorithm

N = (-1)^S \times (1.M)_2 \times 2^E

Worked Problem

Step-by-Step Example

Convert 13.625 to binary scientific notation: 13 = 1101_2 0.625 = 0.101_2 (0.625×2=1.25 -> 1, 0.25×2=0.5 -> 0, 0.5×2=1.0 -> 1) Combined: 1101.101_2 Normalize: 1.101101_2 × 2^3 Sign = +, Exponent = 3, Mantissa = 101101.

Important Rules & Edge Cases

Normalized numbers always have an implicit leading 1 before the binary point.
The exponent shifts the binary point left (negative) or right (positive).