Binary Mantissa Exponent Calculator
Calculate the binary mantissa or significand and base-2 exponent from binary or decimal input. View normalized notation, decimal equivalent, point movement, exponent value, binary precision and step-by-step representation details.
Binary to Mantissa & Exponent
Enter a binary integer or fraction to calculate its normalized significand and exponent.
Decimal to Binary Mantissa & Exponent
Convert a decimal value into normalized binary significand and exponent notation.
What Is a Binary Mantissa Exponent Calculator?
A binary mantissa exponent calculator separates a binary floating-point value into its significant binary digits and a base-2 exponent. The number can then be expressed in binary scientific notation such as 1.10101 x 2^3.
The significant portion is commonly called the significand. The word mantissa is also widely used in calculators, textbooks and search queries, although significand is the more precise term for floating-point representation.
The exponent determines the power of two required to scale the significand back to the original numeric value.
Binary Mantissa and Exponent Formula
Normalized Binary Value
A nonzero binary floating-point value can be expressed using a normalized significand and a base-2 exponent.
Recover Original Value
Convert the binary significand to decimal and multiply it by the power-of-two scale.
Mantissa, Significand and Exponent Explained
Mantissa
Mantissa is a commonly used term for the significant digits of a floating-point number. In modern floating-point terminology, significand is generally preferred.
Significand
The significand contains the meaningful binary digits. A normalized base-2 value normally has one leading binary 1 before the point.
Exponent
The exponent determines how the significand is scaled by a power of two.
Base
Binary floating-point uses base 2, so the scaling factor is always a power of two rather than a power of ten.
Binary Mantissa and Exponent Examples
| Binary Value | Significand | Exponent | Normalized Form | Decimal |
|---|---|---|---|---|
| 1 | 1 | 0 | 1 x 2^0 | 1 |
| 10 | 1 | 1 | 1 x 2^1 | 2 |
| 101 | 1.01 | 2 | 1.01 x 2^2 | 5 |
| 101.1 | 1.011 | 2 | 1.011 x 2^2 | 5.5 |
| 1101.01 | 1.10101 | 3 | 1.10101 x 2^3 | 13.25 |
| 0.1 | 1 | -1 | 1 x 2^-1 | 0.5 |
| 0.101 | 1.01 | -1 | 1.01 x 2^-1 | 0.625 |
| 0.00101 | 1.01 | -3 | 1.01 x 2^-3 | 0.15625 |
Worked Example 1: Find Mantissa and Exponent of 1101.01
Worked Example 2: Binary Fraction Smaller Than 1
Worked Example 3: Decimal 13.25
Worked Example 4: Negative Binary Value
Decimal Values That Need Binary Approximation
Some decimal fractions cannot be represented using a finite number of binary fractional digits. In these cases, the selected precision determines how many binary digits are retained.
| Decimal | Binary Behavior | Exact Finite Binary? |
|---|---|---|
| 0.5 | 0.1 | Yes |
| 0.25 | 0.01 | Yes |
| 0.125 | 0.001 | Yes |
| 0.75 | 0.11 | Yes |
| 0.1 | Repeating binary fraction | No |
| 0.2 | Repeating binary fraction | No |
Binary Exponent Reference Table
| Exponent | Power of Two | Decimal Scale |
|---|---|---|
| -5 | 2^-5 | 0.03125 |
| -4 | 2^-4 | 0.0625 |
| -3 | 2^-3 | 0.125 |
| -2 | 2^-2 | 0.25 |
| -1 | 2^-1 | 0.5 |
| 0 | 2^0 | 1 |
| 1 | 2^1 | 2 |
| 2 | 2^2 | 4 |
| 3 | 2^3 | 8 |
| 4 | 2^4 | 16 |
| 5 | 2^5 | 32 |
Binary Mantissa vs IEEE 754 Fraction Field
A mathematical binary significand is not exactly the same thing as the stored fraction field in IEEE 754.
| Feature | Mathematical Significand | IEEE 754 Fraction Field |
|---|---|---|
| Example | 1.10101 | 10101000… |
| Leading 1 | Shown explicitly | Usually implicit for normal values |
| Exponent | Ordinary signed exponent | Stored as biased exponent field |
| Purpose | Mathematical notation | Computer bit storage |
Mantissa Calculation vs Binary Normalization
Mantissa / Exponent Calculation
Focuses on identifying the significant binary portion and the power-of-two exponent that together represent the number.
Normalization
Focuses on the process of moving the binary point into the required standard position and tracking that movement.
Where Binary Mantissa and Exponents Are Used
Floating-Point Arithmetic
Floating-point systems represent values by combining significant digits with a scale determined by an exponent.
Computer Architecture
Studying significands and exponents helps explain how floating-point processors and registers represent numbers.
Scientific Computing
Exponent-based representation allows values with very different magnitudes to be handled efficiently.
Computer Science Education
Mantissa and exponent calculations are commonly taught alongside binary scientific notation and IEEE 754.
Important Mantissa and Exponent Notes
This calculator uses ordinary mathematical binary scientific notation. It does not treat the exponent as an IEEE 754 biased exponent.
For decimal values such as 0.1 that have repeating binary expansions, the calculated significand depends on the selected binary precision and is therefore an approximation.