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Mantissa — Definition, Formula & Examples

Mantissa is the decimal (fractional) part of a common logarithm. For example, if log⁡1050≈1.6990\log_{10} 50 \approx 1.6990, the mantissa is 0.69900.6990.

Given a common logarithm log⁡10x=n+f\log_{10} x = n + f, where nn is an integer (the characteristic) and 0≤f<10 \le f < 1, the mantissa is the non-negative fractional part ff. The mantissa depends only on the sequence of significant digits in xx, not on the position of the decimal point.

Key Formula

mantissa=log⁡10x−⌊log⁡10x⌋\text{mantissa} = \log_{10} x - \lfloor \log_{10} x \rfloor
Where:
  • xx = A positive real number
  • ⌊⋅⌋\lfloor \cdot \rfloor = The floor function, giving the greatest integer less than or equal to the value

How It Works

Every common logarithm can be split into two parts: the characteristic (the integer part) and the mantissa (the decimal part). The characteristic tells you the order of magnitude — essentially how many digits the number has before the decimal point, minus one. The mantissa encodes the specific digits of the number. Because log⁡10(k⋅10n)=n+log⁡10k\log_{10}(k \cdot 10^n) = n + \log_{10} k, numbers like 3.5, 35, and 350 all share the same mantissa; only their characteristics differ. Before calculators, students looked up mantissas in printed log tables to perform multiplication and division.

Worked Example

Problem: Find the characteristic and mantissa of log₁₀ 350.
Step 1: Compute the logarithm.
log⁡10350≈2.5441\log_{10} 350 \approx 2.5441
Step 2: Identify the characteristic (integer part).
characteristic=⌊2.5441⌋=2\text{characteristic} = \lfloor 2.5441 \rfloor = 2
Step 3: Subtract the characteristic to get the mantissa.
mantissa=2.5441−2=0.5441\text{mantissa} = 2.5441 - 2 = 0.5441
Answer: The characteristic is 2 and the mantissa is approximately 0.5441. Notice that log₁₀ 3.5 ≈ 0.5441 has the same mantissa — only the characteristic changes.

Why It Matters

Understanding the mantissa helps you read logarithm tables and grasp how logarithmic scales (like the Richter scale or decibel scale) separate magnitude from precision. In precalculus, splitting a logarithm into characteristic and mantissa builds intuition for properties of logarithms and scientific notation.

Common Mistakes

Mistake: Taking the mantissa of a negative logarithm by just dropping the minus sign. For example, treating log₁₀ 0.035 ≈ −1.4559 as having mantissa 0.4559.
Correction: Apply the floor function correctly. Since ⌊−1.4559⌋ = −2, the characteristic is −2 and the mantissa is −1.4559 − (−2) = 0.5441. The mantissa is always between 0 and 1.