mantissa

Thân thiện

A student writes the mantissa of a logarithm in a math notebook.

Definition

Noun: The decimal part of a common logarithm (logarithm base 10). It is the part of the logarithm that follows the decimal point and represents the significant digits of the original number, independent of its magnitude or order of scale. The mantissa is always a positive number.

Usage

The mantissa is used in mathematics, specifically in logarithmic calculations and pre-calculator computational methods. It is paired with the characteristic (the integer part of the logarithm). The mantissa defines the specific sequence of digits in a number, while the characteristic determines the number's scale or power of ten.

Examples

• In the common logarithm log(643) ≈ 2.808, the mantissa is .808.
• For the number 0.00643, log(0.00643) ≈ -2.192 = -3 + 0.808. Here, the mantissa is still .808, while the characteristic is -3.
• The logarithms of 64.3, 6.43, and 6430 all share the same mantissa (approximately .808) but have different characteristics.

Advanced Usage

• Historical Computational Use: Logarithm tables were used to look up the mantissa for a given sequence of significant digits. The user would combine this mantissa with a separately calculated characteristic to find the full logarithm.
• In Computing: The term is analogous to the significand or fraction in the IEEE floating-point representation of numbers in computer science, though technical definitions can differ.

Variants and Related Words

• Characteristic (noun): The integer part (which can be positive, negative, or zero) of a common logarithm. Together, the characteristic and mantissa compose the logarithm.
• Significand (noun): In floating-point arithmetic and scientific notation, the significant digits of a number (e.g., in 6.43 × 10², the significand is 6.43). This is a related but distinct concept.

Synonyms

• Fractional part (of a logarithm)
• Decimal part (of a logarithm)

Notes on Meaning

The term "mantissa" is specifically and exclusively used in the context of common logarithms (base 10). It is not correctly applied to natural logarithms (base e). Its defining property is that it is always positive, even when the logarithm itself or its characteristic is negative.

Học thuật

Noun

1. the positive fractional part of the representation of a logarithm; in the expression log 643 = 2.808 the mantissa is .808

Từ đồng nghĩa

• fixed-point part