The mantissa (also known as the significand or coefficient) of a floating-point number represents the significant digits of that number. It's essentially the number's value without regard to the exponent or sign.
Understanding the Mantissa
The mantissa is a key component of how floating-point numbers are represented in computers, using scientific notation (in binary form). Let's break down what that means:
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Scientific Notation: Think of scientific notation in base-10 (decimal), where you have a number like 6.022 x 1023. Here, 6.022 is the mantissa, and 23 is the exponent.
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Floating-Point Representation: In binary, a floating-point number is represented as:
sign * mantissa * base<sup>exponent</sup>- Sign: Indicates whether the number is positive or negative.
- Mantissa: The significant digits of the number. It is typically normalized to have a single non-zero digit to the left of the radix point (the binary point in this case).
- Base: In the context of computers, the base is typically 2.
- Exponent: Indicates the power to which the base is raised.
Example
Let's consider a simple decimal example to illustrate the concept before diving into binary:
Suppose we want to represent the number 123.45 in a floating-point-like format. We could represent it as:
- 2345 x 102
In this simplified example, 1.2345 would be the mantissa (significand), and 2 would be the exponent.
Now, let's consider how the mantissa appears in the binary representation of a floating-point number.
Mantissa in IEEE 754 Standard
The most common standard for representing floating-point numbers is the IEEE 754 standard. This standard defines how the sign, mantissa, and exponent are stored in memory. For example, in single-precision (32-bit) floating-point format:
- Sign Bit: 1 bit
- Exponent: 8 bits (biased)
- Mantissa: 23 bits (plus an implicit leading 1)
The "implicit leading 1" is a clever trick. Since normalized binary numbers always have a leading 1 before the binary point (e.g., 1.xxxx), the IEEE 754 standard doesn't explicitly store this leading 1. This gives you an extra bit of precision. The 23 bits stored are thus the bits after the binary point.
Therefore, the mantissa effectively holds 24 bits of precision (1 implicit + 23 stored).
Importance of the Mantissa
The mantissa directly affects the precision of the floating-point number. The more bits allocated to the mantissa, the more accurate the representation can be. Limited precision can result in rounding errors and other numerical inaccuracies, especially when dealing with very large or very small numbers.
Summary
The mantissa is the part of a floating-point number that contains its significant digits, without regard to the exponent or sign. It directly determines the precision of the floating-point representation.
