Why is 0.1 not represented exactly in floating-point arithmetic, while 0.5 is?

Floating-Point Precision: Why 0.5 is Exact, but 0.1 Isn't

Floating-point arithmetic, while efficient for many computations, faces inherent limitations in precisely representing all decimal numbers. This article explores why 0.5 enjoys exact representation while 0.1 does not.

Understanding the Representation of 0.1

The IEEE 754 standard dictates how floating-point numbers are stored. Let's examine 0.1's representation:

<code>s eeeeeeee mmmmmmmmmmmmmmmmmmmmmmm
0 01111011 10011001100110011001101</code>

• Sign (s): 0 (positive).
• Exponent (eeeee): 123. Subtracting the bias (127), we get an effective exponent of -4.
• Mantissa (mmmmmmmmmmmmmmmmmmmmmm): Interpreted as a binary fraction with an implicit leading "1." The sum of the mantissa bits yields approximately 1.60000002384185791015625.

Multiplying the mantissa by 2^-4 gives roughly 0.100000001490116119384765625. This is a close approximation of 0.1, but not an exact match. The inherent limitations of binary representation prevent a perfect conversion from the decimal 0.1.

The Precise Representation of 0.5

In contrast, 0.5 has a straightforward representation:

<code>s eeeeeeee mmmmmmmmmmmmmmmmmmmmmmm
0 01111110 00000000000000000000000</code>

• Sign (s): 0 (positive).
• Exponent (eeeee): 126. The effective exponent (after bias adjustment) is -1.
• Mantissa (mmmmmmmmmmmmmmmmmmmmmm): 0.

This translates to 1 * 2^-1 = 0.5, a perfect representation.

Conclusion: The Limits of Binary Precision

The difference lies in the ability of the binary system to represent certain decimal fractions exactly. While 0.5 (1/2) is a power of two and thus easily represented in binary, 0.1 (1/10) is not. This results in a slight rounding error when storing 0.1 in floating-point format, leading to the approximation observed. Programmers must be mindful of this inherent limitation when working with floating-point numbers and precision-sensitive applications.

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