Decimal fractions, such as 0.1, present challenges in base 2 representation
Imagine trying to represent the decimal number 0.1 in binary form. This value is equivalent to 1/10. However, when divided by 10 in base-2, the result becomes 0.000110011001100..., displaying a repeating pattern of decimals.
In base-10, accurately expressing numbers like 0.1 is straightforward, but in base-2, exact representation of fractional values based on tenths is impossible. Instead, approximations are required using the available storage capacity for bits.
For instance, if only the first 8 significant binary digits were stored for 0.1, the resulting approximation would be 11001100 (with an exponent of 11). This translates back to 0.000110011 in binary, which computes to 0.099609375 in decimal – not exactly 0.1. This discrepancy signifies the error introduced when converting 0.1 to a floating-point variable with a base of 8 bits, excluding the sign bit.
The mechanics behind storing numerical values in floating-point variables
The IEEE 754 standard outlines a methodology for encoding real numbers into binary format, incorporating a sign and a binary exponent. The application of the exponent occurs within the binary realm, post conversion from decimal to binary.
Varying sizes of IEEE floating-point numbers dictate the allocation of binary digits for the base number and the exponent respectively.
The mismatch seen in computations like 0.1 + 0.2 != 0.3 stems from operating on approximations of these numbers in binary form rather than their precise decimal equivalents. Upon reconverting the outcomes to decimal form, they deviate from the intended values due to this inherent imprecision. Moreover, the resultant sum does not align perfectly with the binary representation of 0.3 either, with the degree of deviation contingent on the precision level set by the floating-point size.
Rounding: a potential remedy, though ineffective in certain scenarios
In circumstances where computational inaccuracies arise from precision loss in binary conversions, rounding can address minor discrepancies during the return transition to decimal notation, fully concealing any errors.
Within the context of 0.1 + 0.2 compared to 0.3, however, rounding fails to rectify the disparity. The addition of the binary estimates of 0.1 and 0.2 results in a figure distinct from the binary approximation of 0.3.