precision-issues

Approximation & rounding

• In computing, real numbers are often stored using floating-point binary, which has a limited number of bits for the mantissa and exponent

• This means not all real numbers can be represented exactly, only approximated

Rounding errors

• Some decimal values cannot be precisely represented in binary

• Example:

Decimal: 0.1
Binary: 0.000110011001100... (recurring)

• Because memory is limited (e.g. 32 or 64 bits), the binary value must be truncated or rounded, which introduces small errors

• Consequence:

  • Calculations using approximated values may accumulate errors

  • Final results may be slightly inaccurate

Underflow

• Underflow occurs when a number is too close to zero to be stored in the available number of bits

• Example:

  • A very small number like 1.2 × 10⁻⁴⁰

  • The exponent is too negative to be stored

  • Result: Stored as 0

• Consequence:

  • Loss of precision

  • Can affect accuracy in scientific or financial calculations

Overflow

• Overflow occurs when a number is too large to fit in the allocated bits for the exponent or mantissa

• Example:

  • A calculation produces 1.2 × 10⁵⁰, but the maximum representable number is 1.2 × 10³⁸

• Consequence:

  • Program may crash or return an infinity, error, or wraparound value

  • Can be especially dangerous in loops or financial applications

Summary table