FließkommaArithmetik

FließkommaArithmetik refers to the arithmetic operations performed on floating-point numbers. These numbers are a way to represent real numbers that can be very large or very small, using a fixed number of digits. Unlike integer arithmetic, floating-point arithmetic is not always exact. This is due to the finite precision with which floating-point numbers are stored in computers.

The most common standard for floating-point representation is IEEE 754. This standard defines formats for single-precision (32-bit) and double-precision (64-bit) numbers, as well as rules for arithmetic operations, rounding, and handling special values like infinity and NaN (Not a Number).

When performing calculations with floating-point numbers, several phenomena can occur. Rounding errors are inherent because many decimal fractions cannot be represented exactly in binary. These small errors can accumulate over a series of calculations, potentially leading to significant deviations from the true result. Another issue is precision loss, which can happen when subtracting two nearly equal numbers.

Overflow and underflow are also concerns. Overflow occurs when a calculation results in a number too large to be represented by the available bits, leading to an infinite value. Underflow happens when a number is too small to be represented, often resulting in zero.

Despite these limitations, floating-point arithmetic is essential for a vast range of scientific, engineering, and financial computations where the range and approximate nature of real numbers are more important than absolute exactness. Understanding its characteristics is crucial for writing reliable and accurate numerical software.