Even when the absolute error is small in value, it can lead to significant consequences depending on the context in which the number is used. For instance, in systems that handle extremely precise measurements—such as in satellite navigation, financial transactions, or medical devices—a small absolute error can disrupt accuracy, trigger incorrect decisions, or cause a chain of faulty computations. A tiny absolute error in a time-critical system could cause synchronisation problems, or in a banking system could accumulate over millions of transactions to create major discrepancies. Additionally, many algorithms use approximated results in further calculations, so even a minor error in an early stage can propagate and magnify in later computations. This is known as error propagation. The issue isn’t just the size of the error but the context in which it occurs and the sensitivity of the system to even small inaccuracies. Hence, all errors, even small ones, must be considered seriously.

Yes, relative error can exceed 100% when the absolute error is greater than the true value. This can happen when the approximated value is significantly wrong—such as storing a value as 0.2 when the actual value is 0.05. The absolute error in this case is |0.05 − 0.2| = 0.15. Dividing this by the true value gives 0.15 ÷ 0.05 = 3.0, which is 300%. In binary representation, this can occur due to extreme rounding errors, underflow, or using too few bits for representing small fractions. A relative error above 100% indicates that the stored value is not only inaccurate but also misleading, potentially rendering any calculations based on it invalid. This is particularly problematic in floating-point representations where very small numbers are approximated with few bits, and rounding errors dominate the stored result. A relative error of over 100% essentially means the approximation has lost almost all fidelity to the true value.

Binary systems often use rounding methods like truncation (cutting off extra bits) or rounding to nearest (adjusting the last bit based on remaining bits). Truncation tends to introduce a consistent bias, as it always underestimates or overestimates depending on the direction of truncation. This results in a larger absolute error on average, especially for recurring binary fractions. Rounding to the nearest, on the other hand, aims to minimise the absolute error by balancing over- and under-estimation, but it can still result in small inaccuracies. In terms of relative error, the impact of rounding depends on the scale of the true value. For small numbers, even a small rounding change can result in a large relative error. In contrast, for large numbers, the same change may have minimal relative impact. Therefore, the choice of rounding method affects not just how large the error is but also how it behaves across different magnitudes, influencing both accuracy and consistency.

Focusing solely on either absolute or relative error can give a misleading picture of accuracy. Absolute error tells you the exact size of the difference, which is useful when dealing with consistent units or when a system is sensitive to raw error values (e.g. control systems, sensor readings). However, it doesn’t scale with the size of the number. A 0.1 error may be trivial if the true value is 1000 but disastrous if it’s 0.01. Relative error, on the other hand, helps you understand how significant the error is relative to the size of the true value. It’s useful for comparing errors across different scales or when evaluating the precision of measurements. Using both together allows a complete assessment: absolute error gives raw impact, while relative error gives contextual significance. In binary systems, this dual perspective is essential to decide whether the approximation is acceptable for a given application or whether higher precision is needed.

Several key factors affect how large absolute and relative errors will be when using fixed-point binary systems. First, the number of bits allocated to represent the number significantly influences precision. More bits after the binary point allow for finer granularity, reducing both types of error. Second, the placement of the binary point (also called the scaling factor) affects how many bits are used for the fractional part versus the whole number part. If too few bits are assigned to the fractional part, small decimal values will be poorly represented, increasing absolute and relative errors. Third, the method of conversion from decimal to binary—whether rounding or truncating—is a major source of error. Lastly, the range of values being represented can impact relative error. If very small values are stored with insufficient bits, even a tiny absolute error may result in an enormous relative error. Thus, precision depends on both system design choices and the nature of the data being stored.