Round-off error, or numerical truncation error, is an error that occurs when an approximation of a number is made during the process of computing. It is an arithmetic error that is caused by a formula that is not exact but is instead rounded off to a certain number of digits or figures that are typically represented by a finite number of bytes.
Round-off error is commonly used in computing when numbers cannot be represented accurately due to a lack of storage space or computational resources. For example, numbers that are stored in a computer memory are typically represented by a finite number of bits, so they can be represented only within a certain range of accuracies. As a result, when a number is rounded off to a certain number of digits, errors, known as round-off errors, accumulate, creating a deviation from the true value of the number.
When round-off errors accumulate, they can be significant in certain computations. For example, when the calculations involve millions of calculations performed in a loop or in other computations, the error can become large enough to skew the calculations in an undesired way.
Fortunately, there are methods that can be used to control or minimize the effects of round-off errors, such as careful checking of the accuracy of the data used and the results obtained. Other approaches include algorithmic strategies, such as adding a small amount of random noise as a stochastic process and using a floating-point representation system that can represent fractions more precisely.
In summary, round-off error or numerical truncation error is an error that occurs when an approximation of a number is made during the process of computing. It often arises due to the limited accuracy of digital representation and can cause errors when precise calculations are required. Fortunately, methods exist that can be used to minimize its effects, allowing much greater control and accuracy in computations.
