**R****ound off error **is the error caused by approximate representation of numbers.

When people talk about round off error, it is the error between the number and its representation. For example 200/3 would be represented as 66.6667 in a six significant digit computer that rounds off the last digit. The last digit has been rounded up from 6 to a 7. The difference between 200/3 and 66.6667, that is, 200/3-66.6667 is the round off error.

If a computer is chopping off as opposed to rounding the last digit, the error caused is *still* called the round off error (caused by chopping). If a computer is using chopping, then for example, 200/3 would be represented as 66.6666 in a six significant digit computer. The difference between 200/3 and 66.6666, that is, 200/3-66.6666 is the round off error.

Where does the myth come from? Because if one is chopping off the number, students think that we are truncating a number, and hence the resulting error should be truncation error. **No! No! That is still round off error. **As a side note, there is something called truncating a number – a number if truncated is just the integer part of the number (example: truncating 20.568 gives 20; truncating 20.03 gives 20).

So what then is truncation error?

**Truncation error **is error caused by truncating a mathematical procedure.

Examples of truncation error abound and include

- In exact differentiation, you need dx approaching zero; in numerical differentiation we can only choose dx=finite.
- In exact integration, one would need infinite number of trapezoids to find the integral; in numerical integration, we can only choose a finite number of trapezoids.
- In the Maclaurin series for transcendental and trigonometric functions, we need infinite number of terms for exact solution; in a numerical solution, we can only choose finite number of terms.

So let us get this straight – **round off error **is caused by representing numbers approximately; **truncation error **is caused by approximating mathematical procedures.

For more details, read the textbook chapter on Sources of Error in Numerical Methods at http://numericalmethods.eng.usf.edu

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://numericalmethods.eng.usf.edu and the textbook on Numerical Methods with Applications available from the lulu storefront.

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