**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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### Autar Kaw

Autar Kaw (http://autarkaw.com) is a Professor of Mechanical Engineering at the University of South Florida. He has been at USF since 1987, the same year in which he received his Ph. D. in Engineering Mechanics from Clemson University. He is a recipient of the 2012 U.S. Professor of the Year Award. With major funding from NSF, he is the principal and managing contributor in developing the multiple award-winning online open courseware for an undergraduate course in Numerical Methods. The OpenCourseWare (nm.MathForCollege.com) annually receives 1,000,000+ page views, 1,000,000+ views of the YouTube audiovisual lectures, and 150,000+ page views at the NumericalMethodsGuy blog. His current research interests include engineering education research methods, adaptive learning, open courseware, massive open online courses, flipped classrooms, and learning strategies. He has written four textbooks and 80 refereed technical papers, and his opinion editorials have appeared in the St. Petersburg Times and Tampa Tribune.
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Great article! I was confused at first.. Thanks!

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Thanks for this! I was so confused, and fell for the myth. Was going into my exams not knowing this at all.

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