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Roundoff Error
A roundoff error is the difference between the calculated approximation of a number and its exact mathematical value.
Learning Objective

Explain the impact roundoff errors may have on calculations, and how to reduce this impact
Key Points
 When a sequence of calculations subject to rounding error is made, these errors can accumulate and lead to the misrepresentation of calculated values.
 Increasing the number of digits allowed in a representation reduces the magnitude of possible roundoff errors, but may not always be feasible, especially when doing manual calculations.
 The degree to which numbers are rounded off is relative to the purpose of calculations and the actual value.
Term

approximation
An imprecise solution or result that is adequate for a defined purpose.
Full Text
Roundoff Error
A roundoff error, also called a rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations, algorithms, or both, especially when using finitely many digits to represent real numbers. When a sequence of calculations subject to rounding errors is made, errors may accumulate, sometimes dominating the calculation.
Calculations rarely lead to whole numbers. As such, values are expressed in the form of a decimal with infinite digits. The more digits that are used, the more accurate the calculations will be upon completion. Using a slew of digits in multiple calculations, however, is often unfeasible if calculating by hand and can lead to much more human error when keeping track of so many digits. To make calculations much easier, the results are often 'rounded off' to the nearest few decimal places.
For example, the equation for finding the area of a circle is
However, when doing a series of calculations, numbers are rounded off at each subsequent step. This leads to an accumulation of errors, and if profound enough, can misrepresent calculated values and lead to miscalculations and mistakes.
The following is an example of roundoff error:
Rounding these numbers off to one decimal place or to the nearest whole number would change the answer to 5.7 and 6, respectively. The more rounding off that is done, the more errors are introduced.
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Key Term Reference
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 magnitude
 Appears in these related concepts: Multiplying Vectors by a Scalar, Newton and His Laws, and Components of a Vector
 relative
 Appears in these related concepts: Relative Deprivation Approach, Addition of Velocities, and Relative Velocity
 series
 Appears in these related concepts: Combination Circuits, Resisitors in Series, and The General Term of a Sequence
Sources
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Cite This Source
Source: Boundless. “Roundoff Error.” Boundless Physics. Boundless, 26 May. 2016. Retrieved 30 Sep. 2016 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/thebasicsofphysics1/significantfiguresandorderofmagnitude33/roundofferror2026030/