Round-off error

A round-off error is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate it when using approximation equations and/or algorithms, especially when using finite digits to represent infinite digits of real numbers.

Example

Notation	Represent	Approximate	Error
1/7	0.142857	0.142857	1/7000000
ln 2	0.69314718055994530941...	0.693147	0.00000018055994530941...
log₁₀ 2	0.30102999566398119521...	0.3010	0.00002999566398119521...
∛ 2	1.25992104989487316476...	1.25992	0.00000104989487316476...
√ 2	1.41421356237309504880...	1.41421	0.00000356237309504880...
e	2.71828182845904523536...	2.718281828459045	0.00000000000000023536...
π	3.14159265358979323846...	3.141592653589793	0.00000000000000023846...

No matter how many digits it has, if the number of digits is finite, the error exists. This kind of error is unavoidable and is named round-off error.

There are, at least, two ways of performing the termination at the limited digit place:

Chopping: It simply chops off the remaining digits.

0.142857 ≈ 0.142 (chopping at the 5^th digits.)

Rounding: It adds 5 to the next digit and then chop it. The result may round up or round down.

0.142857 ≈ 0.143 (rounding at the 5^th digits. This is round up because the next digit, 8, is >= 5)

0.142857 ≈ 0.14 (rounding at the 4^th digits. This is round down because the next digit, 2, is < 5)

Round-off error

Example

See also

See also: Round-off error, Algorithm, Approximation, Cubic root, E (mathematical constant), Equation, Logarithm, Mathematics, Natural logarithm