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SHARJAH INSTITUTE For comments, corrections, etcPlease contact Ahnaf Abbas:
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Sharjah Institute of Technology





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1


Error Arithmetic Handout #1
Topic Interpretation
Given any mathematical expression, it
follows that if the variables have error,
then the result will have an associated
error, as well. This passing of error
from variables to the result is termed
error propogation. It is possible for
computations to magnify this error,
however, the amount the error is
maginfied depends on the operation:
Example 1
suppose the two values 3.55 and 3.54 represent
numbers in the ranges 3.545 to 3.555 and 3.535 to
3.545, respectively. If we calculate 3.55 - 3.54 =
0.01, the actual value could be anywhere in the
range 0 to 0.02. Thus while we are reporting that
the difference is positive, the actual difference may
be zero (or, if the error widths were slightly larger,
the difference may be negative).

Precision and accuracy
Two words we will be using to describe
how good a measurement or
approximation is to an actual value are
precision and accuracy.
Absolute and Relative errors
There are two techniques for
measuring error: the absolute error of
an approximation and the relative error
of the approximation. The first gives
how large the error is, while the second
gives how large the error is relative to
the correct value.
Calculating Absolute Error

Given an approximation
a
of a value
x
,
the absolute error
E
abs
is calculated
using the formula:
E
abs
= | x a |

Calculating Relative Error

Given an approximation
a
of a value
x
,
the relative error
E
rel
is calculated using
the formula:
x
a
x
E
rel

=

Example 2
What are the absolute and relative errors of the
approximation
3.14
to the value ?
E
abs
= |3.14 - | 0.0016
E
rel
= |3.14 - |/|| 0.00051

Example 3

A resistor labeled as
240
is actually
243.32753
.
What are the absolute and relative errors of the
labeled value?
E
abs
= |240 - 243.32753| 3.3
E
rel
= |240 - 243.32753|/|243.32753| 0.014
Note: the
label is the approximation of the actual value.
Example 4
The voltage in a high-voltage transmission line is
stated to be
2.4 MV
while the actual voltage may
range from
2.1 MV
to
2.7 MV
. What is the
maximum absolute and relative error of voltage?
E
abs
= |2.4 - 2.1| = 0.3 MV
E
rel
= |2.4 - 2.1|/|2.1| 0.14

E
abs
= |2.4 - 2.7| = 0.3 MV
E
rel
= |2.4 - 2.7|/|2.7| 0.11
Thus, the maximum absolute error is
0.3 MV
but
the maximum relative error is
0.14
.Note: as before,
the stated voltage is an approximation of the actual
voltage. For comments, corrections, etcPlease contact Ahnaf Abbas:
ahnaf@mathyards.com
Sharjah Institute of Technology





http://www.sit.ac.ae/hnd/index.html
2
Significant figures
Given a relative error E
rel
, find
the largest integer n such that
E
rel
< 0.5 10
-n
.
If the relative error is greater
than 0.5, state that the
approximation does not have
any significant digits.
In general, the number of
significant digits between a
number and its approximation
are equal to the number of
leading digits which are equal,
though this is only a rule of
thumb, and if the most
significant digit is 1 or 2, it is
most useful to ignore it when
counting the number of
significant digits.

Example 5
What is the number of significant digits
of the approximation 3.14 to the value
?
E
rel
= |3.14 - |/|| 0.00051 0.005
= 0.5 10
-2
, and therefore it is correct to
two significant digits.
This example demonstrates a weakness
in the concept of significant digits: in this
example, it would be almost better to
say that 3.14 approximates to almost
or approximately three significant digits.
Example 6
What is the number of significant digits
of the label 240 when the correct value
is 243.32753 ?
E
rel
= |240 - 243.32753|/|243.32753|
0.014 0.05 = 0.5 10
-1
, and therefore
it is correct to one significant digit.
Example 7
To how many significant digits is the
approximation 1.998532 when the actual
value is 2.001959?
E
rel
= |1.998532 - 2.001959|/|2.001959|
0.0017 0.005 = 0.5 10
-2
and
therefore it is correct to two digits.