SIGNIFICANT FIGURES
When using a number in a calculation, there must be assurance that can be used with confidence. The significant figures (or significant digits) represent the use of a level of uncertainty under certain approximations. The concept of significant figures has been developed to formally designate the reliability of a numerical value. The number of significant figures is the number of digits plus one estimated digit that can be used with confidence.
The concept of significant figures has two important implications in the study of numerical methods.
1. The numerical methods obtain approximate results. Therefore it should develop criteria to specify how accurate are the results, with significant figures, we can say that the approach is acceptable if it is correct to four significant figures.
2. Certain quantities such as ℮ and Π represent specific numbers, you can not exactly express a number as they tend to infinity. The concept of significant figures is very important in the definition of accuracy and precision.
When using a number in a calculation, there must be assurance that can be used with confidence. The significant figures (or significant digits) represent the use of a level of uncertainty under certain approximations. The concept of significant figures has been developed to formally designate the reliability of a numerical value. The number of significant figures is the number of digits plus one estimated digit that can be used with confidence.
The concept of significant figures has two important implications in the study of numerical methods.
1. The numerical methods obtain approximate results. Therefore it should develop criteria to specify how accurate are the results, with significant figures, we can say that the approach is acceptable if it is correct to four significant figures.
2. Certain quantities such as ℮ and Π represent specific numbers, you can not exactly express a number as they tend to infinity. The concept of significant figures is very important in the definition of accuracy and precision.
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PRECISION
Refers to the number of significant figures representing a number or extension in the repeated readings of an instrument that measures some physical property.
ACCURACY
It refers to the approach of a number or measure the true value is supposed to represent.
DEFINITIONS OF ERROR
The numerical errors are generated with the use of approximations for representing mathematical operations. These include truncation errors resulting from about exact mathematical procedure and the rounding errors that result from about exact numbers
.
For both types of errors the relationship between the exact result and the approximate or true is given by:
Approximate value = true value + error
ROUNDING ERROR
Occur because the computers just to hold a finite number of significant figures in a calculation. Computers perform this function in different ways. For example if you only saved seven significant figures computers can store and use π as π = 3.141592, omitting the remaining terms and generating a rounding error.
Truncation error
Are those that result when using an approximation rather than an exact mathematical procedure. For example, the approximate speed of a parachutist falling through the differential equation of the form:
dv / (dt) ≈ Δv / Dt = (v (t (i +1))-v (ti)) / (t (i +1) - t (i))
It introduces a truncation error in the numerical solution of differential equation because only approximates the true value of the derivative.
REFERENCES: NUMERICAL METHODS FOR ENGINEERS WITH PERSONAL COMPUTER APPLICATIONS. STEVEN C. Chapra / Raymond P. CANALE
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