Significant Figures
Introduction: The subject of “significant figures” relates to defining what is mean by specific numbers. The topic can relate to specifying project goals but is relevant in perhaps all situations combining business and technical issues.
Significant Figures
A. The subject of “significant figures” relates to defining what is mean by specific numbers.
B. This section includes one convention for the interpretation and documentation of numbers.
C. This convention is associated with a method for deriving the uncertainty of the results of calculations.
D. The interpretation of numbers can be important in any phase of a technical project and in many other situations.
E. In general, there are at least three ways to document uncertainty:
(1) by implication,
(2) with explicit ranges written either using “±” or (low, high), or
(3) using a distribution function and probability theory
F. This section focuses on the former two documentation methods.
G. he term “significant figures” refers to the number of digits in a written number that can be trusted by implication.
H. Factors that can reduce trust include the possibility of round-off errors and any explicit expression of uncertainty.
I. Unless specified otherwise, all digits in a written number are considered significant. Also, whole numbers generally have an infinite number of significant figures unless uncertainty is expressed explicitly.
J. The “digit location” of a number is the power of 10 that would generate a 1 digit in the right-most significant digit.
Example (Significant Figures of Sums and Products) For each question, use the steps outlined above.
y = 2.51 (10.2 ± 0.5). What is the explicit uncertainty of y?
Answer: In Step 1, the range for x1 is (2.505, 2.515) and for x2 is (9.7, 10.7).
In Step 2, the 22 = 4 sums are: 2.505 9.7 = 12.205, 2.505 10.7 = 13.205, 2.515 9.7 = 12.215, and 2.515 10.7 = 13.215.
The ranges in Step 3 are (12.205, 13.215).
Therefore with range (12.2, 13.2) with rounding. This can also be written 12.71 ± 0.5.
Product Question: y = 2.51 × (10.2 ± 0.5). What is the explicit uncertainty of y?
Product Answer: In Step 1, the range for x1 is (2.505, 2.515) and for x2 is (9.7, 10.7). In Step 2, the 22 = 4 products are: 2.505 × 9.7 = 24.2985, 2.505 × 10.7 = 26.8035, 2.515 × 9.ds7 = 24.3955, and 2.515 × 10.7 = 26.9105. The ranges in Step
3 are (24.2985, 26.9105).
Therefore, the product can be written 25.602 with uncertainty range (24.3, 26.9) with rounding. This could be written 25.602 ± 1.3.
Example:(Whole Number)y = 4 people × 2 (jobs/person). What is the explicit uncertainty of y?
Answer: In Step 1, the range for x1 is (4, 4) and for x2 is (2, 2) since we are dealing with whole numbers. In Step 2, the 22 = 4 products are: 4 × 2 = 8, 4 × 2 = 8, 4 × 2 = 8, and 4 × 2 = 8. The ranges in Step 3 are (8, 8). Therefore,
the product can be written as 8 jobs with uncertainty range (8, 8). This could be written 8 ± 0.000.
Note that in multiplication or product situations, the uncertainty range does not usually split evenly on either side of the quoted result. Then, the notation (–, ) can be used. One attractive feature of the “Formal Derivation of Significant Figures”
method proposed here is that it can be used in cases in which the operations are not arithmetic in nature, which is the purpose of the next example.