SG 
VN 
Approximation And Estimation
In this chapter, we will be discussing the below-mentioned topics in detail:
- Significant Figures
- 5 rules to determine if a digit is significant
- Rounding off to a given number of significant figures
- Rounding errors
- Estimation of computations
Degree of Accuracy & Significant Figures
How accurate a number is, also known as its degree of accuracy, is determined by how many significant figures the number has.
Significant figures are also known as non-zero numbers.
| Number | \(1.1 \;m\) | \(112.5 \;cm\) (more accurate) |
| Number Of Significant Figures | \(2\) (i.e. both the \(1\) are significant numbers) |
\(4\) (i.e. \(1, 1, 2\) and \(5\)) |
- A higher degree of accuracy means the number is more accurate.
- With a greater number of significant figures, the degree of accuracy increases.
Hence, a number rounded off to 4 significant figures is more accurate compared to the same number rounded off to 2 significant figures.
| Number | \(14523\) (more accurate) |
\(15000\) |
| Number of Significant Figures | \(5\) | \(2\) to \(5\) |
Hence, 5 significant figures is more accurate as 5 is definite.
5 Rules to determine if a digit is significant
| Rule 1: All non-zero digits are significant. |
Question 1:
State the number of significant figures in each of the following:
- 5378
- 12
- 4.69
Solution:
- Number of significant figures in \(5378 = 4\)
- Number of significant figures in \(12 = 2\)
- Number of significant figures in \(4.69 = 3\)
| Rule 2: All zeros between non-zero digits are significant. |
Question 2:
State the number of significant figures in each of the following:
- 8.029
- 203
- 40.001
Solution:
- Number of significant figures in \(8.029 = 4 \)
- Number of significant figures in \(203 = 3\)
- Number of significant figures in \(40.001 = 5\)
| Rule 3: In a decimal, all zeros before a non-zero digits are not significant. |
Question 3:
State the number of significant figures in each of the following:
- 0.385
- 0.0027
- 0.30607
Solution:
- Number of significant figures in \(0.385 = 3\)
- Number of significant figures in \(0.0027 = 2\)
- Number of significant figures in \(0.30607 = 5\)
| Rule 4: In a decimal, all zeros after non-zero digits are significant. |
Question 4:
State the number of significant figures in each of the following:
- 0.670
- 0.0400
- 3.0250
Solution:
- Number of significant figures in \(0.670 = 3 \)
- Number of significant figures in \(0.0400 = 3\)
- Number of significant figures in \(3.0250 = 5\)
| Rule 5: In a whole number, the zeros at the end may or may not be significant. |
| Round off \(2799.99 \) to the nearest |
Whole Number | \(10\) | \(100\) |
| \(2800\) | \(2800\) | \(2800\) | |
| Number of significant figures | \(4\) | \(3\) | \(2\) |
Intermediate Steps & Rounding Error
Question 5:
The area of a square is \(108.9 \;cm^2\). Find the perimeter of the square.
Solution:
In order to find the perimeter, we need the length of the square. So, to find the length we can square root:
Length \(=\sqrt{108.9}\)
Pressing the calculator, it gives us a value of \(\sqrt{108.9} = 10.4355\) (Truncated value)
Rounding it off, we get \(10.44\).
| Round Off Intermediate Step |
Truncate Intermediate Step |
| \(\begin{align} \text{L} &= \sqrt{108.9}\\ &≈ 10.44\\ \\ \text{Perimeter} &= 10.44 \times 4 \\ &= 41.76 \\ &≈ 41.8 \;cm & \text{(3 significant figures) } \end{align} \) | \(\begin{align} \text{L} &= \sqrt{108.9}\\ &= 10.43\\ \\ \text{Perimeter} &= 10.43 \times 4 \\ &= 41.72 \\ &≈ 41.7 \;cm & \text{(3 significant figures) } \end{align} \) |
| Using calculator, \(\begin{align} \text{Perimeter} &= \sqrt{108.9} \times 4 \\ &= 41.742 \\ &≈ 41.7 \;cm & \text{(3 significant figures) } \end{align} \) Note: Always remember to truncate the intermediate step which means to cut it off after the 5th or 7th significant figure and not to round it off. Only round off in the final answer i.e. in the final step. |
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Note: If the degree of accuracy is not specified, i.e. the question does not say anything about the degree of accuracy, we will always round off to 3 significant figures.
| Continue Learning | |
|---|---|
| Basic Geometry | Linear Equations |
| Number Patterns | Percentage |
| Prime Numbers | Ratio, Rate And Speed |
| Functions & Linear Graphs 1 | Integers, Rational Numbers And Real Numbers |
| Basic Algebra And Algebraic Manipulation 1 | Approximation And Estimation |
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