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 a 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, \(10.44\).
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 5^{th} or 7^{th} significant figure and not to round it off. Only round off in the final answer i.e. in the final step. |
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 |