Study S1 Mathematics Maths - Approximation, Estimation - Geniebook

Approximation And Estimation

In this chapter, we will be discussing the below-mentioned topics in detail:

  • Significant Figures
  1. 5 rules to determine if a digit is significant
  2. Rounding off to a given number of significant figures
  3. 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:

  1. 5378
  2. 12
  3. 4.69

 

Solution:

  1. Number of significant figures in \(5378 = 4\)
  2. Number  of  significant  figures  in \(12 = 2\)
  3. 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:

  1. 8.029
  2. 203
  3. 40.001

 

Solution:

  1. Number of  significant  figures in \(8.029 = 4 \)
  2. Number  of  significant  figures  in  \(203 = 3\)
  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:

  1. 0.385
  2. 0.0027
  3. 0.30607

 

Solution:

  1. Number of  significant  figures  in  \(0.385 = 3\) 
  2. Number of  significant  figures in \(0.0027 = 2\)
  3. 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:

  1. 0.670
  2. 0.0400
  3. 3.0250

 

Solution:

  1. Number of  significant  figures in \(0.670 = 3 \)
  2. Number of significant figures in \(0.0400 = 3\)
  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 5th or 7th 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.

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