In this article, we shall study the concept of significant figures and order of magnitude.

Significant Figures:

The digits, whose values are accurately known in a particular measurement are called significant figures.

The number expressing the magnitude of measurement should be recorded with only as many as digit as properly indicate the accuracy of the measurement. When the calculation is made with these numbers the final answer should contain the same number of significant figures as that in original data. The number of Significant figures depends upon the accuracy of the measuring instrument.

Example: consider measurement of the rod using a metre scale of least count 0.1 cm. Let us assume that the length measured, is 3.5 cm. It means that the length of the rod may be anywhere between 3.4 cm and 3.6 cm. It means we are sure of the digit 3 but we are not sure about the digit 5. But still, the digit 5 is trustworthy. Hence there are two significant figures.

If the length of the same rod is measured using vernier caliper of least count 0.01 cm. Let us assume that the length measured, is 3.52 cm. It means that the length of the rod may be anywhere between 3.51 cm and 3.53 cm. It means we are sure of the digit 3 and 5 but we are not sure about the digit 2. But still, the digit 2 is trustworthy. Hence there are three significant figures.

Hence we can say that the larger the accuracy of the measuring instrument, the larger will be the number of the significant figure will be larger.

Rules for Determining Significant Figures:

• One and only one uncertain digit is to be retained in a measurement. for e.g. When we record reading as 2.65 it means that we are sure of the first two digits but we are not sure about the last digit  Hence there is only one uncertain digit
• When a number is to be rounded off to a specific number of significant figures then, If the number dropped is less than five, the last digit retained is left unchanged. for e.g.  If the number is 12.42 is to be rounded to three significant figures then it is rounded off as 12.4. If number dropped is equal to or greater than five, the last digit retained is increased by 1.  for e.g.  If the number is 17.49 is to be rounded to three significant figures then it is rounded off as 17.5
• All non-zero digits are significant. All zeros occurring between two non-zero digits are significant. The zeros on the left side of the number are not significant. for e.g. in the number 0.00034, the number of significant figures is 2.
• The zeros on the right side of the number are significant because they indicate the accuracy of the measurement. for e.g. in the number 3.400, the number of significant figures is 4.
• All zeros to the right of the last rightmost non-zero digits are not significant. For e.g. in 2020 there are 3 significant figures.
• All zeros to the right of the last non-zero digit are significant if they come from the measurement. For e.g. If the distance between two points is measured 2020 cm (measured to the nearest centimeter) then in this number there are four significant figures.
• If the number contains more digits than the significant figures, the number should be expressed as a power of 10. for e.g. earth mass is known to be correct up to 3 significant figures, hence it is expressed as 5.98 × 10²⁴ kg.

Rules for Calculation Using Significant Figures:

Addition and Subtraction:

Find the number with less number of significant figures and then retain only that number of significant figures in the other numbers and then add.

e.g. consider addition 7.2 + 7.83 + 6.287

7.2 + 7.83 + 6.287 = 21.317 (actual calculation)

7.2 contains less number of significant figures  i.e. 2.

7.2 + 7.8 + 6.3 = 21.3 (significantly)

Rounding off the other two numbers to two significant the answer is 21.3

Multiplication and Division:

The operations are carried out and the result is rounded off to contain the same number of significant figures as those in the factors with less number of significant figures.for

e.g. Consider multiplication 15.62 × 3.83  = 59. 8246 (actual calculation)

3.83 contains less number of significant figures  i.e. 3.

The result  59. 8246 is rounded up to three significant figures as 59.8. Therefore the answer is 59.8.

Example – 01:

The length, breadth, and height of a cuboid are measured as 8.24 m, 8,001 m, and 125.3 cm. Find the volume of the cuboid.

Solution:

l = 8,24 m, b = 8.001 m and h = 125.3 cm = 1.253 m

Volume of cuboid = l × b × h = 8,24 m × 8.001 m × 1.253 m = 82.608084 m³.

8,24 contains less number of significant figures  i.e. 3.

Thus the volume of the cuboid is 82.6  m³.

Example – 02:

46.7 – 10.04 = 36.66 = 36.7

Example – 03:

(3.0 × 10^-8) + (4.5 × 10^-6) = (0.03 × 10^-6) + (4.5 × 10^-6) = (4.53 × 10^-6) = 4.5 × 10^-6 

Example – 04:

(3.9 × 10⁵) – (2.5 × 10⁴) = (39 × 10⁴) – (2.5 × 10⁴) = 36.5 × 10⁴ =  3.65 × 10⁵ =3.7 x 10⁵  

Example – 05:

(4.0 x 10^-4) – (2.5 x 10^-6) = (400 x 10^-6) – (2.5 x 10^-6) = 397.5 x 10^-6 =  3.975 x 10^-4 = 4 x 10^-4

Example – 06:

09996 x 3.54 / 1.638 =  2.160 = 2.16

Example – 07:

09996 x 3.52 / 1.756 =  2.004 = 2.00

Order of Magnitude:

In the present age, the scientific study involves study from very minute particles like subatomic particles to large bodies like a star. To have a quantitative idea of universe man uses units. Thus we have to deal with very small numbers as well as with very large numbers. If these numbers are converted to the nearest power of 10 notation the comparison is easy.

The order of magnitude of a physical quantity is defined as the value of its magnitude rounded off to the nearest integral power of 10.

For e.g. if a man has a weight of 96 kg, then his weight can be rounded off to the nearest integral power of 10 is 100 kg or 10² kg. Hence the order of magnitude of man is 10² kg.

Order of magnitude of Length:

In this universe, the various objects have lengths over a wide range. The following table gives the order of magnitude of their lengths from the size of the nucleus to the size of the universe.

Order of magnitude of Mass:

In this universe, the various objects have masses over a wide range. The following table gives the order of magnitude of their masses from the electron to the universe.

Order of magnitude of Time:

The following table gives the order of magnitude of the time intervals of few events.

To Find Order of Magnitude:

Find the order of magnitude of the following.

Acceleration Due to Gravity:

Acceleration due to gravity = 9.8 m s^-2.

Acceleration due to gravity = 9.8 m s^-2 =  9.8 × 10⁰ m s^-2

Hence the order of magnitude of the acceleration due to gravity is 10¹.

Gravitational Constant:

The gravitational constant = G  = 6.67 × 10^-11 N m² kg^-2

The gravitational constant = G  = 6.67 × 10^-11 N m² kg^-2

Hence the order of magnitude of the gravitational constant is 10^-10 N m² kg^-2.

Charge on Electron:

The charge of electron  = e  = 1.6 × 10^-19 C

The charge of electron  = e  = 1.6 × 10^-19 C

Hence the order of magnitude of the charge of an electron is 10^-19 C.

Value of 1/4πε₀ :

The value of 1/4πε₀  = 9 × 10⁹ N m² C^-2.

Value of 1/4πε₀  = 9 × 10⁹ N m² C^-2.

Hence the order of magnitude of the value of 1/4πε₀ is 10¹⁰ N m² C^-2.

Mass of an Electron:

The mass of an electron  = 9 × 10^-31 kg.

The mass of electron  = 9 × 10^-31 kg.

Hence the order of magnitude of the mass of an electron is 10^-30 kg.

Radius of the Earth:

The radius of the earth  = 6400 km

The radius of the earth  = 6400 km = 6.4 × 10³ m.

Hence the order of magnitude of the radius of the earth is 10⁴ m.

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