Significant figures : The number of significant figures in the measured value of a physical quantity gives the accuracy of its value.

The number of digits in a measurement about which we are reasonably sure, plus the one additional digit which is uncertain are significant.

## Common rules of counting significant figure :

(a) All non-zero digits are significant.

(b) All zeros occurring between two non-zero digits are significant, no matter where the decimal point is, if at all.

(c) In a number less than one, all zeros to the right of decimal point and to the left of the first non-zero digit are not significant. [In 0.002308, the underline zeroes are not significant]

(d) The terminal or trailing zeroes in a number without a decimal point are not significant. [Thus 123 m = 12300 cm = 123000 mm has three significant figures, the trailing zeroes being not significant].

(e) All zeros on the right of the last non-zero digit in the decimal part are significant. [The numbers 3.500 or 0.06900 has four significant figures each]

Solved Example : State the number of significant figures in the following :

(a) 0.007 m^{2}

(b) 2.64 × 10^{24} kg

(c) 0.2370 g cm^{-3}

(d) 6.320 J

(e) 6.032 N m^{-2}

(f) 0.0006032 m^{2}

Solution:

No. of significant figures:

(a) one (b) three

(c) four (d) four

(e) four (f) four

__How to calculate Significant figures in calculations ?__

### Significant figures in Addition & subtraction :

The accuracy of a sum or a difference is limited to the accuracy of the least accurate observation.

**Rule:** Do not retain a greater number of decimal places in a result computed from addition and subtraction than in the observation, which has the **Fewest Decimal Places.**

Solved Example : Add and subtract 428.5 and 17.23 with due regards to significant figures

Solution: we have , SUM = 428.50 +17.23 = 445.73

Difference = 428.50 – 17.23 = 411.27

Rounding off the results of the above sum and difference to the first decimal,

we have Correct sum 445.7 and correct difference 411.3

__Significant figures in Multiplication & Division :__

When the values of different observations are multiplied or divided, the number of digits to be retained in the answer depends upon the number of significant figures in the weakest link.

**Rule: **Do not retain a greater number of significant figures in a result computed from multiplication and or division than the **Least Number of Significant Figures** in the data from which the result is computed.

Solved Example : Multiply 312.65 and 26.4 with due regards to significant figures.

Solution : 312.65 × 26.4 = 8253.960

But as the weakest link i.e. the data 26.4 has only three Significant figures, the correct result of multiplication will be 8250. This is because in 8250, there are three significant figures. Hence, 312.65 x 26.4 = 8250

Solved Example : Each side of a cube is measured to be 7.203 m. what are the total surface area and the volume of the cube to appropriate significant figure ?

Solution: Total surface area = 6 L^{2}

= 6(7.203)^{2} = 311.299254 = 311.3 m^{2}

Volume of cube = L^{3} = (7.203)^{3}

= 373.714 m^{3} = 373.7 m^{3}

Solved Example : The mass of box measured by a grocer’s balance is 2.3 kg. Two gold pieces of 20.15 g and 20.17 g are added to the box. What is (a) total mass of the box (b) the difference in masses of gold pieces to correct significant figures

Solution. Here, mass of the box m = 2.3 kg

Mass of one gold piece, m_{1} = 20.15 g = 0.02015 kg

Mass of other gold piece, m_{2} = 20.17 g = 0.02017 kg

Total mass = m + m_{1} + m_{2} = 2.3 + 0.02015 + 0.02017 = 2.34032 kg.

As the result is correct only upto one place of decimal, therefore, on rounding off

Total mass = 2.3 kg

Difference in mass= m_{2} – m_{1} = 20.17 – 20.15 = 0.02 g (correct upto two places of decimal).

Solved Example : 5.74 g of a substance occupies 1.2 cm^{3}. Express its density by keeping the significant figures in view.

Solution: $\large Density = \frac{Mass}{Volume}$

$\large Density (\rho ) = \frac{5.74}{1.2} = 4.783 g cm^{-3}$

Rounding off, we get ρ = 4.8 g cm^{-3} (least significant)

### Rounding Off

Rule (i). If the digit to be dropped is less than 5, then the preceding digit is left unchanged. For example, x = 7.82 is rounded off to 7.8. Again, x = 3.94 is rounded off to 3.9.

Rule (ii). If the digit to be dropped is more than 5, then the preceding digit is raised by one. For example, x = 6.87 is rounded off to 6.9. Again, x = 12.78 is rounded off to 12.8.

Rule (iii). If the digit to be dropped is 5 followed by digits other than zero, then the preceding digits is raised by one. For example x=32.154 is rounded off to 32.2

Rule (iv). If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is left unchanged, if it is even.

For example: x = 3.250 become 3.2 on rounding off, again x = 12.650 becomes 12.6 on rounding off.

Rule (v). If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd.

For example: x = 3.750 is rounded off to 3.8. Again, x = 16.150 is rounded off to 16.2.