Significant Figures

Imagine you're baking a cake, and the recipe says you need 1.5 cups of sugar. You have only a measuring cup marked in whole numbers. It is important to be accurate to get the cake right. Because significant figures are a measure of precision, they determine which digits in a measured or calculated value are reliable, thus building confidence in the quality of the result. Actually, in this article, we are going to see what are significant figures, and how they provide accuracy in measured values.

The concept of significant figures comes under the chapter Physics and Measurement which is a crucial chapter in Class 11 physics. It is not only important for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, VITEEE and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of two questions have been asked on this concept. And for NEET three questions were asked from this concept.

What is Significant Figure?

The figures of a number that expresses a magnitude to a specified degree of accuracy. All non-zero digits are significant

For Example-

42.3 -Three significant figures

238.4 -four significant figures

33.123 -five significant figures

• Zero becomes a significant figure if it exists between two non-zero digits

For example-

2.09 - Three significant figures

8.206 -four significant figures

6.002 -four significant figures

• For leading zero(s), the zero(s) to the left of the first non-zero digits are not significant.

For example-

0.543 - three significant figures

0.069 - two significant figures

0.002 -one significant figure

• The trailing zero(s) in a number without a decimal point are not significant. But if the decimal point is there then they will be counted in significant figures.

For example-

4.330- four significant figures

433.00- five significant figures

343.000- six significant figures

• Exponential digits in scientific notation are not significant.

For example- 1.32 X 10^-2- three significant figures

Rounding off

While rounding off measurements, we use the following rules by convention:

Rounding off of figures during calculation helps to make the calculation of big digits easier.

(1) If the digit to be dropped is less than 5, then the preceding digit is left unchanged.

Example: x=7.82 is rounded off to 7.8, again x=3.94 is rounded off to 3.9.

(2) If the digit to be dropped is more than 5, then the preceding digit is raised by one.

Example: x = 6.87 is rounded off to 6.9, again x = 12.78 is rounded off to 12.8.

(3) If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one.

Example: x = 16.351 is rounded off to 16.4, again x = 6.758 is rounded off to 6.8.

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

Example: x = 3.250 becomes 3.2 on rounding off, again x = 12.650 becomes 12.6 on rounding off.

(5) 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.

Example: x = 3.750 is rounded off to 3.8, again x = 16.150 is rounded off to 16.2.

Significant Figures in Calculation

1. Rules for addition and subtraction

The result of an addition or subtraction in the number having different precisions should be reported to the same number of decimal places as are present in the number having the least number of decimal places.

For example:-

1) 33.3+3.11+0.313=36.723 but here the answer should be reported to one decimal place as the 33.3 has the least number of the decimal place(i.e only one decimal place), therefore the final answer = 36.7

2) 3.1421+0.241+0.09=3.4731 but here the answer should be reported to two decimal places as the 0.09 has the least number of decimal place(i.e two decimal places), therefore the final answer=3.47

2. Rules for multiplication and division

The answer to a multiplication or division is rounded off to the same number of significant figures as is possessed by the least precise term used in the calculation:-

For example:-

1) 142.06 x 0.23=32.6738 but here the least precise term is 0.23 which has only two significant figures, so the answer will be 33.

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Solved Example Besed On Significant Figures

Example 1: What is true for significant figure

1) The higher no. of significant figures, the higher the accuracy

2) All non-zero digits are significant

3) Both A and B

4) only B

Solution:

Significant figures -

The figures of a number that express a magnitude to a specified degree of accuracy

Higher accuracy means there are higher no of significant figures.

Hence, the answer is the option is (3).

Example 2: Find the true match -

1)1 -Q, 2 - R, 3- P

2)1 - R, 2 -P, 3 - Q

3)1 -P, 2 - R, 3 - Q

4)1 - P, 2 - Q, 3 - R

Solution:

As we have studied all non-zero digits are significant and a zero becomes a significant figure if it exists between two non-zero digits

42.3 -Three significant figure

238.4 -four significant figure

2165.4 -five significant figures

Hence, the correct option is (1).

Example 3: The diameter and height of a cylinder are measured by a meter scale to be 12.6±0.1 cm and 34.2±0.1 cm, respectively. What will be the value of its volume in the appropriate significant figure?

1) 4300±80 cm3
2) 4264.4±81.0 cm3
3) 4264±81 cm3
4) 4260±80 cm3

Solution:

v=πd24 h=4260 cm3Δvv=2Δdd+ΔhhΔv=2×0.1v12.6+0.1v34.2=0.212.6×4260+0.1×426034.2=80∴ Volume =4260±80 cm3

Hence, the answer is the option (4).

Example 4: Which of the following has the maximum no. of significant figures?

1) 234.000
2) 0.000303
3) 234×105
4) 12×10−5

Solution:

Leading Zeros-

0.000303 has 3 significant figures
Exponential digits in scientific notation are not significant.

234×105 has 3 significant figures
12×10−5 has 2 significant figures

Trailing Zeros -

234.000 has 6 significant figures

All zeros to the right of a decimal point are significant

So 234.000 has the maximum number of significant figures.

Hence, the correct option is 1.

Example 5: For the four sets of three measured physical quantities as given below. Which of the following options is correct?

(i) A1=24.36,B1=0.0724,C1=256.2
(ii) A2=24.44,B2=16.082,C2=240.2
(iii) A3=25.2,B3=19.2812,C3=236.183
(iv) A4=25,B4=236.191,C4=19.5

1) A4+B4+C4<A1+B1+C1=A2+B2+C2=A3+B3+C3
2) A1+B1+C1=A2+B2+C2=A3+B3+C3=A4+B4+C4
3) A1+B1+C1<A3+B3+C3<A2+B2+C2<A4+B4+C4

4) None of these

Solution:

A1+B1+C1=24.36+0.0724+256.2=280.6324=280.6A2+B2+C2=24.44+16.082+240.2=280.722=280.7A3+B3+C3=25.2+19.2812+236.183=280.6642=280.7A4+B4+C4=25+236.191+19.5=280.691=281

Answer should be A_1+B_1+C_1<A_2+B_2+C_2=A_3+B_3+C_3<A_4+B_4+C_4

Hence, the answer is option (3).

Summary
Significant figures improve precision and accuracy in measurements and calculations, which is essential for scientific experiments and competitive exams. The greater the number of significant figures, the more precise the measurement.