We have been exploring the topics ‘measurements’ and ‘errors‘ in the previous few posts. Whatever we have learned till now,can be recapitulated as follows –
During various calculations that we carried out during studying these parameters, we oftentimes encountered numbers with more digits after the decimal point. In such cases, we rounded off the digits. e.g. In the previous post , we rounded off the standard deviation values (Refer Post 18). During our course of study, we very often come across such situations where rounding off the digits is very helpful. In this post we shall study the rules of rounding off and getting significant figures.The concept of significant numbers is ubiquitous to all calculations made in Chemistry.
Significant Figures –
Significant figures can be defined as those digits which are exactly known plus the last one which is uncertain.
Significant numbers are the digits which tell us how accurate and precise our measurement is.
e.g. Consider a value 0.00900 kg. How many Significant figures does this value have?
If we rewrite this value in ‘grams’ , we get a value 9.00g. So, the three zeros before the digit ‘9’ were not significant when the quantity was expressed in kilograms as soon as the units changed the zeros were lost. Note that , 0.00900kg = 9.00g. So, essentially both the quantities are same , just the units are different. So, in the above example we have 3 significant figures namely – 9,0,0.
Why are the zeros after 9 significant? They are significant because they tell us the how precise our measurement is. The quantity could be written as 9g or 0.009kg. But addition of those extra two digits(two zeros) tells us that the measurement is more accurate.It simply means that, if we just write ‘9g’ , the weight is not very accurate as this has only one significant digit. A simple,non-digital balance would give us such a rough result . Technically, the weight could not be exactly 9g but somewhere close to it.So,the weight could be 9.02 or 9.04 or any other value close to 9g.
Thus, 0.009kg/9g ⇒ Indicates comparitively rough measurement ⇒Weight measured by a simple, non-didgital balance ⇒ Low Precision.
Whereas, when we write 9.00g , we indicate that the measurement done was more accurate( upto two digits after the decimal point). Here, we are sure that the weight is exact 9.00g .Such accuracy is obtained by a digital balance, which is more accurate than the simple balance. So the last two zeros indicate accuracy and thus are significant along with the non-zero numnber ‘9’.Thus,0.00900kg/9.00g ⇒ Indicates more exact measurement ⇒ Weighed by digital balance ⇒ High precision.
Thus, 0.00900 has three significant figures highlighted in bold font.
Rules for calculating significant figures -(Significant figures are highlighted in Orange color)
- All non-zero digits (1 to 9) and zeros in between non-zero digits are considered significant.
e.g. 0.78902 has five significant figures.
- Zeros immediately before and after the decimal point (leading zeros) are NOT significant.
e.g. 0.00867 has three significant digits. The zero before the decimal and the two zeros just after the decimal point are not significant numbers.
- For zeros which appear after the last non-zero digit (trailing zeros) there could be two possibilities –
Case I – For a number with a decimal point, these trailing zeros are significant.e.g. 0.2300
Case II – For a number without a decimal point, the trailing zeros may or maynot be significant.This is a case of ambiguity and one cannot determine the significance of the zeros correctly.e.g. Distance between two cities is 1830kms … here we cannot predict whether this distance was accurately measured or just a round figure is mentioned.
- Rounding the digits –
Case I – If the first rejected digit > 5 ⇒ Increase the last significant digit by one.
Case II – If the first rejected digit <5 ⇒Keep the last significant digit unchanged.
Case III -If the first rejected digit =5 ⇒ Round so that the result ends with an even number. Thus, if the previous digit is even we keep the it unaltered and if its odd , we add one to it to make it an even number.
- Addition & Subtraction – Retain only as many decimals in the final result as those in the number with fewest decimals.
e.g. 14.22+8.145 +120.4 = 142.765. Here, the number with the fewest decimals is 120.4 (only one place after decimal point).Therefore, we will round the answer to have only one digit after decimal point. Thus, the answer is 142.8
- Multiplication & Division – Retain only as many significant figures as those in the number with fewest significant figures.
e.g. (25×0.524)÷100 =0.131. Here,
25 →2 significant figures.
0.524 →3 significant figures.
100 →3 significant figures. Thus, the number with fewest significant figures is 25. So the answer can have only two signifiacnt figures.Thus, our answer is 0.13.
Though these topics seem to be very unexciting, they are pertinent to many applications in Science and their importance can only be understood in the light of high precision work that is carried out worldwide in almost all branches of science and technology. So, lets continue our discussion on significant numbers in our next post as well and see its applications in Chemistry. Till then ,
Be a perpetual student of life and keep learning…
All images used in this post are created by me and so ‘Image source’ is not provided.
References and Further Reading –