Table of contents  
Introduction  
Scientific Notation  
What are Significant Figures?  
Addition and Subtraction  
Multiplication and Division  
Order of Magnitude 
Many times in the study of chemistry, one has to deal with experimental data as well as theoretical calculations.
There are meaningful ways to handle the numbers conveniently and present the data realistically with certainty to the extent possible like:
Example: We can write 232.508 as 2.32508 × 10^{2} in scientific notation. Similarly, 0.00016 can be written as 1.6 × 10^{–4}.
Thus, we can write 232.508 as 2.32508 × 10^{2} in scientific notation. Note that while writing it, the decimal had to be moved to the left by two places and the same is the exponent (2) of 10 in the scientific notation.
Similarly, 0.00016 can be written as 1.6 × 10^{–4}. Here the decimal has to be moved four places to the right and (– 4) is the exponent in the scientific notation.
Q.1. Which of the following options is not correct?
(a) 8008 = 8.008 x 10^{3}
(b) 208 = 3
(c) 5000 = 5.0 x 10^{3}
(d) 2.0034 = 4
Ans: (d)
Solution:
2.0034 = 4
Q.2. Exponential notation in which any number can be represented in the form, Nx 10^{n }here N is termed as
(a) non –digit term
(b) digit term
(c) numeral
(d) base term
Ans: (b)
Solution:
In exponential notation N × 10^{n}, N is a number called digit term which varies between 1.000 and 9.000….
It is important to understand that the way a measurement is taken affects its accuracy. For example, you could measure the length of a leaf with a ruler that had markings every centimeter (cm). In this example, illustrated below, the leaf is longer than 3 cm and shorter than 4 cm, so you might estimate that the leaf is 3.5 cm long.
On the other hand, if you measured the same leaf with a ruler that had markings every millimeter (mm), as drawn below, you can see that the end of the leaf actually falls between the markings for 3.5 and 3.6 cm (or 35 and 36 mm). Because it’s closer to the 3.5 marking, you might estimate that the leaf is 3.52 cm (or 35.2 mm) long.
Using the second ruler, it’s possible to estimate that the leaf is 3.52 cm long, but it is not possible to measure that accurately with the first ruler. In this way, the number of digits in the measured value gives us an idea of the maximum accuracy of the measurement. These are called significant digits or significant figures.
RULES FOR SIGNIFICANT FIGURES
How are significant figures handled in calculations? It depends on what type of calculation is being performed. If the calculation is an addition or a subtraction, the rule is as follows: limit the reported answer to the rightmost column that all numbers have significant figures in common. For example, if you were to add 1.2 and 4.71, we note that the first number stops its significant figures in the tenths column, while the second number stops its significant figures in the hundredths column. We therefore limit our answer to the tenths column.
We drop the last digit—the 1—because it is not significant to the final answer.
The dropping of positions in sums and differences brings up the topic of rounding. Although there are several conventions, in this text we will adopt the following rule: the final answer should be rounded up if the first dropped digit is 5 or greater, and rounded down if the first dropped digit is less than 5.
For multiplication or division, the rule is to count the number of significant figures in each number being multiplied or divided and then limit the significant figures in the answer to the lowest count. An example is as follows:
The final answer, limited to four significant figures, is 4,094. The first digit dropped is 1, so we do not round up.
Scientific notation provides a way of communicating significant figures without ambiguity. You simply include all the significant figures in the leading number. For example, the number 450 has two significant figures and would be written in scientific notation as 4.5 × 10^{2}, whereas 450.0 has four significant figures and would be written as 4.500 × 10^{2}. In scientific notation, all significant figures are listed explicitly.
Example. Order of magnitude of the following values can be determined as follows:
(a) 49 = 4.9 × 101 » 101 \ Order of magnitude = 1
(b) 51 = 5.1 × 101 » 102 \ Order of magnitude = 2
(c) 0.049 = 4.9 × 10–2 » 10–2 \ Order of magnitude = –2
(d) 0.050 = 5.0 × 10–2 » 10–1 \ Order of magnitude = –1
(e) 0.051 = 5.1 × 10–2 » 10–1 \ Order of magnitude = –1
When doing calculations using significant figures, you will find it necessary to round your answer to the nearest significant digit. There are therefore a few rules of rounding that help retain as much accuracy as possible in the final answer.
AMBIGUOUS ZEROS
So what happens if your calculation or measurement ends in a zero? For example, what if you measured a branch that was 200 cm (not 199 or 201 cm) long? The zeros in a measured value of 200 cm in this case appear ambiguous, since it could suggest that there is only one significant digit.
One way to reduce this ambiguity is to use significant figures with scientific notation.
118 videos470 docs189 tests

5 Days Study Plan: Units and Measurements Doc  3 pages 
Multiplication & Division with Significant Figures Video  09:31 min 
Test: Significant Figures (NCERT) Test  10 ques 
1. What is scientific notation? 
2. What are significant figures? 
3. How do you perform addition and subtraction with significant figures? 
4. How do you perform multiplication and division with significant figures? 
5. What is the order of magnitude in scientific notation? 
118 videos470 docs189 tests

5 Days Study Plan: Units and Measurements Doc  3 pages 
Multiplication & Division with Significant Figures Video  09:31 min 
Test: Significant Figures (NCERT) Test  10 ques 

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