Significant Figures( or digits)

Scientific Notation

In which any number can be represented in the form $N \times 10^n$ (Where n is an exponent having positive or negative values and N can vary between 1 to 10).
e.g. We can write $162.108$ as $1.62108 \times 10^2$ in scientific notation. Similarly, $0.00021$ can be written as $2.1 \times 10^{-4}$
Addition and Subtraction expressed in Scientific Notation
In this case, we need to express both the numbers in same exponent form and then perform addition or multiplication of the number as usual
$1.1 \times 10^4 + 1.23 \times 10^3 = 1.1 \times 10^{4} + .123 \times 10^{4} = 1.223 \times 10^4$
Multiplication and Division
They follow the same rules as followed for exponential numbers

Significant figures

Every experimental observation has some amount of uncertainty associated with it
Precision refers to the closeness of various measurements for the same quantity.
Accuracy is the agreement of a particular value to the true value of the result
Suppose true value of quantity is 3 and two measurement taken are 2.91 , 2.92. They are precise as they close to each other but they are not accurate.
The measurement 2.98 and 3.01 are accurate as they are close to true value

Significant figures are meaningful digits which are known with certainty. Those are certain values and following rules are used to determine the no. of significant figures

Rules for Significant Figures

1)All non – zero digits are significant
E.g.185, significant figure 3
2195, significant figure - 4

2) Zero, proceeding to 1st non – zero digit are not significant. Such zero indicates the position of decimal point.
E.g. 0.03 – significant figure 1

3) Zero between non – zero digits are significant
E.g. 2.005 – significant figure 4
1001 – significant figure 4

4) All zeros placed to the right of a number are significant. For example, 16.0 has three significant figures, while 16.00 has four significant figures. Zeros at the end of a number without decimal point are ambiguous.

5)) In exponential notations, the numerical portion represents the number of significant figures. For example, $0.00045$ is expressed as $4.5 \times 10^{-4}$ in terms of scientific notations. The number of significant figures in this number is 2, while in Avogadro's number ($6.023 \times 10^{23}$ )it is four.

Retention of Significant Figures - Rounding off Figures

The rounding off procedure is applied to retain the required number of significant figures.
1. If the digit coming after the desired number of significant figures happens to be more than 5, the preceding significant figure is increased by one, 5.318 is rounded off to 5.32
2. If the digit involved is less than 5, it is neglected and the preceding significant figure remains unchanged, 4.312 is rounded off to 4.31.
3. If the digit happens to be 5, the last mentioned or preceding significant figure is increased by one only in case it happens to be odd. In case of even figure, the preceding digit remains unchanged. 8.375 is rounded off to 8.38 while 8.365 is rounded off to 8.36.

Addition and Subtraction of Significant Figure:

The result should be reported to the same no. of decimal places as in the term having least no. of decimal places.Your final answer may have no more significant figures to the right of the decimal than the LEAST number of significant figures in any number in the problem.
(1) $10.1 + 1.52 + 2.301 = 4.921$
Answer $4.9$
(2) $2.52 + 1.11 + 2.222 = 5.852$
Answer $5.85$
(3) $5.4 - 2.1$
$= 3.3$

Multiplication and Division of Significant Figure:

Result should be reported up to the same no. of significant figure as present in least precisive number.So The LEAST number of significant figures in any number of the problem determines the number of significant figures in the answer
(1) $2.210 \times 0.011$
$= .024312$
$= 0.024$
(2) $1.01 \times 0.02$
$= 0.0202$
$= 0.02$
(3) $\frac {4.24}{0.2}$
$= 21.2$

Solved Examples

Question 1
Round off up to 3 significant figure the below numbers
(a) 4.135
(b) 5.125
By following the rounding rules ,we get
(a) 4.14
(b) 5.12

Question 2
Express the following in the scientific notation with 2 significant figures-
(a) 0.0023
(b) 586,00
(c) 100.0
(a) $2.3 \times 10^{-3}$
(b) $5.9 \times 10^{-4}$
(c) $1.0 \times 10^2$
Question 3
Round 451.45 to four, three, and two significant digits
(a) 451.4
(b) 451
(c) 450

Dimensional Analysis

During calculations generally there is a need to convert units from one system to other. This is called factor label method or unit factor method or dimensional analysis.
We know that
1 inch = 2.54 x 10-2 m
This is can be written as
1 inch/ 2.54 x 10-2 m =1
2.54 x 10-2 m / 1 inch= 1
The above two are called Unit factors and it can be used to convert unit from one system to another

Quiz Time

Question 1 Which is of these zero is not significant.
A. .012
B. .120
C. .102
D. .201
Question 2 The number of significant digits in the number .132000?
A. 3
B. 6
C. 4
D. 5
Question 3 The number of significant digits in the number 1600
A. 4
B. 3
C. 2
D. 1
Question 4 The scientific notation for the number 0.0000080
A. $8 \times 10^{-6}$
B. $8.00 \times 10^{-6}$
C. $80 \times 10^{-7}$
D. $8.0 \times 10^{-6}$
Question 5 The sum of $4.56+0.298+0.0365+0.00722$
A. 4.90172
B. 4.9017
C. 4.901
D. 4.90
Question 6The value of $6.40 \times 5.23$
A. 33.4
B. 33.47
C. 33.5
D. 33.472

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