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The question, now, is how to write a measurement of 50 cm in accordance with rule 5, so that it has decimal point to indicate that zeros are significant. We make use of scientific notation, which expresses a value in the powers of 10. Hence, we write different experiment values as given here,
$$50\phantom{\rule{1em}{0ex}}cm=5.0X{10}^{1}\phantom{\rule{1em}{0ex}}cm$$
This representation shows that the value has two significant figures. Similarly, consider measurements of 500 cm and 3240 cm as measured by an instrument. Our representation is required to reflect that these values have "3" and "4" significant figures respectively. We do this by representing them in scientific notation as :
$$500\phantom{\rule{1em}{0ex}}cm=5.00X{10}^{2}\phantom{\rule{1em}{0ex}}cm$$
$$3240\phantom{\rule{1em}{0ex}}cm=3.240X{10}^{2}\phantom{\rule{1em}{0ex}}cm$$
In this manner, we maintain the number of significant numbers, in case measurement value involves trailing zeros.
From the discussion above, we observe following important aspects of significant figures :
A physical quantity is generally dependent on other quantities. Evaluation of such derived physical quantity involves mathematical operations on measured quantities. Here, we shall investigate the implication of mathematical operations on the numbers of significant digits and hence on error estimate associated with last significant digit.
For example, let us consider calculation of a current in a piece of electrical conductor of resistance 1.23 (as measured). The conductor is connected to a battery of 1.2 V (as measured). Now, The current is given by Ohm’s law as :
$$I=\frac{V}{R}=\frac{1.2}{1.23}$$
The numerical division yields the value as rounded to third decimal place is :
$$\Rightarrow I=0.976\phantom{\rule{1em}{0ex}}A$$
How many significant numbers should there be in the value of current? The guiding principle, here, is that the accuracy of final or resulting value after mathematical operation can not be greater than that of the operand (measured value), having least numbers of significant digits. Following this dictum, the significant numbers in the value of current should be limited to “2”, as it is the numbers of significant digits in the value of “V”. This is the minimum of significant numbers in the measured quantities. As such, we should write the calculated value of current, after rounding off, as:
$$\Rightarrow I=0.98\phantom{\rule{1em}{0ex}}A$$
We have already dealt the case of division. We take another example of multiplication. Let density of a uniform spherical ball is 3.201 $gm/{\mathrm{cm}}^{3}$ and its volume 5.2 $gm/c{\mathrm{cm}}^{3}$ . We can calculate its mass as :
The density is :
$$m=\rho V=3.201X5.2=16.6452\phantom{\rule{1em}{0ex}}gm$$
In accordance with the guiding principle as stated earlier, we apply the rule that the result of multiplication or division should have same numbers of significant numbers as that of the measured value with least significant numbers.
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