1.4 Significant figures  (Page 5/5)

Rule 4 : The “even” preceding digit (uncertain digit of the significant figures) is left unchanged, if the digit following it is 5. For example, a value of 2.525 is rounded as “2.52” to have three significant figures or to have two decimal places.

Rule 5 : If mathematical operation involves intermediate steps, then we retain one digit more than as specified by the rules of mathematical operation. We do not carry out “rounding off” in the intermediate steps, but only to the final result.

Rule 6 : In the case of physical constants, like value of speed of light, gravitational constant etc. or in the case of mathematical constants like “π”, we take values with the precision of the operand having maximum precision i.e. maximum significant numbers or maximum decimal places. Hence, depending on the requirement in hand, the speed of light having value of “299792458 m/s” can be written as :

$$c=3X{10}^{8}m/s\left(\text{1 significant number}\right)$$

$$c=3.0X{10}^{8}m/s\left(\text{2 significant number}\right)$$

$$c=3.00X{10}^{8}m/s\left(\text{3 significant number}\right)$$

$$c=2.998X{10}^{8}m/s\left(\text{4 significant number}\right)$$

Note that digit "9" appearing in the value is rounded off in the first three examples. In fourth, the digit "7" is rounded to "8".

Scientific notation

Scientific notation uses representation in terms of powers of 10. The representation follows the simple construct as given here :

$$x=a{10}^b$$

where “a” falls between “1” and “10” and “b” is positive or negative integer. The range of “a” as specified ensures that there is only one digit to the left of decimal. For example, the value of 1 standard atmospheric pressure is :

$$1atm=1.013X{10}^6\text{Pascal}$$

Similarly, mass of earth in scientific notation is :

$$M=5.98X{10}^{24}Kg$$

Order of magnitude

We come across values of quantities and constants, which ranges from very small to very large. It is not always possible to remember significant figures of so many quantities. At the same time we need to have a general appreciation of the values involved. For example, consider the statement that dimension of hydrogen atom has a magnitude of the order of “-10”. This means that diameter of hydrogen atom is approximately "10" raised to the power of "-10" i.e. ${10}^{-10}$ .

$$d\sim {10}^{-10}$$

Scientific notation helps to estimate order of magnitude in a consistent manner, if we follow certain rule. Using scientific notation, we have :

$$x=a{10}^b$$

We follow the rule as given here : If “a” is less than or equal to “5”, then we reduce the value of "a" to “1”. If “a” is greater than “5” and less than “10”, then we increase the value of “a” to "10".

In order to understand the operation, let us compare the order of magnitude of the diameter of a hydrogen atom and Sun. The diameters of Sun is :

$$d_S=6.96X{10}^{8}m$$

Following the rule, approximate size of the sun is :

$$ds\prime =10X{10}^8={10}^{9}m$$

Thus, order of magnitude of Sun is “9”. On the other hand, the order of magnitude of hydrogen atom as given earlier is “-10”. The ratio of two sizes is about equal to the difference of two orders of magnitude. This ratio can be obtained by deducting smaller order from bigger order. For example, the relative order of magnitude of sun with respect to hydrogen atom, therefore, is 9 – (-10) = 19.

This means that diameter of sun is about ${10}^{19}$ greater than that of a hydrogen atom. In general, we should have some idea of the order of magnitudes of natural entities like particle, atom, planets and stars with respect to basic quantities like length, mass and time.