Every measurement involves errors. Thus, the result of measurement should be reported in a way that indicates the precision of measurement. Normally, the reported result of measurement is a number that includes all digits in the number that are known reliably plus the first digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures.
All the non-zero digits are significant.
All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all.
If the number is less than 1, the zero(s) on the right of decimal point but to the left of the first non-zero digit are not significant. [In 0.00 2308, the underlined zeroes are not significant].
The terminal or trailing zero(s) in a number without a decimal point are not significant. [Thus 123 m = 12300 cm = 123000 mm has three significant figures, the trailing zero(s) being not significant.]
The trailing zero(s) in a number with a decimal point are significant. [The numbers 3.500 or 0.06900 have four significant figures each.]
There can be some confusion regarding the trailing zero(s). Suppose a length is reported to be 4.700 m. It is evident that the zeroes here are meant to convey the precision of measurement and are, therefore, significant. [If these were not, it would be superfluous to write them explicitly, the reported measurement would have been simply 4.7 m]. Now suppose we change units, then 4.700 m = 470.0 cm = 4700 mm = 0.004700 km
Since the last number has trailing zero(s) in a number with no decimal, we would conclude erroneously from observation above that the
number has two significant figures, while in fact, it has four significant figures and a mere change of units cannot change the number of significant figures.
To remove such ambiguities in determining the number of significant figures, the best way is to report every measurement in scientific notation (in the power of 10).
For a number greater than 1, without any decimal, the trailing zero(s) are not significant.
For a number with a decimal, the trailing zero(s) are significant.
`text(Rules for Arithmetic Operations with Significant Figures :)`
In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.
In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.
`text(Rounding off the Uncertain Digits :)`
A number `2.74ul6` rounded off to three significant figures is 2.75, while the number 2.743 would be 2.74. The rule by convention is that the preceding digit is raised by 1 if the insignificant digit to be dropped (the underlined digit in this case) is more than 5, and is left unchanged if the latter is less than 5.
But what if the number is `2.74ul5` in which the insignificant digit is 5. Here, the convention is that if the preceding digit is even, the insignificant digit is simply dropped and, if it is odd, the preceding digit is raised by 1. Then, the number `2.74ul5` rounded off to three significant figures becomes 2.74. On the other hand, the number 2.735 rounded off to three significant figures becomes 2.74 since the preceding digit is odd.
Every measurement involves errors. Thus, the result of measurement should be reported in a way that indicates the precision of measurement. Normally, the reported result of measurement is a number that includes all digits in the number that are known reliably plus the first digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures.
All the non-zero digits are significant.
All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all.
If the number is less than 1, the zero(s) on the right of decimal point but to the left of the first non-zero digit are not significant. [In 0.00 2308, the underlined zeroes are not significant].
The terminal or trailing zero(s) in a number without a decimal point are not significant. [Thus 123 m = 12300 cm = 123000 mm has three significant figures, the trailing zero(s) being not significant.]
The trailing zero(s) in a number with a decimal point are significant. [The numbers 3.500 or 0.06900 have four significant figures each.]
There can be some confusion regarding the trailing zero(s). Suppose a length is reported to be 4.700 m. It is evident that the zeroes here are meant to convey the precision of measurement and are, therefore, significant. [If these were not, it would be superfluous to write them explicitly, the reported measurement would have been simply 4.7 m]. Now suppose we change units, then 4.700 m = 470.0 cm = 4700 mm = 0.004700 km
Since the last number has trailing zero(s) in a number with no decimal, we would conclude erroneously from observation above that the
number has two significant figures, while in fact, it has four significant figures and a mere change of units cannot change the number of significant figures.
To remove such ambiguities in determining the number of significant figures, the best way is to report every measurement in scientific notation (in the power of 10).
For a number greater than 1, without any decimal, the trailing zero(s) are not significant.
For a number with a decimal, the trailing zero(s) are significant.
`text(Rules for Arithmetic Operations with Significant Figures :)`
In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.
In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.
`text(Rounding off the Uncertain Digits :)`
A number `2.74ul6` rounded off to three significant figures is 2.75, while the number 2.743 would be 2.74. The rule by convention is that the preceding digit is raised by 1 if the insignificant digit to be dropped (the underlined digit in this case) is more than 5, and is left unchanged if the latter is less than 5.
But what if the number is `2.74ul5` in which the insignificant digit is 5. Here, the convention is that if the preceding digit is even, the insignificant digit is simply dropped and, if it is odd, the preceding digit is raised by 1. Then, the number `2.74ul5` rounded off to three significant figures becomes 2.74. On the other hand, the number 2.735 rounded off to three significant figures becomes 2.74 since the preceding digit is odd.