Lecture 2 Summary

The concept of significant figures ("sig-figs") and how to properly treat them in calculations is important in any experimental science. In any measurement, there is a degree of uncertainty, and the values reported from a measurement ought to reflect that uncertainty.

Here is a summary of rules for how the proper number of significant figures for the results of calculations are determined:

(1) Addition and subtraction: Express all numbers in scientific notation and with the same exponent (power of ten). Arrange the numbers with the decimal points lined up, just as is normally done in adding and subtracting numbers by hand. The significant figures of the result is determined by the last digit which is furthest to the left. In other words, the number of digits to the right of the decimal point in the result should be equal to the number of digits to the right of the decimal point in the quantity with the fewest such digits.

(2) Multiplication and division: This is the simplest rule to put into words. The significant figures of the result is determined by how many sig figs the number with the fewest sig figs in the calculation has. In other words, the number with the fewest sig figs limits the sig figs of the result.

(3) Logarithms and exponentials: The number of sig figs of the logarithm of a number will be determined by giving the result the same number of sig figs to the right of the decimal point as were in the number whose log is being taken. Thus, the number of sig figs generally increases when the log of a number is taken. In the logarithm of a number, the digits to the right of the decimal point are called the mantissa, and the number to the left of the decimal point is called the characteristic. Suppose the number whose logarithm is to be taken is written in scientific notation, with - as is customary - one digit to the left of the decimal point in the digit term. Then the power of ten in the exponential term determines the characteristic of the result, while the digit term determines the mantissa, which should contain the same number of digits. with one digit to the left of the decimal point. For the antilog (exponential) function, as in raising 10 to the power of some number, the number of digits in the mantissa determines the number of significant figures in the result. Thus, when taking an antilog, the number of significant figures will in general decrease.