SIGNIFICANT FIGURES - Mathematical Operations

ADDITION AND SUBTRACTION:
When adding or subtracting numbers, count the NUMBER OF DECIMAL PLACESto determine the number of significant figures. The answer cannot CONTAIN MORE PLACES AFTER THE DECIMAL POINT THAN THE SMALLEST NUMBER OF DECIMAL PLACES in the numbers being added or subtracted.
Example:
23.112233 | (6 places after the decimal point) | 1.3324 | (4 places after the decimal point) | + 0.25 | (2 places after the decimal point) | 24.694633 | (on calculator) | 24.69 | (rounded to 2 places in the answer) |
Note: There are 4 significant figures in the answer.
MULTIPLICATION AND DIVISION:
When multiplying or dividing numbers, count the NUMBER OF SIGNIFICANT FIGURES. The answer cannot CONTAIN MORE SIGNIFICANT FIGURES THAN THE NUMBER BEING MULTIPLIED OR DIVIDED with the LEAST NUMBER OF SIGNIFICANT FIGURES.
Example:
23.123123 | (8 significant figures) | x 1.3344 | (5 significant figures) | 30.855495 | (on calculator) | 30.855 | (rounded to 5 significant figures) |
The rules governing fundamental operations involving significant figures are:

In addition and subtraction, identify first the s.f. in the decimal point of each given measurement. second, add or subtract(depends on the operation given) the given measurement and finally, your answer must have the least decimal point as the least accurate given measurement.
For example: 2.51 + 3.98000
The given 2.51 contains 2 decimal place while 3.98000 contains 5 decimal place so your answer must only have the less decimal point as the given. Your answer here is 6.49 not 6.49000.

In multiplication and division, identify first the number of significant figures, then do the given operation if it is multiplication or division. your answer must have the least accurate measurement. for example: 2.89x4.987 is 14.41243. since the least number of the significant figure in my example is 2.94 which contains 3 significant figures. Therefore, your answer must have the same no. of the significant figure. Round off your answer the answer in my example will become 14.4 only

remember: all non-zero digits are significant (1-9) all leading zeros- are zeros before the non-zero digit is not significant all sandwich zeros these are zeros between non-zero digits are significant all trailing or final zeros - zeros after the non-zero digit are not significant if it is in the left side of the decimal point because it only indicates or used as placeholders only but if the trailing zeros are in the right side of the decimal point and is written after a non-zero digit it is considered as significantMath Review for Chemistry
Significant Figures
Significant figures (sometimes called significant digits) provide a simple method for indicating the precision of a measurement. Simply stated, the more digits the more precise the measurement. A higher level of precision corresponds to having more information. The goal is to reflect accurately the amount of information that is known, by expressing measured values with an appropriate number of digits. Writing too few significant figures or too many significant figures, a much more common error when calculators are used, misleads the reader.
Generally, the value will need to be rounded to the appropriate number of significant figures. This may involve adding zeros to the number or removing digits by rounding. It is sometimes convenient to express the correct number of significant figures using scientific notation.
Rules for Counting Significant Figures 1. Exact numbers * - Considered to have unlimited precision (infinite number of significant figures). * - Integers are exact. One could count 15 books on a shelf. This number is exact. One cannot have 15.2 books. * - By definition there are exactly 60 seconds in 1 minute and 3 feet in 1 yard. * - Many unit conversion factors are considered exact and do not alter precision. 2. Nonzero digits * - All nonzero digits in a number are considered significant. 3. Zeros may or may not be significant * - Significant when between nonzero digits. * - Significant when at the end of a number that includes a decimal point. * - Not significant when before the first nonzero digit. * - Not significant when at the end of a number without a decimal point. Examples: * 805 has three sig figs but 805.0 has four.
6.10 has three sig figs but 6.100 has four.
4.013 has four sig figs.
900.2 has four sig figs.

0.40 has two sig figs.
0.0027 has two sig figs.
0.0302 has three sig figs.
0.8400 has four sig figs.

2400 has two sig figs.
But, 2.400×103 has four sig figs and 2.40×103 has three sig figs.
2.4×103 has two sig figs.
Rules for Rounding to Any Number of Significant Figures 1. Rounding down * - When the first digit following those to keep is in the range 0-4, that digit and all digits following to the right are dropped. 2. Rounding up * - When the first digit following those to keep is in the range 5-9, that digit and all digits following to the right are dropped. Then the last digit to be kept is increased by one. Examples (rounding to three digits): * 3.0132 = 3.01 (32 dropped because 3 is in the range 0-4)
6.1487 = 6.15 (87 dropped, 4 increased to 5 because 8 is in the range 5-9)
40.5046 = 40.5 (046 dropped because 0 is in the range 0-4)
928.69 = 929 (69 dropped, 8 increased to 9 because 6 is in the range 5-9) Examples (rounding to four digits): * 3.0132 = 3.013 (2 dropped because 2 is in the range 0-4)
6.1487 = 6.149 (7 dropped, 8 increased to 9 because 7 is in the range 5-9)
40.5046 = 40.50 (46 dropped because 4 is in the range 0-4)
928.69 = 928.7 (9 dropped, 7 increased to 8 because 9 is in the range 5-9)
Rules for Addition/Subtraction 1. Add or subtract the numbers. 2. Round result to the same number of decimal places as least precise value.
Rules for Multiplication/Division 1. Multiply or divide the numbers. 2. Round result to the same number of sig figs as least precise value. Examples:
Write the sum of 1.586 + 2.31 with the correct number of sig figs. * Step 1) 1.586 + 2.31 = 3.896 * Step 2) 3.896 = 3.90 (two decimal places because 2.31 has two)
The answer is 3.90
Write the difference of 0.954 - 0.3109 with the correct number of sig figs. * Step 1) 0.954 - 0.3109 = 0.6431 * Step 2) 0.6431 = 0.643 (three decimal places; 0.954 has three)
The answer is 0.643
Write the product of 2.10 × 0.5896 with the correct number of sig figs. * Step 1) 2.10 × 0.5896 = 1.23816 * Step 2) 1.23816 = 1.24 (three sig figs because 2.10 has three)
The answer is 1.24
Write the quotient of 16.15 ÷ 2.7 with the correct number of sig figs. * Step 1) 16.15 ÷ 2.7 = 5.98148148 * Step 2) 5.98148148 = 6.0 (two sig figs because 2.7 has two)
The answer is 6.0 Multi-step Calculations:
When calculations involve multiple operations, the normal algebraic order of operations applies. First expressions enclosed are evaluated, then multiplications and divisions are performed, and then additions and subtractions are performed.
Only the final result should be rounded, but determination of significant figures usually requires determination of the number of significant figures for the intermediate results. During intermediate steps, the result should not be rounded, though in practice it is acceptable to round these results to the number of significant figures of the most precise value.
The end result, as usual, should be rounded to the number of significant figures (or decimal places) of the least precise component of the calculation. Examples:
Write the result of 0.83 + 2.10 × 0.5896 with the correct number of sig figs. * Step 1a) 2.10 × 0.5896 = 1.238 (multiplication before addition) * Step 1b) 0.83 + 1.238 = 2.068 * Step 2) 2.068 = 2.07 (two decimal places; 0.83 has two)
The answer is 2.07
Rules for Logarithms 1. Calculate the logarithm of the number. 2. Round result keep the same number of decimal places as there were sig figs in original value. Examples: * log(4.500×103) = 3.6532125 = 3.6532 (4 decimal places) * log(4.50×103) = 3.6532125 = 3.653 (3 decimal places) * log(4.5×103) = 3.6532125 = 3.65 (2 decimal places)
The rules for expressing sig figs with logarithms will only rarely be encountered in introductory chemistry. The main context for logarithms is the calculation of pH in acid-base chemistry. Often, pH is reported with one decimal place corresponding to a concentration measurement with a precision of one sig fig.