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Rules for Significant Figures The following applies for numbers that are measured: 1) All non-zero digits in a number are significant. a. A non-zero digit is any digit not zero: 1,2,3,4,5,6,7,8, and 9. 2) All leading zeros are not significant. a. Leading zeros are the zeros in front of a number (on the left side). For example, the number 0.00236 has 3 leading zeros in front of 236. These zeros are not significant. 3) Trailing zeros are significant only if a decimal point is present somewhere in the number. a. Trailing zeros are the zeros at the end of a number (on the right side). For example, the number 2000 has 3 trailing zeros. b. For example, the zeros in 25.0 and 1.300 are significant because a decimal point is present in both cases. c. The zeros in 350 and 1000 are not significant because a decimal point is not present in both cases. d. The trailing zeros in 0.00420 and 800. are significant because a decimal point is present in both cases. 4) Any zero between two other significant digits is always significant. a. For example, all of the zeros in 801 and 10.05 are significant. 5) A line over a zero indicates that it is significant. _ a. For example, there are 3 sig figs in 3500. The last zero (in the ones place) is not significant. 6) When multiplying or dividing, the answer will have the same amount of sig figs as the value with the least number of sig figs. If the answer has too many decimal places, round to the number of sig figs it should have. a. For example, 12 / 3.00 = 4 4.0 i. 12 has two significant figures and 3.00 has three, so the answer should only have two. b. For example, 3.20 x 5.04 = 16.128 16.1 i. The starting values only have three significant figures each, so round the answer to three significant figures, 16.1. 7) When adding and subtracting values, find the smallest decimal place that all values have in common and are significant and that is the last significant decimal place in your answer. Add the values like normal and round to the significant decimal place. c. For example, 5.23 (significant to hundredths place) + 3.484 (significant to thousandths place) 8.714 8.71 i. The answer is limited to the hundredths place because 5.23, so the answer is rounded to 8.72 (hundredths place). d. For example, 1050 (significant to tens place) -326 (significant to ones place) 724 720 i. The answer is limited to the tens place because of 1050, so the answer is rounded to 720 (tens place). 8) When performing a calculation that involves only multiplication or division, do the whole calculation and round at the end. Do not do any intermediate rounding. 9) When performing a calculation that involves only addition or subtraction, do the whole calculation and round at the end. Do not do any intermediate rounding. 10) When performing a calculation that involves both multiplication or division and addition or subtraction, you must round in between steps. That is, you must round between moving from a multiplication or division step to an addition or subtraction step. Rules for rounding When making calculations with measured values, the answer will need to be rounded to the correct number of significant figures according to the following rules. 1) Round down whenever the digit following the last significant figure is a 0,1,2,3, or 4. a. For example, 30.24 becomes 30.2. 2) Round up whenever the digit following the last significant figure is a 6,7,8,9 or a 5 with any nonzero digit at any place after it. a. For example, 22.49 becomes 22.5. b. For example, 54.7511 rounded to the tenths place becomes 54.8. 3) If the digit following the last significant figure is a 5 with either zeros or nothing after it, round to the even number. a. For example, 79.3500 rounded to the tenths place becomes 79.4. The value is rounded up to the even number. b. For example, 32.65 becomes 32.6. The value is rounded down to the even number 6 instead of up to 7.