Rounding Off Numbers

Rounding Off Numbers In correcting a number to express the proper number of sig. fig., we often have to drop off unwanted digits. The rules for rounding off numbers are explained in your textbook and/or lab manual. Here is a summary: Rules for rounding off numbers: If the digit immediately to the right of the last sig. fig. is more than 5, you round up. If the digit immediately to the right of the last sig. fig. is less than 5, you round down. 35.76 in 3 sig. fig. is 35.8 (round up because it is more than half-way between 35.7 and 35.8) 35.74 in 3 sig. fig. is 35.7 (round down because it is less than half-way between 35.7 and 35.8) If the digit immediately to the right of the last sig. fig. is equal to 5, you round up if the last sig. fig. is odd. You round down if the last sig. fig. is even. You round up if 5 is followed by nonzero digits, regardless of whether the last sig. fig. is odd or even. 24.35 in 3 sig. fig. is 24.4 (round up because last sig. digit is 3, an odd number) 24.25 in 3 sig. fig. is 24.2 (round down because last sig. digit is 2, an even number) 24.258 in 3 sig. fig. is 24.3 (round up because the digits 58 means it is past halfway to 24.3) After rounding off, if the resulting number has ambiguous zeroes, it should be recorded in scientific notation to avoid ambiguity. 34821.0 in 2 sig. fig. is 35000 where the three zeroes may or may not be significant. The correct answer is 3.5 x 10 4 . Try these problems: 72.49 in 3 sig. fig. is ___________ 299000 in 2 sig. fig. is ___________ 45.52 in 3 sig. fig. is ___________ 92,528 in 4 sig. fig. is ___________ 120.05 in 4 sig. fig. is ___________ 13,052 in 3 sig. fig. is ___________ 239.5 in 3 sig. fig. is ___________ 2.448 x 104 in 3 sig. fig. is ___________ 28.149 in 3 sig. fig. is ___________ 32000.000 in 3 sig. fig. is ___________ 63500 in 2 sig. fig. is ___________ 89999 in 3 sig. fig. is ___________ Answers are shown below: 72.49 in 3 sig. fig. is 72.5 299000 in 2 sig. fig. is 3.0 x 105 45.52 in 3 sig. fig. is 45.5 92,528 in 4 sig. fig. is 9.253 x 104 120.05 in 4 sig. fig. is 120.0 13,052 in 3 sig. fig. is 1.31 x 104 239.5 in 3 sig. fig. is 2.40 x 102 2.448 x 104 in 3 sig. fig. is 2.448 x 104 28.149 in 3 sig. fig. is 28.1 32000.000 in 3 sig. fig. is 3.20 x 104 63500 in 2 sig. fig. is 6.4 x 104 89999 in 3 sig. fig. is _9.00 x 104 When to Use Scientific Notation A number should be expressed in scientific notation (with only one nonzero digit to the left of the decimal) under these conditions: A number with ambiguous zeroes (tailing zeroes in a number without a decimal) To remove the ambiguity, it MUST be expressed in sci. not. e.g. It is not clear whether 35000 has 2, 3, 4 or 5 sig.fig. It is not clear whether the three "tailing zeroes" are significant or not. If you mean 35000 to have 3 sig. fig., then it MUST be expressed as 3.50 x 10 4 . A number that is very small (as a rule of thumb, less than 0.01). It is tedious and riskier to copy numbers with a string of avoidable zeroes. e.g. 0.000 000 83 SHOULD be expressed as 8.3 x 10 - 7 A number that is in exponential form for any reason e.g. 324.3 x 10 - 8 SHOULD be expressed as 3.243 x 10 - 6 Some students indiscriminately express all their numbers in scientific notation because they are too lazy to figure out when it is necessary. Although it is not "wrong" to do so, you should learn when it is appropriate. For example, it would not be appropriate to tell someone to weigh out "2.5 x 10 grams of salt" when "25 grams of salt" would do equally well. 20.0 x 5.0 = _________________ (42 x 10 3) x 2 = ____________ 0.004 ? 800 = _______________ 22 x 2.0 = _________________