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Student Academic Learning Services Page 1 of 5 Significant Digits What are significant digits? Significant digits are a way to give solutions in a way that is as accurate as the question. They present a set of rules for rounding off answers to an appropriate number of decimal places. In general, the number of decimal places we end up using is dictated by how accurate our measurements are. This insures that the final answer is just as accurate as the measurements that went into determining it. For example, if we measure the lengths of the sides of a rectangle as 2.5 cm and 1.5 cm respectively then we may conclude that the area of the rectangle is 3.75 cm (2.5 X 1.5). BUT, if both our measurements for side length were only accurate to one decimal place then how can our area calculation be accurate to two decimal places? The answer is that it can’t. What we need is a method for figuring out what accuracy we should be rounding our answers to. This is what the study of significant digits provides for us. Which digits are significant? There is a set of rules for determining which digits of a number are significant. This is important because later we will need to know how many significant digits a number has. A digit IS significant if: Rule 1 Example Rule 1 Explanation: It is NOT zero Rule 2 Example Rule 2 Explanation: It is zero and is contained between two significant digits. Rule 3 Example Rule 3 Explanation: It is a zero that is to the right of All numbers are significant the decimal and does not because the extra zeros indicate change the value of the that the number is more accurate number. then it would have been without the zeroes. www.durhamcollege.ca/sals Student Services Building (SSB), Room 204 905.721.2000 ext. 2491 This document last updated: 12/22/2010 Student Academic Learning Services Page 2 of 5 A digit is NOT significant if: Rule 4) Example Rule 4 Explanation The zeros are place holders 5300 Only the first two digits are UNLESS it is specifically significant in the first example, stated that the zeros are 402000 and the first three in the second significant. example. The extra zeroes are acting as place holders, and unless we are told otherwise, we assume that the zeroes are not significant. Rule 5) Example Rule 5 Explanation The zeroes come to the left of 0.0005342 All the zeroes to the left of the all the significant digits. non-zero digits are also merely 0.0656 placeholders. Therefore there are four significant digits in the first example and three in the second example. Multiplying and Dividing Significant digits The number of significant digits in the product or quotient should be equal to the number of significant digits in the factor with the least amount of significant digits. Examples: and BUT when we use significant digits the answers change. and The numbers with the least significant digits in the first example are both 36 and 420. These two numbers have both two significant digits where as the second example contains a number with a single significant digit. Therefore, we can see that the answer for the first example has two significant digits. The second example has an answer with only one significant digit because the number 3 only has one significant digit. Our answer has changed due to the use of significant digits. Adding and Subtracting Significant digits For adding and subtracting the rule states that the sum must be rounded to the least number of decimal places of all the terms being added. www.durhamcollege.ca/sals Student Services Building (SSB), Room 204 905.721.2000 ext. 2491 This document last updated: 12/22/2010 Student Academic Learning Services Page 3 of 5 This means that a sum cannot be more accurate than any of the terms added to make the sum. Also, note that this rule applies regardless of the number significant digits in any of the terms. www.durhamcollege.ca/sals Student Services Building (SSB), Room 204 905.721.2000 ext. 2491 This document last updated: 12/22/2010 Student Academic Learning Services Page 4 of 5 For instance: and But when we use significant digits the answers change. and In the first example, the least accurate measurement is five. It is only accurate to the nearest unit. Therefore, our answer for example 1 can only be accurate to the nearest unit. In the second example, the least accurate number is 5000, so we must round our answer to the nearest thousand. Combining Addition and Multiplication When you must evaluate an expression with both addition and multiplication, the rules of BEDMAS still apply, but you must determine the correct number of significant digits at every step along the way to the final answer Practice Problems Evaluate each of the following expressions with the correct significant digits. If there is doubt, always assume a zero is not significant: 1) 150.0 x 14.5 x 1.0032 2) 750 2.0 + 8 3) 1432 + 12.5 Answers: 4) (100 - 92.5) x 5500 1) 2180 5) 4.35 x 5.56 x 12.43 100.0 2) 390 3) 1440 4) 0 www.durhamcollege.ca/sals (SSB), Room 204 Student Services Building 5) 3.01 6) 144.5 2491 905.721.2000 ext. This document last updated: 12/22/2010 Student Academic Learning Services Page 5 of 5 6) 4.3521 x 3.545 15.89 + 143.5 www.durhamcollege.ca/sals Student Services Building (SSB), Room 204 905.721.2000 ext. 2491 This document last updated: 12/22/2010