SIGNIFICANT FIGURES The following rules will be used to determine the number of significant figures in a measured or calculated number 1 All nonzero digits are always significant examples 1 23 3 s f 1567 4 s f 2 Zeros between nonzero digits are always significant examples 17009 5 s f 6 008 4 s f 3405 4 s f 3 Leading zeros zeros to the left of the first nonzero digit are never significant examples 00987 3 s f 0 00001234 4 s f 4 Trailing zeros zeros to the right of the last nonzero digit in a number with a decimal point will always be counted as significant examples 345 980 6 s f 12300 5 s f 0 0009800 4 s f 5 1 Trailing zeros in a number without a decimal point generally will not be significant examples 2000 1 s f 101200 4 s f 5 2 If one or more of the trailing zeroes are significant then it is best to convert the number to scientific notation and include only the significant zeroes Remember every digit included in a number written in scientific notation is considered to be significant Consider the following possibilities for the number 109000 a None of the trailing zeroes are significant 3 s f 109000 written with 3 s f 1 09 x 105 b Only the first trailing zero is significant 4 s f 109000 written with 4 s f 1 090 x 105 c The first two trailing zeroes are significant 5 s f 109000 written with 5 s f 1 0900 x 105 d All three trailing zeroes are significant 6 s f 109000 written with 6 s f 109000 notice the decimal point or using scientific notation it would be 1 09000 x 105 Practice with Significant Figures version 2003 I Counting significant figures 1 2 State the number of significant figures in each of the following numbers and give the rule you used in each case Number significant figures Based on Rule 345 56 5 s f 1 4500 2 s f 5 1 239 1300 7 s f 4 00004976 4 s f 3 1 200 x 10 12 4 s f 5 2 25 0086 6 s f 2 Your turn Note I do not expect you to remember the rule number it is much more important that you have the rules for counting significant figures memorized so you will know which rule to apply when counting significant figures Number 1000000 1000000 1200 003 0012300 1457 00045321 1457 1 00200 Significant figures State the Rule II Rounding off calculated results 3 Round off each of the following numbers to 4 significant figures a 12346 b 12343 1235 decider number 6 greater than 5 12350 or 1 235x104 decider number 3 less than 5 Do you see why 12350 is correct and 12350 or 1235 are not In the first case 12350 is 5 s f rule 4 In the second case the number 1235 would change the magnitude of the original number That is the original number is about twelve thousand and 1235 is about twelve hundred The original magnitude of the number must be retained c d e f f 1 00345 1 003 decider number 45 less than 50 1 00350 1 004 decider number 50 equals 50 even remains even 1 00850 1 008 decider number 50 equals 50 odd raised to even 1 00851 1 009 decider number 51 greater than 50 000023445 00002344 decider number 45 less than 50 2 344 x 10 5 g 25300821 2 530 x 107 decider number 0821 less than 5000 h 199999 2 000 x 105 decider number 99 greater than 50 4 Your turn Round off the following to 5 significant figures a 1 00345 b 0023087610 c 12300000 d 56 340502 e 9 2003498 f 39999999 III Determining the number of significant figures in the result of a calculation 5 Rule 1 The result of a multiplication or division should contain the same number of significant figures as was present in the number with the fewest significant figures 2 0044 5 s f x 12 54 4 s f 25 135176 25 14 4 s f 00342 3 s f x 1 230 4 s f 000074085 0000741 3 s f 56 781 5 s f 6 Rule 2 The result of an addition or subtraction should contain the same number of decimal places d p as the number with the fewest decimal places 3 123 3 d p 123 23 2 d p 126 353 126 35 2 d p 456 1 1 d p 476 25 2 d p 20 15 20 2 1 d p 7 Notice how the number of significant figures and decimal places change in the following calculation as the different rules come into play 3 123 3 d p 123 23 2 d p 126 353 456 1 1 d p 476 25 2 d p 8 20 15 126 35 2 d p 5 s f 20 2 1 d p 3 s f Your turn 002345 x 5 234 123 137 49 23 023 x 18 000 x 00087638 3 870 x 10 88 312 81 1 88 312 81 1 4 6 25495 6 25 3 s f