Significant Figures and Scientific Notation
Precision is very important in science when measurements are being taken. Errors can be minimized with the use of precise
equipment…which includes the reporting of measured data in significant figures. When measuring, significant figures are all the
numbers reported, with certainty, by the equipment, plus one estimated digit(see example). There are five rules with regards to reporting
significant figures.
Rule 1- In numbers that do not contain zeros, all the digits are significant.
3.14
three sig figs
567.2
four sig figs
765
three sig figs
Rule 2- All zeros between signficant digits are significant.
4005
four sig figs
5.09
three sig figs
12.073
five sig figs
Rule 3- Zeros to the left of the first nonzero digit serve only to fix the position of the decimal point and are not significant.
0.0087
two sig figs
0.0427
three sig figs
0.000001
one sig fig
Rule 4- In a number with digits to the right of the decimal point, zeros to the right of the last nonzero digit are significant.
53.0
three sig figs
53.00
four sig figs
0.02100
four sig figs
Rule 5-In a number with no decimal point that ends in one or more zeros, the zeros that end the number may or may not be significant.
3600
two or four sig figs depending on how the number was obtained. YES(counted or defined) No(measured)
Scientific Notation can alleviate some of the confusion surrounding significant figures. It uses a base ten logarithmic scale to simplify
numbers. The nature of the number is not changed…just its format. In Scientific Notation, all digits are considered significant.
4600000
4.6 x 106
two sig figs
0.00542
5.42 x 10-3
three sig figs
143.20
1.4320 x 102
five sig figs
.000000073
7.3 x 10-8
two sig figs
Rounding - Many times calculators will display excess digits and need to be rounded. Rounding numbers to remove insignificant digits
has rules too.
Rule 1- If the first digit to the right of the last significant figure is less than 5, that digit and all that follow are dropped.
68.7543 rounded to four significant figures is 68.75
Rule 2- If the first digit to be dropped is greater than 5(or equal to 5 with non zero digits trailing), the last retained digit is raised one unit.
68.7543 rounded to three significant figures is 68.8
Rule 3- If the first digit to be dropped is a 5 followed by zeros or no other digits, then an odd/even rule is applied.
68.75 rounded to three significant figures is 68.8 because 7 is an odd number and we round up for odd.
68.85 rounded to three significant figures is 68.8 because 8 is an even number and we do not change when it is
Addition/Subtraction- the answer should not have digits beyond the last digit position that is common to all the numbers being added
or subtracted.
This answer must be rounded to 100.1
34.6
345.56
18.8
This answer must be rounded to 68
- 245.5
+15
100.06(calculator answer)
68.4(calculator answer)
Multiplication/Division- To correctly round numbers after multiplying or dividing, the answer must have the same number of
significant figures as the number with the fewest significant figures.
3.5 x 211.3 = 739.55(on calculator) must be rounded to two sig figs = 740 or 7.4 x 10
3.45 ÷ 210 = .016428571(calculator) = 1.6 x 10-2
Worksheet
Using the five rules for significant figures, determine how many significant figures each of the following numbers has.
1. 45.9090
6. 765.2
11. 0.008700
2. 0.0078
7. 0.0780
12. 376
3. 880
8. 9806
13. 990.00
4. 909.07
9. 34
14. 76320.01
5. 123400
10. 80320
15. 053.00
Round the following numbers to the specified number of significant figures using the rules for rounding.
3 sig figs
4 sig figs
5 sig figs
16. 173.46
21. 0.012345
26. 8996.55
17. 8951
22. 34.6753
27. 303.03425
18. 39.245
23. 998.1450
28. .6666666
19. 264.52
24. 1001.6
29. .3333333
20. 264.5
25. 55.945
30. 7652.359
Round the following numbers to the specified number of significant figures and convert them into scientific notation.
3 sig figs
4 sig figs
5 sig figs
31. 343599900
36. .00084235
41. 991456789132
32. 33000
37. 300.455
42. .00003265984812
33. 5555555.5
38. 6459.86
43. 1.0023564
34. .000000123
39. 789.153
44. 26548.00852
35. .08967
40. 12.434
45. 0.00000003985
Add /subtract the following numbers, round the answer to the correct number of significant figures, then write in scientific
notation.
46. 45.6 + 14.55 =
47. 18.4 + 7.3 + 90 =
48. 111 + 120 + 909 =
49. .787 - 3.23 =
50. .2565 + 5.26 + 95.36 =
51. .265 + 918 - .989 =
Multiply/divide the following numbers, round the answer to the correct number of significant figures, then write in scientific
notation.
52. 48.59 x 32.4 =
53. 332 ÷ 584 =
54. 30.5 x 50.3 =
55. 1.45049 ÷ .343
56. 96.2 x 35 =
57. 1.113 ÷ 123 =
Word Problems
58. Nancy measured each of her fingers on one hand: 11.1 cm 13.4 cm 10 cm 7.3cm What is her average finger length? (hint: use
only the rule for addition and subtraction)
59.Density is the amount of mass that is found in a particular volume…expressed as mass ÷ volume. If a scientist found that a particular
fluid had a mass of 54.5 grams within a volume of 111.56 mL, what would the liquids density be?
60. A train is moving at 65mph. What distance has it covered if it has been travelling for 2.75 hours?