2E Significant Figure Rules

When adding, subtracting, multiplying or dividing decimals it is important to know where to round the answer to. Most science classes base their rounding on something called significant figures. When taking science classes, it is vital to know the rules for significant figures.

Let’s start by defining what numbers in a decimal are significant and which are not. This will help us determine how many significant figures a number has.

Significant Figure Rules

• Non-zero numbers (1,2,3,4,…,9) are always significant. For example 2.5834 has five significant figures.
• Zeros between non-zero numbers are also considered significant. For example 20.0003 has six significant figures.
• Zeros at the end of a whole number are not considered significant. For example 31,000 has only two significant figures.
• Zeros to the right of a decimal point and before any non-zero numbers are not considered significant. For example 0.000027 has only two significant figures.
• In a decimal, zeros to the right of a non-zero number are significant. For example 0.3400 has four significant figures.

Significant figures and scientific notation relate well to one another. You will only use significant numbers in the scientific notation representation of the number. For example we said that 0.000027 has only two significant figures. Notice the scientific notation for this number $2.7\times {10}^{-5}$only uses the two significant figures.

Now you try a couple of examples with your instructor.

Rounding Rules with significant figures

1. When adding or subtracting decimals, identify the number that has the least number of place values to the right of the decimal. Round the sum or difference to the same place value. For example $$5.217+1.38+2.5=9.097\approx 9.1$$ (Notice the 2.5 has only 1 decimal place to the right of the decimal. Since this was the least amount we rounded our answer to the same accuracy.)
2. When multiplying or dividing decimals, identify the number that has the least significant figures. Round your answer so that it has the same number of significant figures. For example $$12.831\times 17.4=223.2594\approx 223$$ (notice that 12.831 has 5 significant figures and 17.4 has 3 significant figures. Since the least is 3, we will want our answer to have 3 significant figures. So we had to round to the ones place to have 3 significant figures.)

Now you try a couple more examples with your instructor.

Practice Problems Section 2E

Determine the number of significant figures for each of the following.

1. 7.569

   Check answer

   4

2. 3.0062

3. 700,000

   Check answer

   1

4. 8.200

5. 0.00051

   Check answer

   2

6. 9,260,000

7. 358.1

   Check answer

   4

8. 0.000572

9. 23.1

   Check answer

   3

10. 90

11. 5.0174

    Check answer

    5

12. 21,350

13. 33.7934

    Check answer

    6

14. 1.00006

15. 28,700

    Check answer

    3

16. 0.000309

17. 12,400

    Check answer

    3

18. 1.3750

19. 0.000480

    Check answer

    3

20. 35,481.2

21. 0.0003031

    Check answer

    4

22. 55,600,000

23. 850

    Check answer

    2

24. 84.007

25. 23.409

    Check answer

    5

26. 200,000

27. 7.1190

    Check answer

    5

28. 0.35

29. 0.000005

    Check answer

    1

30. 0.000000403

Add or subtract the following. Then use significant figure rounding rules to round the answer to the appropriate place.

31. $$3.8+7.249$$

    Check answer

    11.0

32. $$3.8269-0.097$$

33. $$0.7852+2.3$$

    Check answer

    3.1

34. $$9.5-0.3174$$

35. $$72.431+5.56$$

    Check answer

    77.99

36. $$19.31-12.8$$

37. $$75-54.62$$

    Check answer

    20

38. $$1.0893+0.641$$

39. $$11.07-8.43255$$

    Check answer

    2.64

40. $$12-7.113$$

41. $$128.4+89.927+13$$

    Check answer

    231

42. $$13.4-5.8+7.149$$

43. $$65.7-19.31+8.5$$

    Check answer

    54.9

44. $$300+72.4+258.1$$

45. $$980.3-746.2-35$$

    Check answer

    199

46. $$6.314+0.996-2.16$$

47. $$0.08751-0.004468+1.05$$

    Check answer

    1.13

48. $$13.4-5.8+7.149$$

Multiply or divide the following in order from left to right. Then use significant figure rounding rules to round the answer to the appropriate place.

49. $$7.631÷0.25$$

    Check answer

    31

50. $$0.94\times 1.47$$

51. $$9\times 1.633$$

    Check answer

    10

52. $$8.415÷0.05$$

53. $$\frac{0.4489}{0.33}$$

    Check answer

    1.4

54. $$0.4\times 5.993$$

55. $$77\times 0.0581$$

    Check answer

    4.5

56. $$\frac{8.832}{0.07}$$

57. $$23.611÷0.025$$

    Check answer

    940

58. $$9.4\times 3.88$$

59. $$13.4÷0.2\times 57$$

    Check answer

    4,000

60. $$4.2÷0.25\times 1.7$$

61. $$7.2\times 0.4\times 1.11$$

    Check answer

    3

62. $$5.5÷0.11\times 0.006$$

63. $$50.2÷0.8\times 90.33$$

    Check answer

    6,000

64. $$0.099\times 1.3\times 42.88$$

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.