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1.2 SIGNIFICANT DIGITS / SIGNIFICANT FIGURES Accuracy – Precision – The precision of any measurement depends upon the precision of the instrument used. The digits in an answer which imply more accuracy or precision than the measurements justify are not significant and should dropped so that those digits which remain truly imply the precision of the original measurements. The remaining digits are called significant digits or significant figures. Significant digits or significant figures consist of : A QUANTITY contains both a _____________________ and a ______________________. Measured numbers Exact numbers Determining which zeros are significant -- because zeros must be written both as placeholders and as indicators of the precision of the measurement, we must learn how to distinguish between them. The following rules are used to determine the number of significant digits: 1. 2. 3. 4. 5. All non-zero numbers are significant Sandwich zeroes are significant Leading zeroes are NEVER significant. Trailing zeroes are significant if decimal point is present. Exact numbers have as many significant figures as needed. Rules for Rounding: 1. If the eliminated digit is less than 5, leave alone. 2. If the eliminated digit is 5 or greater, round up. In Class Practice: Determine the number of significant digits in each of the following: 1. 0.02 ________ 11. 142 ________ 2. 0.020 ________ 12. 0.073 ________ 3. 501 ________ 13. 1.071 ________ 4. 501.0 ________ 14. 10810 ________ 5. 5000 ________ 15. 5.00 ________ 6. 5000. ________ 16. 55.320 ________ 7. 6051.00 ________ 17. 1.010 ________ 8. 0.0005 ________ 18. 154 ________ 9. 0.1020 ________ 19. 8710 ________ 10001 20. 1.0004 ________ 10. ________ Round each of the following to three significant digits. 1. 88.473 ____________ 9. 69.95 ____________ 2. 8505 ____________ 10. 0.000056794 ____________ 3. 976450 ____________ 11. 67.048 ____________ 4. 699.5 ____________ 12. 3.002 ____________ 5. 123.98 ____________ 13. 0.0300 ____________ 1.2 6. 0.00086321 ____________ 14. 90100 ____________ 7. 12.17 ____________ 15. 54.009 ____________ 8. 8040 ____________ SCIENTIFIC (EXPONENTIAL) NOTATION Putting Ordinary Numbers into Scientific Notation: Scientists (and those studying science) frequently must deal with numbers that are very_______________ or very _______________. Have you met Avogadro's number (6.02 x 10 23)? Or have you calculated the wavelength of red light (6.10 x 10-7 m)? If those numbers weren't written the way they are, all of us who must deal with them would be spending much of our time just counting the zeros that separate the figures from the decimal point. To avoid that kind of time wasting, a method of writing very large and very small numbers was invented. The rules for writing numbers in scientific notation are 1. The first figure is a number from 1 to less than 10. 2. The first figure is followed by a decimal point and then the rest of the figures. 3. Then multiply by the appropriate power of 10. In class practice: Write each of these numbers in scientific notation: 17 = ____________________ 0.000000614 = ____________________ 3= ____________________ 0.0037004 = ____________________ 5.000 = ____________________ 0.00000038 = ____________________ 215 = ____________________ 0.01010 = ____________________ 7,000,631 = ____________________ 0.00000000001 = ____________________ Putting Scientific Notation into Ordinary Numbers: If the number ends with a positive exponent, move the decimal point to the right. If the number ends with a negative exponent, move the decimal point to the left. In class practice: Write each of the following as ordinary numbers. 2.926847212 x 109 = 6 ___________________ 4.000 x 10-2 = -5 ___________________ 4.29 x 10 = ___________________ 4.92x10 = ___________________ 3.286 x 104 = ___________________ 8.429 x 10-1 = ___________________ 2 5.92000 x 10 = 3 4.37521 x 10 = ___________________ ___________________ -2 ___________________ -4 ___________________ 5.376 x 10 = 2.986 x 10 = 3. 1.2 Significant Figures and Scientific Notation – HW 1 I. Give the number of significant figures in each. 1. 967 6. 2.700 11. 0.0076009 2. 967,000 7. 304 12. 670,000 3. 9.67 8. 9. 13. 0.00872 4. 0.00967 9. 90 14. 8.53x103 5. 4050 10. 7.805 II. Round each of the following to four significant figures. 15. 2.16347x105 ______________ 17. 7.2513 _____________ 16. 4.000574x105 ____________ 18. 375.6523 ____________ III. Round each of the following to two significant figures. 19. 3.512 ____________ 21. 2.751x108 ____________ 20. 25.631 ____________ 22. 3.9814 x105 ____________ IV. Round each of the following to the nearest whole number. 23. 56.912 ____________ 25. 0.5182 ____________ 24. 3.4125 ____________ 26. 112.511 ____________ V. Round each of the following to the nearest hundredth. 27. 54.7421 ____________ 29. 79.2588 ____________ 28. 100.0925 ____________ 30. 0.9114 ____________ VI. Express each of the following numbers in scientific notation. (Remember to pay attention to zeros. include them if they are significant, do not include them If they are not) 31. 325 ________________ 36. 0.361 _________________ 32. 70 __________________ 37. 0.0428 ___________________ 33. 96,400 __________________ 38. 0.00573 _________________ 34. 5,921 __________________ 39. 0.0005438 35. 6,587,324,000 ________________ 40. 0.00005673 _________________ ___________________ VII. Write each of the following as ordinary numbers. (Watch for significant zeroes) 41. 3.64 x104 46. 2.97 x10-4 __________________ 42. 3.9734 x105 ._________________ 47. 3.88 x10-2 _________________ 43. 6.285x103 _________________ 48. 5.65 x10-1___________________ 44. 6.7978x100 _________________ 49. 3.7283 x 10-4 _________________ 45. 5.8643 x102 _________________ 50. 4.763x10-3 __________________ _________________