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SIGNIFICANT figures

SIGNIFICANT figures. Two types of numbers: exact and inexact . Exact numbers are obtained by counting or by definitions – a dozen of wine, hundred cents in a dollar All measured numbers are inexact . Learning objectives. Define accuracy and precision and distinguish between them

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SIGNIFICANT figures

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  1. SIGNIFICANT figures Two types of numbers: exact and inexact. Exact numbers are obtained by counting or by definitions – a dozen of wine, hundred cents in a dollar All measured numbers are inexact.

  2. Learning objectives • Define accuracy and precision and distinguish between them • Make measurements to correct precision • Determine number of SIGNIFICANT FIGURES in a number • Report results of arithmetic operations to correct number of significant figures • Round numbers to correct number of significant figures

  3. All analog measurements involve a scale and a pointer • Errors arise from: • Quality of scale • Quality of pointer • Calibration • Ability of reader

  4. ACCURACY and PRECISION • ACCURACY: how closely a number agrees with the correct value • PRECISION: how closely individual measurements agree with one another – repeatability • Can a number have high precision and low accuracy?

  5. 2.0 2.1 2.2 2.3 2.4 2.5 Significant figures are the number of figures believed to be correct • In reading the number the last digit quoted is a best estimate. Conventionally, the last figure is estimated to a tenth of the smallest division 2.3 6

  6. 2.0 2.1 2.2 2.3 2.4 2.5 The last figure written is always an estimate • In this example we recorded the measurement to be 2.36 • The last figure “6” is our best estimate • It is really saying 2.36 ± .01

  7. 97 98 99 100 Precision of measurement (No. of Significant figures) depends on scale – last digit always estimated • Smallest Division = 1 • Estimate to 0.1 – tenth of smallest division • 3 S.F. 99.6

  8. 70 80 90 100 Lower precision scale • Smallest Division = 10 • Estimate to 1 – tenth of smallest division • 2 S.F. 96

  9. 0 100 Precision in measurement follows the scale • Smallest Division = 100 • Estimate to 10 – tenth of smallest division • 1 S.F. 90

  10. Measuring length • What is value of large division? • Ans: 1 cm • What is value of small division? • Ans: 1 mm • To what decimal place is measurement estimated? • Ans: 0.1 mm (3.48 cm)

  11. Scale dictates precision • What is length in top figure? • Ans: 4.6 cm • What is length in middle figure? • Ans: 4.56 cm • What is length in lower figure? • Ans: 3.0 cm

  12. Measurement of liquid volumes • The same rules apply for determining precision of measurement • When division is not a single unit (e.g. 0.2 mL) then situation is a little more complex. Estimate to nearest .02 mL – 9.36 ± .02 mL

  13. Reading the volume in a burette • The scale increases downwards, in contrast to graduated cylinder • What is large division? • Ans: 1 mL • What is small division? • Ans: 0.1 mL

  14. RULES OF SIGNIFICANT FIGURES • Nonzero digits are always significant 38.57 (four) 283 (three) • Zeroes are sometimes significant and sometimes not • Zeroes at the beginning: never significant 0.052 (two) • Zeroes between: always 6.08 (three) • Zeroes at the end after decimal: always 39.0 (three) • Zeroes at the end with no decimal point may or may not: 23 400 km (three, four, five)

  15. Scientific notation eliminates uncertainty • 2.3400 x 104 (five S.F.) • 2.340 x 104 (four S.F.) • 2.34 x 104 (three S.F.) • 23 400. also indicates five S.F. • 23 400.0 has six S.F.

  16. Note: significant figures and decimal places are not the same thing • 38.57 has four significant figures but two decimal places • 283 has three significant figures but no decimal places • 0.0012 has two significant figures but four decimal places • A balance always weighs to a fixed number of decimal places. Always record all of them • As the weight increases, the number of significant figures in the measurement will increase, but the number of decimal places is constant • 0.0123 g has 3 S.F.; 10.0123 g has 6 S.F.

  17. Significant figure rules • Rule for addition/subtraction: The last digit retained in the sum or difference is determined by the position of the first doubtful digit 37.24 + 10.3 = 47.5 1002 + 0.23675 = 1002 225.618 + 0.23 = 225.85 • Position is key

  18. Significant figure rules • Rule for multiplication/division: The product contains the same number of figures as the number containing the least sig figs used to obtain it. 12.34 x 1.23 = 15.1782 = 15.2 to 3 S.F. 0.123/12.34 = 0.0099675850891 = 0.00997 to 3 S.F. • Number of S.F. is key

  19. Rounding up or down? • 5 or above goes up • 37.45 → 37.5 (3 S.F.) • 123.7089 → 123.71(5 S.F.); 124 (3 S.F.) • < 5 goes down • 37.45 → 37 (2 S.F.) • 123.7089 → 123.7 (4 S.F.)

  20. Scientific notation simplifies large and small numbers • 1,000,000 = 1 x 106 • 0.000 001 = 1 x 10-6 • 234,000 = 2.34 x 105 • 0.00234 = 2.34 x 10-3

  21. Multiplying and dividing numbers in scientific notation • (A x 10n)x(B x 10m) = (A x B) x 10n + m • (A x 10n)/(B x 10m) = (A/B) x 10n - m

  22. Adding and subtracting • (A x 10n) + (B x 10n) = (A + B) x 10n • (A x 10n) - (B x 10n) = (A - B) x 10n

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