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Approximations & Rounding. http://sst.tees.ac.uk/external/u0000504. Rounding. It is important to recognise the errors inherent in measurement Errors can propagate with calculation - as you have already seen

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Approximations & Rounding

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Approximations & Rounding

http://sst.tees.ac.uk/external/u0000504


Rounding

  • It is important to recognise the errors inherent in measurement

  • Errors can propagate with calculation - as you have already seen

  • When reporting figures it is important to only report to a justified degree of precision

  • The process of representing figures to an appropriate degree of precision is called rounding


Exercise 1

  • Round the following figures to the nearest whole number

    • 285.4 285.5 285.6285.0

  • Answers

    • 285286286285

  • When rounding to a whole number leave out the decimal point.


Exercise 2

  • Round the numbers below to the precision given

    • 345632 to the nearest 10 000

    • 0.063 to the nearest hundredth

    • 746.813 to the nearest 10

    • 95.8661 to the nearest tenth

    • 79.96 to the nearest tenth

  • Answers

    • 340 000, 0.06, 750, 95.9, 80.0


Exercise 3

  • Round the following numbers to three decimal places

    • 0.04567, 23.84521, 0.009763, 63567.23567

  • Now round the same numbers to three significant figures.

  • Answers

    • 0.046, 23.845, 0.010, 636567.236

    • 0.0457, 23.8 0.00976, 63600


Summary 1

  • All numbers representing measurements are approximations and should be rounded

  • If the final number is less than 5 round down, if it is 5 or more, round up.

  • Significant figures are counted from the leftmost non-zero digit.

  • With decimals, include a trailing zero if necessary to indicate precision

  • The degree of precision should be indicated in parentheses after the number e.g.

    • 0.010 (3 d.p.),0.00976 (3 s.f.)


Rounding and arithmetic

  • As you have seen earlier, arithmetic operations on measured values can have an impact, usually adverse, on the measurement errors

  • It is therefore important to be aware of the precision of the measurements and to take this in when quoting the results of calculated values.


Performing and checking calculations

  • Carry out the following calculation

  • Are you sure you have the right answer?

  • Carry out a check


Performing and checking calculations

  • This gives approximately 36

  • The actual answer is 39.21260646 (10 s.f.)

  • or is it?


Rounding with calculations

  • All the original values were based on measurements which were subject to error.

  • Let’s take a look at the values

    • 2p - A pure number

    • 0.638 - correct to 3 s.f.

    • 27.1 - correct to 3 s.f.

    • 1.28 - correct to 3 s.f.

    • 96.1 - correct to 3 s.f.

  • Since all values are correct to 3 s.f. at best, the result of the calculation must be quoted to no more than 3 s.f.

  • Hence the answer = 39.2 (3 s.f.)


Exercise 4

  • Four sticks of length 0.46 cm,27.6 cm, 3 cm, 0.12 cm are placed end to end. What is the total length?

  • 14.18 g of element A combined with 1.20g of element B using a balance correct to 0.01 g. After calculation, the mole ratio of A:B was found to be 4.0033778? What is the correct value of the mole ratio?

  • Answers:

    • 31 cm, 4.00


Beware rounding too soon!

  • The wavelength, l of monochromatic light passing through a diffraction grating can be found from

    • 2l = d sinq

    • Where d = slit width and q = angle of diffraction

  • In a particular case, the angle of diffraction of light passing through a grating having 600 slits/mm was 45.2° 0.1°. Calculate the slit width correct to 2 s.f.


Solution

  • d = 1 mm/600

    • 1.666666666 x 10-3

  • sin q = sin 45.2

    • 0.7095707365

  • Hence l = 1.666666666  10-3 x 0.7095707365  2

    • 5.9x10-4 mm


Exercise 5

  • A common procedure is to calculate d and sinq, write them down to 2 s.f. and then calculate l

  • Thus l = 1.7  10-3 x 0.71  2

    • 6.0 x 10-4 mm

  • A difference of 1.0 x 10-5 mm


Effect of early rounding

  • Let’s compare the error involved with the error in the original measurement

  • The measured angle, q has a much greater error than d

  • Error in q

    • 0.1/45.2 = 2.1 x 10-3  0.2%

  • Error in final answer

    • (6.0 - 5.9)/5.9 = 0.017  2%

  • Thus the calculation error is approx. 10 times the measurement error.


Summary 2

  • The accuracy of a multiplication or division can no better than that of the least accurate quantity in the calculation.

  • Only round your answers after the final calculation has been completed.


Finish


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