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

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Approximations rounding

Approximations & Rounding

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


Rounding

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

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

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

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

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

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

Performing and checking calculations

  • Carry out the following calculation

  • Are you sure you have the right answer?

  • Carry out a check


Performing and checking calculations1

Performing and checking calculations

  • This gives approximately 36

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

  • or is it?


Rounding with calculations

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

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

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

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

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

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

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

Finish


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