Approximations & Rounding

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 • 285.4 285.5 285.6 285.0 • Answers • 285 286 286 285 • 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.

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Approximations & Rounding