Approximations &amp; Rounding

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# Approximations & Rounding - PowerPoint PPT Presentation

Approximations &amp; 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

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
• 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
• 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.
• 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?
• 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%