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Approximations & RoundingPowerPoint Presentation

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
- 285.4 285.5 285.6 285.0

- 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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