Approximations & Rounding

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Transcript Approximations & Rounding

Approximations & Rounding
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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


285.5
285.6
285.0
286
286
285
Answers


285.4
285
When rounding to a whole number leave out the
decimal point.
Exercise 2

Round the numbers below to the precision given


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
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
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

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
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
2  0.638  27.1 1.28
K
96.1
2

Are you sure you have the right answer?

Carry out a check
Performing and checking calculations

 
2  3  0.6  302 1 6  6 101  9 102
K

100
102

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






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2  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

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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

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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