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
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
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 6 101 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
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
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.
Finish