SIGNIFICANT figures
Finding the significance in a number and its
importance in measurements
Learning objectives





Define accuracy and precision
Make measurements to correct precision
Determine number of SIGNIFICANT
FIGURES in a number
Report results of arithmetic operations to
correct number of significant figures
Round numbers to correct number of
significant figures
Exact and inexact

Two types of numbers: exact and
inexact.

Exact numbers are obtained by counting or
by definitions – a dozen of wine, hundred
cents in a dollar

All measured numbers are inexact.
All analog measurements involve a
scale and a pointer

Errors arise from:
–
–
–
–
Quality of scale
Quality of pointer
Calibration
Ability of reader
ACCURACY and PRECISION


ACCURACY: how closely a
number agrees with the
correct value
PRECISION: how closely
individual measurements
agree with one another –
repeatability
–
Can a number have high
precision and low accuracy?
Significant figures are the number of
figures believed to be correct

In reading the number the last digit quoted is a best estimate.
Conventionally, the last figure is estimated to a tenth of the
smallest division
2.3 6
2.0
2.1
2.2
2.3
2.4
2.5
The last figure written is always an
estimate



In this example we recorded the
measurement to be 2.36
The last figure “6” is our best estimate
It is really saying 2.36 ± .01
2.0
2.1
2.2
2.3
2.4
2.5
Precision of measurement (No. of
Significant figures) depends on scale –
last digit always estimated
Smallest Division = 1

Estimate to 0.1 – tenth of smallest division

3 S.F.
99.6

97
98
99
100
Lower precision scale



Smallest Division = 10
Estimate to 1 – tenth of smallest division
2 S.F.
96
70
80
90
100
Precision in measurement follows the
scale



Smallest Division = 100
Estimate to 10 – tenth of smallest division
1 S.F.
90
0
100
Measuring length

What is value of large
division?
–

What is value of small
division?
–

Ans: 1 cm
Ans: 1 mm (1 cm = 10 mm)
To what decimal place is
measurement estimated
(1/10 smallest division)?
–
–
Ans: 0.1 mm
(3.48 ± 0.01cm = 34.8 ±
0.1 mm)
Scale dictates precision

What is length in (a)?
–

What is length in (b)?
–

Ans: 4.56 cm
What is length in (c)?
–

Ans: 4.6 cm
Ans: 3.0 cm
What is length if ruler in
(b) is used?
–
Ans: 3.00 cm
Measurement of liquid volumes


The same rules apply
for determining
precision of
measurement
When division is not a
single unit (e.g. 0.2 mL)
then situation is a little
more complex.
Estimate to nearest .02
mL – 9.36 ± .02 mL
Reading the volume in a burette


The scale increases
downwards, in contrast
to graduated cylinder
What is large division?
–

Ans: 1 mL
What is small division?
–
Ans: 0.1 mL
RULES OF SIGNIFICANT FIGURES


Nonzero digits are always significant
(four) 283 (three)
Zeroes are sometimes significant and
sometimes not
–
–
–
–
38.57
Zeroes at the beginning: never significant 0.052 (two)
Zeroes between: always
6.08 (three)
Zeroes at the end after decimal: always
39.0 (three)
Zeroes at the end with no decimal point may or may
not: 23 400 km (three, four, five)
Scientific notation eliminates
uncertainty





2.3400 x 104 (five S.F.)
2.340 x 104 (four S.F.)
2.34 x 104
(three S.F.)
23 400. also indicates five S.F.
23 400.0 has six S.F.
Note: significant figures and decimal
places are not the same thing




38.57 has four significant figures but two decimal
places
283 has three significant figures but no decimal
places
0.0012 has two significant figures but four decimal
places
A balance always weighs to a fixed number of
decimal places. Always record all of them
–
–
As the weight increases, the number of significant figures in
the measurement will increase, but the number of decimal
places is constant
0.0123 g has 3 S.F.; 10.0123 g has 6 S.F.
Significant figure rules


Rule for addition/subtraction: The last digit
retained in the sum or difference is
determined by the position of the first
doubtful digit
37.24 + 10.3 = 47.5
1002 + 0.23675 = 1002
225.618 + 0.23 = 225.85
Position is key
Significant figure rules


Rule for multiplication/division: The product
contains the same number of figures as the number
containing the least sig figs used to obtain it.
12.34 x 1.23 =
15.1782
= 15.2 to 3 S.F.
0.123/12.34 = 0.0099675850891
= 0.00997 to 3 S.F.
Number of S.F. is key
Rounding up or down?

5 or above goes up
–
–

37.45 → 37.5 (3 S.F.)
123.7089 → 123.71(5 S.F.); 124 (3 S.F.)
< 5 goes down
–
–
37.45 → 37 (2 S.F.)
123.7089 → 123.7 (4 S.F.)