Transcript Significant Figures: RULES

The Rules of the Game
Over hundreds of years ago, physicists and
other scientists developed a traditional way of
expressing their observations.

International System of Units (SI) – metric
system.



The amount of information that the
measuring instrument can provide.
The precision of a measuring tool depends on
how readable the tool is.
Ruler example
◦ 14.5 m is all you can read off of a cm ruler
◦ To say that the line was 14.5678 would give the
impression that a very precise tool had been used
not a ruler.



When measuring with a mechanical instrument
(ruler, triple beam balance etc), record all the digits
that are marked on the instrument’s scale and
estimate (and only one) more digit.
When measuring with an electronic instrument,
record all the digits on the readout. Consider the
last digit to be approximate.
Round calculated answers only once, at the end of
the calculation, so that the number of significant
digits reflects the precision of the original
measurements.




Every experimental
measurement has a
degree of uncertainty.
The volume, V, at right
is certain in the 10’s
place, 10mL<V<20mL
The 1’s digit is also
certain,
17mL<V<18mL
A best guess is needed
for the tenths place.
5
1





2
3
We can see the markings between 1.6-1.7cm
We can’t see the markings between the .6-.7
We must guess between .6 & .7
We record 1.67 cm as our measurement
The last digit an 7 was our guess...stop there
6
4 cm
What is
stick?
1)
2)
3)
the length of the wooden
4.5 cm
4.54 cm
4.547 cm
? 8.00 cm
8



Do you see why Measured Numbers have
error…you have to make that Guess!
All but one of the significant figures are known
with certainty. The last significant figure is only
the best possible estimate.
To indicate the precision of a measurement, the
value recorded should use all the digits known
with certainty.
Below are two measurements of the mass of
the same object. The same quantity is being
described at two different levels of precision
or certainty.
1) All non- zero digits are significant
Ex. (34 cm --- 2 sig fig)
2) Zeros in zero sandwiches are significant
Ex. (2009 --- 4 sig fig)
3) In order for zeros at the end of a measurement to be
significant, there must be a decimal point.
Ex. (23.000 -- 5 sig fig but 2300 ---2 sig fig)
4) Zeros to the left of the first non-zero are NOT
significant. They are just place holders.
Ex. (0.00124 ---3 sig fig)
Measurement # of Sig Figs
1)
2)
3)
4)
5)
6)
1400.0
300
0.0050
6001.30
11232.0
5.00



When we are using measurements in
different calculations in Chemistry and
Physics and even Biology, we need to
account for the level of precision. To do so,
scientists use significant figures.
The last sig fig in a measurement is always
the doubtful digit. But that is not always
clear. For example
12340cm or 12340.cm or 12340.0cm


If Jenn measures a line to be 12.0 cm, what
number is she doubtful about and how many
sig. figs. are there?
If Darren measures a mass to be 1300 g,
what number is he doubtful about and how
many sig. figs. are there?



Scientists use scientific notation as a method
to show the proper amount of sig. figs.
300 km can be written as 3 x 102 km.
(102 is 100 – so 3 x 100 =300)
300 km (with 2 sig. figs) can be written as 3.0
x 102 km. Notice we clearly have 2 sig. figs.

When multiplying by 10+x we can move the
decimal x times to make the number bigger.
Ex. 4 x 105 N = 400000 N
Ex. 3.00 x 104 kg = 30000 kg (3 sig. figs –
shown in sci. notation)

When multiplying by 10-x, we can move the
decimal x times to make the number smaller.

6.01 x 10-2 m = 0.0601 m (3 sig. figs)

3.10 x 10-5 N = 0.0000310 N (3 sig. figs.)



Often when doing arithmetic on a pocket
calculator, the answer is displayed with
more significant figures than are really
justified.
How do you decide how many digits to
keep?
Simple rules exist to tell you how.


Once you decide how many digits to retain,
the rules for rounding off numbers are
straightforward:
RULE 1. If the first digit you remove is 4 or
less, drop it and all following digits. 2.6271
becomes 2.6 when rounded off to two
significant figures because the first dropped
digit (a 2) is 4 or less.


RULE 2. If the first digit removed is greater
than 5, round up by adding 1 to the last digit
kept. 4.5832 is 4.6 when rounded off to 2
significant figures since the first dropped
digit (an 8) is 5 or greater.
RULE 3. If the first digit removed is equal to
5 and the digit before it is an even number,
drop it and all following digits. If the digit
before it is an odd number, round up by
adding 1 to the last digit kept.



* If there are any numbers written after the
five, then you must round up.
Example: 1.245 is 1.24 when rounding to 3
sig figs
Example: 1.245000001 is 1.25 because there
is a number after the 5.
RULE 1. In carrying out a multiplication or
division, the answer cannot have more
significant figures than either of the original
numbers.
RULE
2. In carrying out an addition or
subtraction, the answer cannot have more
digits after the decimal point than either
of the original numbers.
Make the following into a 3 Sig Fig number
1.5587
.0037421
1367
128,522
1.6683 106
Your Final number
1.56
must be of the same
.00374 value as the number
you started with,
1370
129,000 and not
129,000 129. It is 129000
because there is a 5
6
1.67 10 with numbers after it.
For example you want a 4 Sig Fig number
Number after 5, so
rounds up
4965.03
4966
780,582
8 is dropped, it is >5;
780,600 Note you must include
the 0’s
1999.5
2000.
5 is dropped it is = 5;
note you need a 4 Sig Fig
25.5
+34.270
59.770
59.8
32.72
- 0.0049
32.7151
32.72
320
+ 12.5
332.5
330
.56 + .153 = .713
.71
82000 + 5.32 = 82005.32 82000
10.0 - 9.8742 = .12580
.1
10 – 9.8742 = .12580
0
Look for
the last
important
digit
32.27  1.54 = 49.6958
49.7
3.68  .07925 = 46.4353312
46.4
1.750  .0342000 = 0.05985
.05985
7
3.2650106  4.858 = 1.586137  107 1.586 10
6.0221023  1.66110-24 = 1.000000 1.000

When deciding the number of significant
digits kept for calculations involving a series
of operations, round for each set of rules
(multiplication/division and
addition/subtraction).
Ex: 3.56g + 1.3g= 4.86g = 4.9g/1mL
3.3mL -2mL
1.3 mL
= 5g/mL