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 ◦ 14.57 cm 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

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, a 7 was our guess...stop there

6

1 2 3 4 cm

What is the length of the wooden stick? 1) 4.5 cm 2) 4.54 cm 3) 4.547 cm

8

8.00 cm ?

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) 1400.0

2) 300

3) 0.0050

4) 6001.30

5) 11232.0

6) 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. 2.4271 becomes 2.4 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 5 or greater, 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.

If a calculation has several steps, it is best to round off at the end.

RULE 3. If the first digit removed is 5, the preceding digit stays the same if it is an even number, and increases by one if it is an odd number. This is called rounding off to the nearest even number. 4.5832 becomes 4 when rounded off to 1 significant figure since the first dropped digit is a 5. The number 3.56 also becomes 4 when rounded to 1 significant digit.

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

1.56

.00374

1370

128,000

1.67 106

Your Final number

must be of the same

value as the number

you started with,

128,000 and not 128

For example you want a 4 Sig Fig number

4965.03

780,582

1999.5

0 is dropped, it is <5

8 is dropped, it is >5; Note you must include the 0’s

5 is dropped it is = 5; therefore the end result should be even.

4965

780,600

2000.

25.5 32.72 320

+34.270 - 0.0049 + 12.5

59.770 32.7151 332.5

59.8 32.72 332

.56 + .153 = .713

82000 + 5.32 = 82005.32

10.0 - 9.8742 = .12580

10 – 9.8742 = .12580

.71

82000

.1

0

Look for the last important digit

32.27 1.54 = 49.6958

3.68 .07925 = 46.4353312

1.750 .0342000 = 0.05985

3.2650 106 4.858 = 1.586137 107

6.022 1023 1.661 10-24 = 1.000000

49.7

46.4

.05985

1.586 107

1.000