Significant Figures - Ms. De Wolf's Class Sitemsdewolflcchs.weebly.com/.../2/1/0/6/21063404/significant_figures.pdf · "read between the lines" for last digit. ... Significant Figures

Significant Figures

Numbers and Uncertainty

• Numbers express uncertainty.

• Exact numbers contain no uncertainty. They are obtained by counting objects (integers) or are defined, as in some conversion factors (ex min hours).

• Inexact numbers contain uncertainty. They are obtained from measurements. The un-certainty in a measurement is shown by the number of digits recorded.

• Every measurement has UNITS.

• Every measurement has UNCERTAINTY.

• Uncertainty refers to the errors in measurement

• The uncertainty of the measurement is determined by the scale of the measuring device.

– The smaller the unit you use to measure with, the more precise the measurement is

• 2 types of errors can lead to uncertainty in measurements

Measurements and Uncertainty

• Systematic error: come from the measuring instrument

– There is something wrong with the instrument or its data handling system\

– the instrument is wrongly used by the experimenter

• Examples:

– worn out instrument. For example, a plastic tape measure becomes slightly stretched over the years, resulting in measurements that are slightly too high.

– An incorrectly calibrated or tared instrument, like a scale that doesn’t read zero when nothing is on it.

– A person consistently takes an incorrect measurement. For example, they might think the 3/4″ mark on a ruler is the 2/3″ mark.

2 types of Error

• Random error: caused by unknown and unpredictable changes in the experiment. These changes may occur in the measuring instruments or in the environmental conditions.

• Example: You measure the mass of a ring three times using the same balance and get slightly different values: 17.46 g, 17.42 g, 17.44 g

• Minimizing random errors:

– Take more data.

1) a mechanical instrument (ruler, triple beam balance, graduated cylinder etc) record all the digits that are marked on the instrument’s

scale and estimate one more (and only one more) digit. "read between the lines" for last digit. That digit is an estimate,

and contains uncertainty. Example:

A few rules to observe in the lab:

• 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.

2) an electronic instrument

record all the digits on the readout. Consider the last digit to be approximate.

3) 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.

• What is the difference?

• Precision- how close individual measurements agree with each other.Accuracy- how close individual measurements agree with the true or accepted value.

Precision vs. Accuracy

• When making a measurement you must always estimate 1 place past the smallest division.

• Ruler example

– 9.5 cm is all you can read off of a cm ruler, you can estimate the 2nd decimal and say the pencil is 9.50 cm but to say that the pencil was 9. 5012 would give the impression that a very precise tool had been used not a ruler.

10

Significant figuresSignificant Figures are used to indicate the precision of a measured

number or to express the precision of a calculation with measured

numbers.

In any measurement

the digit farthest to the

right is considered to be

estimated.

0 1 2

1.32.0

Which digits are significant?

• Rule #1: All non-zero digits are significant

• 24 has two sig figs, 24.1 has 3 sig figs

• Rule #2: All zeros bounded (trapped) by non-zero integers are significant

• 2004 has four sig figs, 20.04 also has 4 sig figs

• Rule #3: Zeros placed before other digits (leading zeros) are not significant

• 0.024 has 2 sig figs

• Rule #4: Zeros at the end of a number are significant ONLY if they come after a decimal point

• 2.40 has three sig figs, 240 only has 2 sig figs

• Hint: Change the number to scientific notation. It is easier to see

• Example: 2.4 𝑥 102 has 2 sig. figs., while 2.40 𝑥 102 has 3 sig. figs.

Measurement # of Sig Figs

1) 1400.0

2) 300

3) 0.0050

4) 6001.30

5) 11232.0

6) 5.00

5

1

2

6

6

3

• 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?

QUESTIONS

Significant Figures in Calculations

RULE 1. MULTIPLICATION AND DIVISION

• the answer cannot have more significant figures than either of the original numbers.

•RULE 2. ADDITION AND SUBTRACTION

•the answer cannot have more digits after the decimal point than either of the original numbers.

•The last digit retained is set by the first doubtful digit.

3.18 𝐿+ 0.01315 𝐿

Two digits after the decimal

Two digits after the decimal

3.19315 𝐿

3.19 𝐿

Five digits after the decimal

• Rule 3: Rounding

– If the digit removed is 4 or less, drop it and all following digits • Ex. 2.427 becomes 2.4 when rounded to 2 significant figures

– If the digit removed is 4 or greater, round the digit preceding number up• Ex. 4.5832 becomes 4.6 when rounded to 2 significant figures

Rule 4: Working with exact and inexact numbers:

Exact numbers don't have sig figs because they don’t introduce uncertainty.

Just use sig figs in inexact numbers

0.16 km x 60 min = 9.6 kmmin hr hr

60 min/hour is an exact conversion

Rule 5: 5. If your calculator gives you fewer sig figs than the value should have, add zeroes

Example

0.465 x 0.200 = 0.0930

4.389 – 2.589 = 1.800

Rule 6: If your calculator gives you more digits to the left of the decimal than are significant, use scientific notation.

75.3 x 24.8 x 675 = 1260522 = 1.26 x 106

Examples of RoundingRound the following to 4 significant figures

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; note

you need a 4 Sig Fig

4965

780,600

2000.

Practice: RoundingMake the following into a 3 Sig Fig number

1.5547

.0037421

1367

128,522

1.6613 106

1.55

.00374

1370

129,000

1.66 106

Your Final number

must be of the same

value as the number

you started with,

129,000 and not 129

Addition/Subtraction Practice

• Add/Subtract the following. Remember to record your answer using significant figures

25.5+34.27059.77059.8

32.72- 0.0049

32.715132.72

320+ 12.5

332.5330

Look for the last important digit

Addition and Subtraction

.56 + .153 =

82000 + 5.32 =

10.0 - 9.8742 =

10 – 9.8742 =

• Add/Subtract the following. Remember to record your answer using significant figures

0.713 0.71

82005.32 82000

0.12580 0.1

0.12580 0

Multiplication and division

• 32.27 1.54 =

• 3.68 .07925 =

• 1.750 .0342000 =

• 3.2650106 4.858 =

• 6.0221023 1.66110-24 =

• Multiply/divide the following. Remember to record your answer using significant figures

49.6958 49.7

46.4353312 46.4

0.05985 0.05985

1.586137 107 1.586 x 107

1.000000 1.000