Significant Figures

• When using our calculators we must determine the correct answer; our calculators are mindless drones and don’t know the correct answer.
• There are 2 different types of numbers
• Exact
• Measured
• Exact numbers are infinitely important
• Measured number = they are measured with a measuring device (name all 4) so these numbers have ERROR.
• When you use your calculator your answer can only be as accurate as your worst measurement…Doohoo ☺

Chapter Two

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Exact Numbers

An exact number is obtained when you count objects or use a defined relationship.�

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Counting objects are always exact� 2 soccer balls� 4 pizzas

Exact relationships, predefined values, not measured� 1 foot = 12 inches� 1 meter = 100 cm

For instance is 1 foot = 12.000000000001 inches? No

1 ft is EXACTLY 12 inches.

Learning Check

1. Exact numbers are obtained by?

a. using a measuring tool

b. counting

c. definition

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2. Measured numbers are obtained by?

a. using a measuring tool

b. counting

c. definition

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Solution

1. Exact numbers are obtained by

b. counting

c. definition

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2. Measured numbers are obtained by

a. using a measuring tool

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Learning Check

Classify each of the following as an exact or a

measured number.

1 yard = 3 feet

The diameter of a red blood cell is 6 x 10^-4 cm.

There are 6 hats on the shelf.

Gold melts at 1064°C.

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Solution

Classify each of the following as an exact (1) or a

measured(2) number.

1 - This is a defined relationship.

2 - A measuring tool is used to determine length.

1 - The number of hats is obtained by counting.

2 - A measuring tool is required.

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2.4 Measurement and Significant Figures

• 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.

Chapter Two

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What is the Length?

• 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

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Learning Check

What is the length of the wooden stick?

1) 4.5 cm

2) 4.54 cm

3) 4.547 cm

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8.00 cm or 8.1 or 7.9

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Measured Numbers

• 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.

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Precision vs Accuracy

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Chapter Two

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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.

Note the 4 rules

When reading a measured value, all nonzero digits should be counted as significant. There is a set of rules for determining if a zero in a measurement is significant or not.

• RULE 1. Zeros in the middle of a number are like any other digit; they are always significant.

Thus, 94.072 g has five significant figures.

• RULE 2. Zeros at the beginning of a number are not significant;

they act only to locate the decimal point. Thus, 0.0834 cm has three significant figures, and 0.029 07 mL has four.

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Chapter Two

• RULE 3. Zeros at the end of a number and after the decimal point are significant.
• It is assumed that these zeros would not be shown unless they were significant. 138.200 m has six significant figures. If the value were known to only four significant figures, we would write 138.2 m.
• RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant.
• We cannot tell whether they are part of the measurement or whether they act only to locate the unwritten but implied decimal point.

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Chapter Two

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Practice Rule #1 Zeros

45.8736

.000239

.00023900

48000.

48000

3.982×10⁶

1.00040

6

3

5

5

2

4

6

• All digits count
• Leading 0’s don’t
• Trailing 0’s do
• 0’s count in decimal form
• 0’s don’t count w/o decimal
• All digits count
• 0’s between digits count as well as trailing in decimal form

• Scientific notation is helpful for indicating how many significant figures are present in a number that has zeros at the end but to the left of a decimal point.
• The distance from the Earth to the Sun is 150,000,000 km. Written in standard notation this number could have anywhere from 2 to 9 significant figures.
• Scientific notation can indicate how many digits are significant. Writing 150,000,000 as 1.5 x 10⁸ indicates 2 and writing it as 1.500 x 10⁸ indicates 4.
• Scientific notation can make doing arithmetic easier.

Chapter Two

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2.6 Rounding Off Numbers

• 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.

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Chapter Two

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• 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.

Chapter Two

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Practice Rule #2 Rounding

Make the following into a 3 Sig Fig number

1.5587

.0037421

1367

128,522

1.6683 ×10⁶

1.56

.00374

1370

129,000

1.67 ×10⁶

Your Final number must be of the same value as the number you started with,

129,000 and not 129

Examples of Rounding

For example you want a 4 Sig Fig number

4965.03

780,582

1999.5

0 is dropped, it is <5

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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

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780,600

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2000.

RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers.

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Chapter Two

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• 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.

Chapter Two

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Multiplication and division

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

Addition/Subtraction

25.5 32.72 320

+34.270 ‑ 0.0049 + 12.5

59.770 32.7151 332.5

59.8 32.72 330

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Addition and Subtraction

__ ___ __

.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

Mixed Order of Operation

8.52 + 4.1586 × 18.73 + 153.2 =

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(8.52 + 4.1586) × (18.73 + 153.2) =

239.6

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2180.

= 8.52 + 77.89 + 153.2 = 239.61 =

= 12.68 × 171.9 = 2179.692 =