Motion Unit Measurements

Significant Figures in Calculations

“A man with a watch knows what time it is. A man with two watches is never sure”

(Unknown)

I weigh some sugar on my kitchen scales and some more sugar on my lab balance. The results are shown below. How much sugar should I say I have, in total, without being misleading?

Kitchen scales: 2.2 kilograms

Lab scale: 101.237grams

(1kg = 1000g)

Significant Figures in Measurement

It is important to be honest when reporting a measurement, so that it does not appear to be more accurate than the equipment used to make the measurement allows. We can achieve this by controlling the number of digits, or significant figures, used to report the measurement.

Significant Figures in Measurement

The numbers reported in a measurement are limited by the measuring tool

Significant Figures • Significant figures are the meaningful figures

in our measurements and they allow us to generate meaningful conclusions

• Numbers recorded in a measurement are significant. – All the certain numbers plus first estimated

number– e.g. 2.85 cm

• We need to be able to combine data and still produce meaningful information

Where we are going….

3.54 + 3.4

Counting Significant Figures

All non-zero digits in a measured number are ALWAYS significant.

Number of Significant Figures

38.15 cm5.6 ft65.6 lb122.55 m

Sandwiched Zeros

Complete: Zeros between nonzero numbers

are ALWAYS significant

Number of Significant Figures

50.8 mm

2001 min

0.702 lb

0.00405 m

Leading Zeros

All zeros to the left of the first non-zero (leading zeros in decimal

numbers) are not significant.

Number of Significant Figures

0.008 mm

0.0156 oz

0.0042 lb

0.000262 mL

Trailing Zeros

If there is a decimal, zeros to the right of the last non zero are ALWAYS significant (trailing zeros in numbers without decimals are significant if they are serving as place holders).

Number of Significant Figures

25.000 in.

20.0 yr

48.600 gal

20 005 000. 00 g

Trailing Zeros

It is difficult to decide significance of zeros to the right of non zero digits in a number without a decimal. This depends on the instrument used. A line or decimal is often used to identify significance.

Otherwise if there is no decimal they are not significant.

Number of Significant Figures

25 000 in.

200 yr

48.600 gal

20 005 000 g

Learning Check

A. Which answers contain 3 significant figures?1) 0.4760 2) 0.00476 3) 4760

B. All the zeros are significant in

1) 0.00307 2) 25.300 3) 2.050 x 103

C. 534,675 rounded to 3 significant figures is

1) 535 2) 535,000 3) 5.35 x 105

Solution

A. Which answers contain 3 significant figures?

2) 0.00476 3) 4760

B. All the zeros are significant in

2) 25.300 3) 2.050 x 103

C. 534,675 rounded to 3 significant figures is

2) 535,000 3) 5.35 x 105

Learning Check

In which set(s) do both numbers contain the same number of significant figures?

1) 22.0 and 22.00

2) 400.0 and 40

3) 0.000015 and 150,000

Solution

In which set(s) do both numbers contain the same number of significant figures?

3) 0.000015 and 150,000

State the number of significant figures in each of the following:

A. 0.030 m 1 2 3

B. 4.050 L 2 3 4

C. 0.0008 g 1 2 4

D. 3.00 m 1 2 3

E. 2,080,000 bees 3 5 7

Learning Check SF3

A. 0.030 m 2

B. 4.050 L 4

C. 0.00008 g 1

D. 3.00 m 3

E. 2,080,000 bees 3

Solution

Significant Numbers in Calculations

A calculated answer cannot be more precise than the measuring tool.

A calculated answer must match the least precise measurement.

Significant figures are needed for final answers from

1) adding or subtracting

2) multiplying or dividing

Adding and Subtracting

The answer has the same number of decimal places as the measurement with the fewest decimal places.

25.2 3 sig figs (one decimal places)

+ 1.34 3 sig figs (two decimal places)

26.54

answer 26.5 one decimal place

Solution

A. 2.19 X 4.2 =

B. 4.311 ÷ 0.07 =