SIGNIFICANT FIGURE NOTES

I. Uncertainty in Measurement

A. Making measurements

1) Each time a measurement is made, a scale has to be read.

Example: Read the position of the arrow below on the scale.

The scale has a mark for every 1/10th place(.1), so we can see that it is between .5 and .6 cm.

2) Estimate the next smallest decimal place, or 1/100th’s place. That is where the number should be written out to and rounded off.

Example: Since the arrow is more than half way across between .5 and .6, the hundredth’s place must be greater than 5. Now estimate it and write down the whole number:

Measurement =

3) Different people might read it as slightly different from you, .56 or .58 cm for example, but any of these measurements is better than just writing down .5 cm

4) The last, estimated digit in the measurement is the last significant digit in the number

Read these scales correctly:

Scale A: marked to the .01mm place

Scale B: marked to 10m place

Scale A should be read to the 1/1000th place(.001) as about .082 or .083mm;

Scale B should be read to the 1’s place as about 18 or 19m.

5) The number of significant digits a measurement has depends on how many numbers we read on the scale.

6) When measuring liquids like water, read the scale at the bottom of the meniscus, or curve.

Example: What is the volume of water below?

The scale is marked to the 1’s place, so you should read to the 1/10th’s(.1) place.

Measurement =

B. Using Measurements

1) Any calculation using measurements has to be rounded off correctly.The number of significant digits in the answer depends on how many are in each number used to get the answer.

2) Significant Figure Rules

Here is how to look at a number and tell how many significant digits it has:

a) all digits other than zeros are significant.

Example: 25 g has 2 sig.fig.

5,471g has 4 sig. fig.

12.5cm has 3 sig.fig.

b) zeros between nonzero digits are significant.

Example: 309 g has 3 sig.fig.

40.06g has 4 sig.fig.

c) final zeros to the right of the decimal point are significant.

Example: 6.00 mL has 3 sig.fig.

2.350g has 4 sig.fig.

d) in numbers smaller than 1, any zero at the beginning of the number IS NOT significant.

Example: 0.005mL has 1 sig.fig.

0.060cm has 2 sig.fig.

Determine the sig.fig. in each of the following measurements:

Measurement No. of sig.fig.

1.2 mg

1.03 kg

0.003 g

50.10 m

0.000000070 s

3.00 cm

3) Rounding off Rules for Calculations

a) Addition and Subtraction

After + or - operations, round the final answer to the decimal place of largest uncertainty.

(the least number of decimal places)

Example: What is 3.514 - 2.13 cm?

The unrounded answer is 1.384, but the place of greatest uncertainty is in the 1/100th’s place(2.13 has 2 decimal places, and 3.514 has 3 decimal places), so it rounds off to 1.38 cm.

b) Since the number we drop is a 4, we don’t change the number that we kept (the 8).

1.384 cm rounds to 1.38 cm

If the number we dropped were a 5 or greater, we would round the number we kept up to the next number(a 9).

1.386 would round to 1.39 cm

Example: What is 3.56 cm + 2.6 cm + 6.12 cm?

The unrounded answer is 12.28 cm, but the place of greatest uncertainty is in the 1/10th’s place(2.6 only has one decimal place), so the answer has to be rounded off. We round 12.28 off to 12.3 cm.

4) multiplication and division

After x or ÷ numbers, round the final answer off so that it has the same number of sig.fig. as the least certain number you used(the number with the least sig.fig.)

Example: What is 4.29 cm x 3.24 cm?

The unrounded answer is 13.8996 cm², but both numbers used have 3 sig.fig. So the answer should be rounded to 3 sig.fig. also.

13.8996 rounds to 13.9 cm²

Example: What is 8.5g/4.26mL?

[the sign / means divided by]

The calculator answer is 1.9953052 g/mL, but one of the numbers used only has 2 sig.fig.(the 8.5), so the answer should only have 2 sig.fig.

The answer 1.9953052 should be rounded to 2.0 g/mL.

Do these calculations and round the calculator answer to the correct sig.figs.

Problem Calculator Rounded

Answer Answer

0.0012m x 12.7m

512g/63 mL

15cm- .582cm

.5 g - .25 g

12L+ 5.12L + .003L

6.111mL- 2.4L

10,013cm x 22 cm

C) Factor-Label Method of Unit Conversion(Dimensional Analysis)

1) Used for problems where you need to change the units of a measurement.

2) Requires an equality,

(like 1 minute = 60 sec, or 1 cm = 10 mm).

Either you already know it or you make one up using powers of 10.

Example: Change 1.5 hours to time in minutes.

• First, determine the conversion factor you need.

You know that 1 hour = 60 min.

Next, write down the original time with the units, multiplied by a conversion factor fraction containing both 1 hr. and 60 min. Put 60 min.on top, 1hr.on bottom.

1.5 hr. x 60 min = 1 hr.

• Notice that the units of hr. are the same and diagonal. They can be cancelled out. That leaves 1.5 x 60 min

which = 90 min, which is the answer. So if we place the same units diagonally, we have it!!!! Example: How many milligrams are in 4.72 grams?

1) first find the equality

From your notes, 1g = 1000 mg

2) write down the original measurement, multiplied by a conversion factor fraction with the units diagonal.

Which is on top?

4.72 g x _______

3) cancel out the diagonal units

= 4.72 x _______

= ___________, the answer to how many mg are in 4.72 g.

Example: How many Liters are in 6.111 milliliters?

1) get equality(from notes)

2) set up fraction diagonally and multiply

3) cancel units and do the math. This time, we have to divide by 1000 because it is on the bottom of the fraction.

Example: How many meters are in 4.11 cm?