Name ____________________________

Measurements, Significant Figures, and Scientific Notation Use this reading guide to help you complete pg. 13-26 in your course pack. These notes can accompany the notes on page 11-12 on the course pack.

Measurements v. Exact Numbers:Measurement is the collection of quantitative data. All measurements are approximations because measuring devices cannot give perfect data. There is always some uncertainty – the space between the tick marks. Therefore, significant figures and the rules of significant figures must be used when dealing with measurements. Significant figures are the number of digits used to express a measured or calculated quantity. Examples of measurements: mass, volume, length, height, temperature, etc.

Some numbers are exact because they are known with complete certainty. Examples of exact numbers are conversion factors (ex. there are exactly 12 inches in 1 foot) and counts (ex. there are exactly 22 students in the class). Exact numbers have an infinite number of significant digits; therefore, they can be ignored in determining the amount of significant figures that should be reported for an answer to a calculation. Reading Scales and Recording Measurements:The proper handling and the interpretation of measurements are essential in chemistry – and in any scientific endeavor. To use measurements correctly, you must recognize that measurements are not numbers. They always contain a unit (ex. g, mL, etc.) and some inherent error (the “space between the tick marks”).

When reading scales, one must follow these two rules:1. Record the number(s) you know for sure based on the scale’s tick

marks.

2. Record 1 (and only 1) estimate for the space between the tick marks.Note: Your estimate can be “0” if you think the substance is exactly on the

tick mark!

The numbers you record based on the two rules above are called significant figures.

Here is an example:

What do we know for sure about the length of the rectangle? It is between 4.8 and 4.9. We know it is at least 4.8. But we have to estimate one number for the space

between the tick marks. It looks like it might be half way between the tick marks, so 4.85. The five is my estimate. The measurement is 4.85, which has 3 significant figures.Why are these rules important?How you report the data you collect indicates the precision with which the data was gathered. If you report too many significant figures, you are being dishonest about the data collected. You are saying the data is more precise than it really is! If you report too few significant figures, you are reporting that your data is less precise than it really is (Why do that?).

Here is an example to illustrate the important of significant figures:

Using the top ruler, the length of the rectangle is 72.8 mm. The number 72 is based on the tick marks and the number 8 is the one estimate for the space between the tick marks. The measurement has 3 significant figures.

Using the bottom ruler, the length of the rectangle is 72.86 mm. The number 72.8 is based on the tick marks and the number 6 is the one estimate for the space between the tick marks. The measurement has 4 significant figures.

Which measurement would you trust more? The bottom ruler! It has more tick marks; therefore, it is more precise. The number of significant figures in the measurement indicates that. It would be dishonest and a misrepresentation of the data to report more than 3 significant figures for the top ruler. It would also be dishonest to report more than 4 significant figures for the bottom ruler.

Let’s look at some more examples of reading scales (and check your answers on page 13 and 14)…

What do we know for sure based on the tick marks? The arrow is between 2225 and 2226. So, we know it is 2225. Now, we need to estimate 1 number for the space between the tick marks. It looks like it might be halfway between the tick marks. So, 2225.5 is the best answer. The answer has 5 significant figures (4 from the tick mark + 1 estimate).

What do we know for sure based on the tick marks? The arrow is between 114 and 115. So, we know it is 114. Now, we need to estimate 1 number for the space between the tick marks. It looks like it is closer to 114 than 115. So, 114.2 is an acceptable answer. The answers 114.1 and 114.3 would also be acceptable. The last number is just an estimate! The answer has 4 significant figures (3 from the tick mark + 1 estimate).

What do we know for sure based on the tick marks? The arrow is between 1100 and 1200. So, we know it is 1,1_ _ . Now, we need to estimate 1 number for the space between the tick marks. It looks like it is closer to 1110 than 1200. My estimate will be 2. I can’t record my answer as 112 though. 112 is not in the thousands place. I need to add a zero as a placeholder (to keep my answer in the thousands place). So, 1,120 is an acceptable answer. The answers 1,110 and 1,130 would also be acceptable. The last number is just an estimate! The answer has 3 significant figures (2 from the tick mark + 1 estimate….. the placeholder zero does not count!).

The arrow is between 0.06 and 0.07. We need to add 1 estimate. So, an acceptable answer would be 0.065. How many significant figures does this answer have? It looks like it could be 4! BUT, if I change the answer to scientific notation I have

6.5 x 10-2. The zeros just disappeared! They, too, are placeholders and do not count as a significant figures. The measurement 0.065 has 2 significant figures.

The arrow is between 0.025 and 0.026. We need to add 1 estimate. It looks like the arrow might be exactly on the line. If that is the case, your estimate can be the number 0. So, an acceptable answer would be 0.0260. The measurement has 3 significant figures (2 from the tick mark and 1 estimate). The leading zeros do not count as significant because they are placeholders (2.60 x10-2).

Be careful with the tick marks on this graduated cylinder! The tick marks represent every 5. The arrow is between 175 and 180. We know for sure we have 17_. We must estimate 1 number for the space between the tick marks. 177 or 178 would be acceptable answers. There are 3 significant figures in the measurement.

The arrow is between 38 and 39. So we know

38 based on the tick marks, but we must estimate 1 number. 38.5, 38.4, 38.6 are acceptable answers. There are 3 significant figures in the measurement.

The arrow is between 2.3 and 2.4. It looks like it might be right above 2.3. Now, we need to estimate 1 number. 2.31 or 2.30 are acceptable answers. There are 3 significant figures in the answer.

The water level is between 100 and 150. So we know 1_ _. We need to estimate only one number. So, 130 with the 3 being the estimate is an acceptable answer. We need to add a zero as a placeholder. There are 2 significant

The arrow is

between 14.4 and 14.5 (Be

careful with the tick marks!). It

looks like the arrow might be

right on the tick mark. So 14.40 is an acceptable answer with “0” as the estimate.

The measurement

contains 4 significant

figures.

Significant Figures and the Rules of Zeros:

Significant figures are the number of digits used to express a measured or calculated quantity following the rules outlined previously. All non-zero numbers (1-9) are significant. But, after working through a couple of examples, it is evident that zeros can be tricky! Sometimes they count as significant and sometimes they do not. Here are the rules to quickly determine which zeros are significant and which are not:

1. Sandwiched zeros (zeros found between non-zero numbers) = always count as significant!

Examples: 707 (3 SF) / 7,007 (4 SF) / 7,000,543 (7 SF)

2. Leading zeros (zeros found in front of non-zero numbers) = never count as significant!

Examples: 0.007 (1 SF) / 0.544 (3 SF) / 0.0050043 (5 SF)

3. Trailing zeros with a decimal present (zeros found after non-zero numbers; there is a decimal present somewhere in the number) = always count as significant!

Examples: 7765.0 (5 SF) / 0.5400 (4 SF) / 500. (3 SF)

4. Trailing zeros with no decimal present (zeros found after non-zero numbers; there is no decimal present anywhere in the number) = never count as significant!

Examples: 7,760 (3 SF) / 5400 (2 SF) / 500 (1 SF)

Let’s look at some examples of counting the amount of significant figures in a number using the rules outlined above….

22,568 - 5 SFs because all non-zero numbers are significant589,000 – 3 SFs because trailing zeros with no decimal present are not significant0.035 – 2 SFs because leading zeros are not significant760 – 2 SFs because trailing zeros with no decimal present are not significant0.2686 – 4 SFs because leading zeros are not significant85.00 – 4 SFs because trailing zeros with a decimal present are significant0.00458 – 3 SFs because leading zeros are not significant8,000,000 – 1 SF because trailing zeros with no decimal present are not significant760. – 3 SFs because trailing zeros with a decimal present are significant5,000.5 – 5 SFs because sandwiched zeros are significant

Scientific Notation:

Scientific Notation is an easy way of expressing very small and very large numbers in science. For example, we might be dealing with trillions and trillions of particles. So, we need to be able to easily write out the number of particles. Here are the rules for writing a number in scientific notation correctly….

1. Always have one number before the decimal (and only one number!)

2. The coefficient should always express the correct number of significant figures.

3. The base should always be 10 and the exponent should be a non-zero integer (positive or negative).

Let’s look at some examples….

For a LARGE number:

For a SMALL number:

Let’s look at even more examples….

7,500 = 7.5 x 103

One number before the decimal = 7 All significant figures must be present in the coefficient = 7 and

5 The decimal needs to be moved 3 places = exponent of 3 We have a big # = exponent is positive

986.55 = 9.8655 x 102

One number before the decimal = 9 All significant figures must be present in the coefficient = ALL

are SF! The decimal needs to be moved 2 places = exponent of 2 We have a big # = exponent is positive

387,000,000 = 3.87 x 108

One number before the decimal = 3 All significant figures must be present in the coefficient = 3, 8,

and 7 The decimal needs to be moved 8 places = exponent of 8 We have a big # = exponent is positive

0.698 = 6.98 x 10-1

One number before the decimal = 6 All significant figures must be present in the coefficient = 6, 9,

and 8 The decimal needs to be moved 1 places = exponent of 1 We have a small # = exponent is negative

0.00000038 = 3.8 x 10-7

One number before the decimal = 3 All significant figures must be present in the coefficient = 3 and

8 The decimal needs to be moved 7 places = exponent of 7 We have a small # = exponent is negative

7,000,631 = 7.000631 x 106

One number before the decimal = 7 All significant figures must be present in the coefficient = ALL

are SF! The decimal needs to be moved 6 places = exponent of 6 We have a big # = exponent is positive

741, 900 = 7.419 x 105

One number before the decimal = 7 All significant figures must be present in the coefficient = 7, 4,

1, and 9 The decimal needs to be moved 6 places = exponent of 5

We have a big # = exponent is positive

Let’s go in the opposite direction….

1. Look at the exponent and move the decimal that many places.2. If the exponent is positive, move to the RIGHT. If the exponent is

negative move the LEFT.3. Add zeros when needed. 4. Make sure your answer had the correct number of significant figures

(look at the coefficient).

5.900 x 105 = 590,000

Move the decimal 5 places to the right. You need to add in two more zeros. Your number has 4 significant figures. You may show this by underscoring the last SF.

6.080 x 10-3 = 0.006080

Move the decimal 3 places to the left. Make sure you have the 4 significant figures you start with!

9.0 x 106 = 9,000,000

Move the decimal 6 places to the right. You number has 2 significant figures. You may show this by underscoring the last SF.

2.68 x 10-15 = 0.00000000000000268

Move the decimal 15 places to the left. You number should have 3 significant figures.

3.00 x 1010 = 30,000,000,000

Move the decimal 10 places to the right. You number has 3 significant figures. You may show this by underscoring the last SF.

2.97 x 10-4 = 0.000297

Move the decimal 3 places to the left. You number should have 3 significant figures.

Calculations with Significant Figures:

We often perform calculations with the data we collect in lab. It is also important to maintain honest and accurate significant figures as we perform these calculations.

For addition and subtraction:Always round the result to the same number of decimal places as the poorest measurement. Pick out the measurement with the fewest numbers after the decimal. Round your answer to the fewest numbers after the decimal. Units do not change.

Example: 75.36 m + 7.2 m + 3.469 m = ??

^ 2 #s ^ 1 #s ^ 3 #s after the decimal Calculator answer = 86.029

The measurement above with the fewest amount of numbers after the decimal is 7.2 (only 1 number after the decimal). So, the answer should be rounded to have only 1 number after the decimal. Include units!

Rounded answer = 86.0 m

Example: 486 mL + 89.25 mL + 5.8 mL = ??

^ 0 #s ^ 2 #s ^ 1 # after the decimal Calculator answer = 581.05

The measurement above with the fewest amount of numbers after the decimal is 486 (0 numbers after the decimal). So, the answer should be rounded to have 0 numbers after the decimal. Include units!

Rounded answer = 581 mL

Example: 132 g - 64.38 g = ??

^ 0 #s ^ 2 #s after the decimal

Calculator answer = 67.62

The measurement above with the fewest amount of numbers after the decimal is 132 (0 numbers after the decimal). So, the answer should be rounded to have 0 numbers after the decimal. Include units!

Rounded answer = 68 mL (round up!)For multiplication and division:Always round the result to the same number of significant figures as the poorest measurement. Pick out the measurement with the fewest number of significant figures. Round your answer to the fewest number of significant figures. Remember: units multiply and divide, too!

Example: 40.0 kPa x 847.35 mL = ?? 35.7 kPa

Calculator answer = 949.4117647

The measurement above with the fewest amount significant figures is 40.0 and 35.7 (both have 3 SFs). So, the answer should be rounded to have 3 significant figures. Check out your units: kPa cancels out (units divide) and you are left with mL!

Rounded answer = 949 mL

Example: 10.8 g = ?? 4.0 cm3

Calculator answer = 2.7

The measurement above with the fewest amount significant figures is 4.0 with 2 significant figures. So, the answer should be rounded to have 2 significant figures (Oh, look… it already has 2 SFs!) Check out your units: nothing cancels!

Rounded answer = 2.7 g/cm3

Example: 35 cm x 21.2 cm x 89.55 cm = ??

Calculator answer = 66,446.1

The measurement above with the fewest amount significant figures is 35 with 2 significant figures. So, the answer should be rounded to have 2 significant figures. Check out your units: units multiply!

Rounded answer = 66,000 cm3

Note: The 4, 4, and 6 in the calculator answer must be turned into placeholder zeros to have only 2 SFs in the answer!

For multi-step calculations:Never round intermediate results. Round at the end. Rounding and Significant Figures:

Follow standard rounding rules…

Let’s practice some rounding….

Round the following numbers to 3 significant figures:1. First start by underscoring the place of the 3rd significant figure. 2. Then, decide to round up or down!3. Add placeholder zeros when necessary to keep place (hundreds,

thousands, etc).

98.89 Followed by a 9, round up! 98.9!

70,678 Followed by a 7, round up! Add placeholder zeros to stay in thousands place!

70,700

5.0099 Followed by a 9, round up! 5.01

7,000,631 Followed by a 0, round down! Must add placeholder zeros!

7,000,000 (Note: There is no real way to show this number has 3 SFs without changing it to scientific notation = 7.00 x 106. So, you may underscore the last significant figure to indicate 3 SFs.)

240.000 Followed by a 0, round down!

240. (Note: Without a decimal, the number 240 has only 2 SFs. By adding a decimal, you show 3 SFs.)

2.538 Followed by a 8, round up!

2.54