chapter 1 : measurements significant figures and calculations

CHAPTER 1 : MEASUREMENTS

SIGNIFICANT FIGURES AND CALCULATIONS

Significant figures and calculations

Significant figures in a measurement include all of the digits that are known, plus one more digit that is estimated.

Significant Figures •Any digit that is not zero is significant

2.234 kg 4 significant figures •Zeros between non-zero digits are significant.

607 m 3 significant figures • Leading zeros (to the left) are not significant.

0.07 L 1 significant figure.

0.00520 g 3 significant figures Trailing ( to the right) only count if there is a

decimal in the number.

5.0 mg 2 significant figures.

50 mg 1 significant figure.

Two special situations have an unlimited number off Significant figures:

1.. Counted items

a) 23 people, or 425 thumbtacks

2 Exactly defined quantities

b) 60 minutes = 1 hour

Practice #1

How many significant figures in the following?

1.0070 m

17.10 kg

100,890 L

3.29 x 103 s

0.0054 cm

3,200,000 mL

5 dogs

5 sig figs

4 sig figs

5 sig figs

3 sig figs

2 sig figs

2 sig figs

unlimitedThis is acounted value

Rounding Calculated Answers Decide how many significant figures are needed Round to that many digits, counting from the left Is the next digit less than 5? Drop it. Next digit 5 or greater? Increase by 1

3.016 rounded to hundredths is 3.02 • 3.013 rounded to hundredths is 3.01 • 3.015 rounded to hundredths is 3.02 • 3.045 rounded to hundredths is 3.04 • 3.04501 rounded to hundredths is 3.05

Addition and Subtraction The answer should be rounded to the

same number of decimal places as the least number of decimal places in the problem. Examples:

4.8

-3.965 0.835 0.8

1 decimal places

3 decimal places

M 761.50 14.334 10.44 10789 8024.50 203.514

762

14.3

10.4

10800

8020

204

Make the following have 3 sig figs:

Multiplication and Division Round the answer to the same number of

significant figures as the least number

of significant figures in the problem.

Multiplication and Division: # sig figs in the result equals the number in the least precise measurement used in the calculation.

6.38 x 2.0 = 12.76 13 (2 sig figs)

Addition and Subtraction: The number of decimal places in the result equals the number off decimalplaces in the least precise measurement. 6.8 + 11.934 =18.734 18.7 (3 sig figs) 89.332 + 1.1 = 90.432 round off to 90.4 one significant figure after decimal point 3.70 -2.9133 = 0.7867 two significant

figures after decimal point round off to 0.79

Scientific Notation

What is scientific Notation?

Scientific notation is a way of expressing really big numbers or really small numbers.

It is most often used in “scientific” calculations where the analysis must be very precise.

Why use scientific notation?

For very large and very small numbers, these numbers can expressed in a more concise form.

Numbers can be used in a computation with far greater ease.

Scientific notation consists of two parts:

A number between 1 and 10

A power of 10

N x 10x

Changing standard form to scientific notation.

EXAMPLE

5 500 000 = 5.5 x 106

We moved the decimal 6 We moved the decimal 6 places to the left.places to the left.

A number between 1 and 10A number between 1 and 10

EXAMPLE #2

0.0075 = 7.5 x 10-3

We moved the decimal 3 We moved the decimal 3 places to the rightplaces to the right.

A number between 1 and 10A number between 1 and 10

Numbers less than 1 Numbers less than 1 will have a negative will have a negative exponent.exponent.

EXAMPLE #3

CHANGE SCIENTIFIC NOTATION TO STANDARD FORM

2.35 x 2.35 x 101088

= 2.35 x 100 000 000= 2.35 x 100 000 000

= 235 000 000= 235 000 000

Standard formStandard form

Move the decimal 8 places to the rightMove the decimal 8 places to the right

EXAMPLE #4

9 x 10-5

= 9 x 0.000 01

= 0.000 09

Move the decimal 5 places to the leftMove the decimal 5 places to the left

Standard formStandard form

TRY THESE

Express in scientific notation 1) 421.96 2) 0.0421 3) 0.000 56 4) 467 000 000

TRY THESE

Change to Standard Form 1) 4.21 x 105

2) 0.06 x 103

3) 5.73 x 10-4

4) 4.321 x 10-5

To change standard form to scientific notation… Place the decimal point so that there is

one non-zero digit to the left of the decimal point.

Count the number of decimal places the decimal point has “moved” from the original number. This will be the exponent on the 10.

Continued…

If the original number was less than 1, then the exponent is negative. If the original number was greater than 1, then the exponent is positive.

Types of Errors

Random errors- the same error does not repeat every time.

• Blunders • Human Error

Systematic Errors – These are errors caused by the way

in which the experiment was conducted. In other words, they are caused by flaws in equipment or experimental.

Can be discovered and corrected.

Examples:

You measure the mass of a ring three times using the same balance and get slightly different values: 12.74 g, 12.72 g, 12.75 g. ( random error )

The meter stick that is used for measuring, has a millimetre worn off of the end therefore when measuring an object all measurements are off.

( systematic error )

Accuracy or Precision

• PrecisionReproducibility of resultsSeveral measurements afford the same resultsIs a measure of exactness

• AccuracyHow close a result is to the “true” valueIs a measure of rightness

Accuracy vs Precision

π Accuracy Precision

3 NO NO

7.18281828 NO YES

3.14 YES NO

3.1415926 YES YES

Metric Conversions Ladder Method

KILO1000Units

HECTO100

Units

DEKA10

UnitsDECI

0.1Unit

CENTI0.01Unit

MILLI0.001Unit

Meters

LitersGram

s

Ladder Method

How do you use the “ladder” method?

1st – Determine your starting point.2nd – Count the “jumps” to your ending point.3rd – Move the decimal the same number of jumps in the same direction.

4 km = _________ m

12

3

How many jumps does it take?

Starting Point

Ending Point

4.1

__

. 2

__

. 3

__

.= 4000 m

Try these conversions using the ladder method.

1000 mg = _______ g 1 L = _______ mL 160 cm = _______ mm

14 km = _______ m 109 g = _______ kg 250 m = _______ km

Conversion Practice

Compare using <, >, or =.

56 cm 6 m 7 g 698 mg

Write the correct abbreviation for each metric unit.

1) Kilogram _____ 4) Milliliter _____ 7) Kilometer _____

2) Meter _____ 5) Millimeter _____ 8) Centimeter _____

3) Gram _____ 6) Liter _____ 9) Milligram _____

Try these conversions, using the ladder method.

10) 2000 mg = _______ g 15) 5 L = _______ mL 20) 16 cm = _______ mm

11) 104 km = _______ m 16) 198 g = _______ kg 21) 2500 m = _______ km

12) 480 cm = _____ m 17) 75 mL = _____ L 22) 65 g = _____ mg

13) 5.6 kg = _____ g 18) 50 cm = _____ m 23) 6.3 cm = _____ mm

14) 8 mm = _____ cm 19) 5.6 m = _____ cm 24) 120 mg = _____ g

Metric Conversion Challenge