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
5 sig figs
17.10 kg
100,890 L
3.29 x 103 s
0.0054 cm
3,200,000 mL
5 dogs
4 sig figs
5 sig figs
3 sig figs
2 sig figs
2 sig figs
unlimited
This is a
counted 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:
1 decimal places
4.8
3 decimal places
-3.965
0.835
0.8
Make the following have 3 sig figs:
M 761.50
 14.334
 10.44
 10789
 8024.50
 203.514

762
14.3
10.4
10800
8020
204
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 decimal
places 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
places to the left.
A number between 1 and 10
EXAMPLE #2
0.0075
 = 7.5 x 10-3

Numbers less than 1
will have a negative
exponent.
We moved the decimal 3
places to the right.
A number between 1 and 10
EXAMPLE #3

CHANGE SCIENTIFIC NOTATION TO
STANDARD FORM
2.35 x 108
= 2.35 x 100 000 000
= 235 000 000
Standard form
Move the decimal 8 places to the right
EXAMPLE #4
9 x 10-5
= 9 x 0.000 01
= 0.000 09
Standard form
Move the decimal 5 places to the left
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
• Precision
Reproducibility of results
Several measurements afford the same
results
Is a measure of exactness
• Accuracy
How close a result is to the “true” value
Is a measure of rightness
Accuracy vs Precision
π
3
Accuracy
Precision
NO
NO
7.18281828
NO
YES
3.14
YES
NO
3.1415926
YES
YES
Metric Conversions
Ladder Method
Ladder Method
1
2
3
KILO
1000 HECTO
DEKA
100
Units
10
Units
Units
Meter
s
Liters
Gram
s
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.
DECI
0.1
Unit
CENTI
0.01
MILLI
Unit
0.001
Unit
4 km = _________ m
Starting
Ending
Point
Point
How many jumps does it take?
4.
__ __ __
. 1 .2 . 3
= 4000 m
Conversion Practice
Try these conversions using the ladder
method.
1000 mg = _______ g
1 L = _______ mL
14 km = _______ m
109 g = _______ kg
160 cm = _______ mm
250 m = _______ km
Compare using <, >, or =.
56 cm
6m
7g
698 mg
Metric Conversion Challenge
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