Math Unit
Measurement
• When making any measurement, always
estimate one place past what is actually
known.
Example
• For example, if a meter stick has
calibrations (markings) to the 0.1 cm,
the measurement must be estimated to
the 0.01 cm.
• If you think it’s perfectly on a line, estimate
the last digit to be zero
– For example, if you think it’s on the 2.1 cm
line, estimate it to 2.10 cm.
Electronic Devices
• When making a measurement with a
digital readout, simply write down the
measurement. The last digit is the
estimated digit.
Significant Digits
• Significant digits are all digits in a
number which are known with certainty
plus one uncertain digit.
5 Rules for Counting Significant
Digits in a Measurment
1. All nonzero numbers are significant.
– 132.54 g has 5 significant digits.
2. All zeros between nonzero numbers are
significant.
130.0054 m has 7 significant digits
3. Zeros to the right of a nonzero digit but to
the left of an understood decimal point
are not significant unless shown by
placing a decimal point at the end of the
number.
190 000 mL has 2 significant digits
190 000. mL has 6 significant digits
4. All zeros to the right of a decimal point
but to the left of a nonzero digit are NOT
significant.
0.000 572 mg has 3 significant digits
5. All zeros to the right of a decimal point
and to the right of a nonzero digit are
significant.
460.000 dm has 6 significant digits
Shortcut
• If the number contains a decimal point,
draw an arrow starting at the left through
all zeros and up to the 1st nonzero digit.
The digits remaining are significant.
Exact Numbers
Exact numbers have an infinite (∞)
number of significant digits.
3 types of numbers with (∞) number of
sig digs:
1. Definitions (12 eggs = 1dozen)
2. Counting numbers (there are 24
desks in this room
3. Numbers in a formula (2pr)
Try these
•
•
•
•
0.002 5
1.002 5
0.002 500 0
14 100.0
• If the quantity does not contain a decimal
point, draw an arrow starting at the right
through all zeroes up to the 1st nonzero
digit. The digits remaining are significant.
Try these
•
•
•
•
225
10 004
14 100
103
Remember – Atlantic Pacific
• Decimal Point Present, start at the Pacific.
• Decimal Point Absent, start at the Atlantic.
How many significant digits do
these have?
•
•
•
•
•
1.050
20.06
13
0.303 0
373.109
•
•
•
•
•
420 000
970
0.002
0.007 80
145.55
Rounding Rules
Round up if the digit immediately to the right
of the digit you are rounding to is
•
Greater than 5
•
•
Round 0.236 to 2 significant digits
5 followed by another nonzero number
•
•
Round 0.002351 to 2 significant digits
Round 0.00235000000001 to 2 significant digits
Kepp the digit the same if the digit
immediately to the right of the digit you are
rounding to is
• Less than 5
• round 1.23 to 2 significant digits
What if the digit to the right of
the number you are rounding to
is 5 and there’s nothing after it?
• That means you are perfectly in the
middle.
• Half of the time you must round up and
half of the time you must round down.
• There are 2 rules for this
Look to the digit to the right of the
number you are rounding to.
• If it is even – keep the same.
– Round 0.8645 to 3 significant digits
• If it is odd – round up.
– Round 0.8675 to 3 significant digits.
Round These to 3 significant digits
•
•
•
•
•
279.3
32.395
18.29
42.353
0.008 752
•
•
•
•
18.77
7.535
32.25
5 001
Applying significant digits to
arithmetic operations
Addition and Subtraction
• Look at the numbers being added or
subtracted and identify which one has the
lowest number of decimal places.
Calculate the answer. Round the answer
to the lowest number of decimal places.
Examples
• 14.565 + 7.32 = 21.885
• 7.32 has only 2 decimal places, so the
answer should be rounded to 21.88
• 143.52 – 100.6 = 42.92
• 100.6 has only 1 decimal place, so the
answer should be rounded to 42.9
Multiplication and Division
• Look at the numbers being multiplied or
divided and identify which one has the
lowest number of significant digits.
Calculate the answer. Round the answer
to the lowest number of significant
digits.
Examples
• 172.6 x 24.1 = 4159.66
• 24.1 has only 3 significant digits, so the
answer should be rounded to 4160
• 172.6 ÷ 24.1 = 7.161 82
• 24.1 only has 3 significant digits, so the
answer should be rounded to 7.16
Practice
•
•
•
•
•
•
•
•
Add 5.34 cm, 9.3 cm, and 12 cm.
Subtract 4.31 cm from 7.542 cm.
Subtract 1.512 g from 16.748 g.
Add 2.572 5 m, 14.55 m and 0.035 m.
Multiply 176.335 and 0.003 2.
Divide 475.90 by 35.
Multiply 0.000 565, 1.579 52, and 45.006 86.
Multiply 1 456.00 by 0.035 0 and divide that by
17.070.
Percent Error
• This is a way of expressing how far off an
experimental measurement is from the
accepted/true value.
• Final Exam Question
Formula
Value accepted - Value experimental
Value accepted
X 100
Scientific Notation
• It is used for extremely large or small
numbers.
• The general form of the equation is:
m x 10n
• With the absolute value of m ≥ 1 and < 10
Practice
•
•
•
•
12 300
-1 456
0.005 17
-0.000 6
•
•
•
•
6.650 x 102
3.498 x 105
-2.208 x 10-3
1.1650 x 10-4
Arithmetic Rules for Scientific
Notation
• Follow the same rules for math operations
with scientific notation as you would with
standard notation.
Addition and Subtraction
•
•
•
•
(3.37 x 104) + (2.29 x 105)
(9.8 x 107) + (3.2 x 105)
(8.6 x 104) – (7.6 x 103)
(2.238 6 x 109) – (3.335 7 x 107)
Multiplication and Division
•
•
•
•
(1.2 x 103) x (3.3 x 105)
(7.73 x 102) x (3.4 x 10-3)
(9.9 x 106)  (3.3 x 103)
(1.55 x 10-7)  (5.0 x 10-4)
Temperature Conversion
• Temperature is defined as the average
kinetic energy of the particles in a sample
of matter.
• The units for this are oC and Kelvin (K).
Note that there is no degree symbol for
Kelvin.
Kelvin Scale
• The Kelvin scale is based on absolute
zero.
• This is the theoretical temperature when
motion stops.
• Heat is a measurement of the total kinetic
energy of the particles in a sample of
matter.
• The units for this are the calorie (cal) and
the Joule (J).
Formulas
• T(K) = t(oC) + 273.15
• t(oC) = T(K) - 273.15
Dimensional Analysis
• Dimensional analysis is the algebraic
process of changing from one system of
units to another.
You must develop the habit of including
units with all measurements in calculations.
Units are handled in calculations as any
algebraic symbol:
 Numbers added or subtracted must have
the same units.
 Units are multiplied as algebraic symbols.
For example: 10 cm x 10 cm = 10 cm2
 Units are cancelled in division if they are
identical.
 For example, 27 g ÷ 2.7 g/cm3 = 10 cm3.
Otherwise, they are left unchanged. For
example, 27 g/10. cm3 = 2.7 g/cm3.
Conversion Factor
• These are fractions obtained from an
equivalence between two units.
• For example, consider the equality 1 in. =
2.54 cm. This equality yields two
conversion factors which both equal one:
•
and
2.54 cm
1 in
2.54 cm
1 in
Convert 5.08 cm to inches
• 5.08 cm x
1 in
2.54 cm
= 2.00 in
Convert 6.53 in to cm
• 6.53 in x
2.54 cm
1 in
= 16.6 cm
Here are some common
English/metric conversions. You
will not need to memorize these.
1 mm = 0.039 37 in
1 cm = 0.393 7 in
1 m = 39.37 in
1 Km = 0.621 4 mi.
1 quart = 946 ml
1 quart = 0.946 L
1 in = 2.54 cm
1 yd = 0.914 40 m
1 mile = 1.609 Km
1 pound = 453.6 g
1 ounce = 28.35 g
Converting within metric units
• In section 2-5 of your textbook, you
learned the relationship between metric
prefixes and their base units. You need to
have these relationships memorized to do
these problems.
• When you write your conversions factor,
always use the number 1 with the unit
with the prefix and meaning of the
prefix with the base unit.
Examples
1 mg
0.001g
1 ns
10 -9 s
1000 m
1 Km
Try to set up these conversion
factors
W
GW
cm
m
Tb
b
L
L
dL
mL
L
pg