Transcript Document

Lab Math 1
Exponents, Scientific Notation and the
Metric System
Exponents
• An exponent is used to show that a
number has been multiplied by itself a
certain number of times.
24 =2 x 2 x 2 x 2 = 16
• The number that is multiplied is called the
base and the power to which the base is
raised is the exponent.
• By definition, any number raised to the
power of 0 is 1.
Manipulation of
Exponents
• To multiply two numbers in the same
base, add the exponents.
53 x 52 = 55
106 x 10-4 = 102
• To divide two numbers in the same
base, subtract the exponents.
53/56 = 5 3-6 = 5-3
Manipulation of
Exponents
• To raise an exponential number to a
higher power, multiply the two
exponents.
(53)2 = 56
(106)-4 = 10-24
Manipulation of
Exponents
• To multiply or divide numbers with
exponents that have different
bases, convert the numbers to their
corresponding values without
exponents, and then multiply or
divide.
52 x 42 = (5 x 5)(4 x 4) = 400
103/2-2 = 1000/-0.25 = 4000
Manipulation of
Exponents
• To add or subtract numbers with
exponents (whether their bases are
the same or not), convert the
numbers with exponents to their
corresponding values without
exponents.
43 + 23 = 64 + 8 = 72
Base 10
• Base 10 is the most commonly
used system of exponents.
• Base 10 underlies percentages,
and the decimal system, orders
of magnitude, scientific notation
and logarithms.
• Scientific Units also use Base
10.
Base 10 Numbers > 1:
• The exponent represents the
number of places after the
number (and before the decimal
point).
• The exponent is positive
• The larger the positive
exponent, the larger the number.
Use of Exponents in
Base 10
1,000,000 =
100,000 =
thousand
10,000 =
1,000 =
100 =
10=
1=
106 = one million
105 = one hundred
104 = ten thousand
103 = one thousand
102 = one hundred
101 = ten
100 = one
Base 10 Numbers < 1:
• The exponent represents the
number of places to the right of
the decimal point including the
first nonzero number.
• The exponent is negative.
• The larger the negative
exponent, the smaller the
number.
Use of Exponents in
Base 10
0.1 =
10-1 = one tenth
(10%)
0.01=
10-2 = one hundredth
(1%)
0.001 = 10-3 = one thousandth (0.1%)
0.0001 = 10-4 = one ten thousandth
0.00001= 10-5 = one hundred thousandth
0.00001= 10-6 = one millionth
Orders of Magnitude
• One order of magnitude is 101 or
10 times.
• A number is said to be orders of
magnitude bigger or smaller
than another number.
• 102 is two orders of magnitude
smaller than 104.
Scientific Notation
• Useful for very small and very large
numbers.
• A number written in scientific notation
is written as a number between 1
and 10 raised to a power.
• The first part is called the coefficient
and the second part is 10 raised to
some power.
Conversion to Scientific
Notation
• For numbers greater than 10, move
the decimal point to the left so there
is one nonzero digit to the left of the
decimal point. This gives the first
part of the notation.
• Count how many places were
moved. This is the exponent. It is
positive.
5467 = 5.467 x 103
Conversion to Scientific
Notation
• For numbers less than 1, move the
decimal point to the right so there is
one nonzero digit to the left of the
decimal point. This gives the first
part of the notation.
• Count how many places were
moved. This is the exponent. It is
negative.
0.5467 = 5.467-3
Scientific Notation
Number
Coefficient Exponent
1000
1X
103
100,000,000
1X
108
0.0000000000000
000000000602
6.02 X
10-23
Multiplication in
Scientific Notation
• To multiply numbers in scientific
notation, use two steps:
– Multiply the coefficients together.
– Add the exponents to which 10 is raised.
(2.5 x 102)(3.0 x 103) =
(2.5 x 3.0)(102+3) =
7.5 x 105
Division in Scientific
Notation
• To divide numbers in scientific notation,
use two steps:
• Divide the coefficients.
• Subtract the exponents to which 10 is
raised.
(6.0 x 102)/(3.0 x 10-4) =
(6.0 / 3.0)(102-4) =
2.0 x 10-2 =
0.02
Addition/Subtraction in
Scientific Notation
• If the numbers are the same
exponent, just add or subtract the
coefficients.
3.0 x 104
+ 4.5 x 104
7.5 x 104
Addition/Subtraction in
Scientific Notation
• If the numbers are different
exponents, convert both to
standard notation and perform the
calculation.
(2.05 x 102) – (9.05 x 10-1) =
205 - 0.905 = 204.095
Addition/Subtraction in
Scientific Notation
• If the numbers have different exponents,
convert one number so they have 10 raised
to the same power and perform the
calculation.
(2.05 x 102) – (9.05 x 10-1) =
2.05
x 102
-0.00905 x 102
2.04095 x 102
Common Logarithms
• Common logarithms (also called
logs or log10) are closely related to
scientific notation.
• The common log of a number is the
power to which 10 must be raised to
give that number.
• The antilog is the number
corresponding to a given logarithm.
• pH uses natural logarithms.
Common Logarithms of
Powers of Ten
Number
10,000
1,000
100
10
1
0.1
0.01
Name
Power
Ten thousand 104
One thousand 103
One hundred
102
Ten
101
One
100
One tenth
10-1
One hundredth 10-2
Log
4
3
2
1
0
-1
-2
International System of
Units
Length
meter
Mass
kilogram
Time
second
Electric current
ampere
Temperature
Kelvin
K
Amount of substance
mole
Luminous intensity
candela
m
kg
s
A
mol
cd
Meter is the Unit of
Length
• The meter is the length of the path traveled
by light in vacuum during a time interval of
1/299 792 458 of a second.
• The meter was intended to equal 10-7 or one
ten-millionth of the length of the meridian
through Paris from pole to the equator.
• The first prototype was short by 0.2 millimeters
because researchers miscalculated the
flattening of the earth due to its rotation.
• Platinum Iridium Bar was cast to this length.
Kilogram is the Unit of
Mass
• A kilogram is equal to the mass of the
international prototype of the kilogram.
• At the end of the 18th century, a kilogram
was the mass of a cubic decimeter of
water. In 1889, scientists made the
international prototype of the kilogram out
of platinum-iridium, and declared: This
prototype shall henceforth be considered
to be the unit of mass.
Liter is a Volume Unit
• A liter (abbreviated either l or L)
is equal to 1 dm3 = 10-3 m3
• Liters can be liquid or air.
Time Units
•
•
•
•
Minute
min 1 min = 60 s
Hour
h
1 h = 60 min = 3600 s
Day
d
1 d = 24 h = 86,400 s
Second can be abbreviated " (a double
tick).
• Minute can be abbreviated ´ (a single
tick).
Temperature
• The kelvin, unit of thermodynamic
temperature, is the fraction 1/273.16 of
the thermodynamic temperature of the
triple point of water.
• Temperature T, is commonly defined in
terms of its difference from the reference
temperature T0 = 273.15 K, the ice point.
• This temperature difference is called a
Celsius temperature, symbol t, and is
defined by the quantity equation
• t= T- T0.
Mole is the Unit of
Amount of Substance
• A mole is the amount of substance of
a system which contains as many
elementary entities as there are atoms
in 0.012 kilogram of carbon 12.
• Physicists and chemists have agreed
to assign the value 12, exactly, to the
"atomic weight of the isotope of
carbon with mass number 12 (carbon
12, 12C).
Moles and Avogadro's
Number
• "Avogadro's Number" is an honorary
name attached to the calculated
value of the number of atoms,
molecules, etc. in a gram molecule
of any chemical substance.
• 12 grams of pure carbon, whose
molecular weight is 12, will contain
6.023 x 1023 molecules.
Avocado
Avogadro
Moles
• You should specify if you have a
mole of atoms, molecules, ions,
electrons, or other particles, or
specified groups of such particles.
Not a Gram Mole
Metric Prefixes (Big)
•
•
•
•
•
•
•
•
•
•
1024
1021
1018
1015
1012
109
106
103
102
101
yotta
zetta
exa
peta
tera
giga
mega
kilo
hecto
deka
Y
Z
E
P
T
G
M
k
h
da
Metric Prefixes (Small)
•
•
•
•
•
•
•
•
•
•
10-1
10-2
10-3
10-6
10-9
10-12
10-15
10-18
10-21
10-24
deci
centi
milli
micro
nano
pico
femto
atto
zepto
yocto
d
c
m
µ
n
p
f
a
z
y
Use of the Prefixes for
Mass
•
•
•
•
•
•
•
Kilogram
Gram
Milligram
Microgram
Nanogram
Picogram
Femtogram
Kg
g
mg
µg
ng
pg
fg
103 g
1
g
10-3 g
10-6 g
10-9 g
10-12 g
10-15 g