Transcript Document

Scales
•Linear
•Logarithmic
Number Notation
•Exponential
•Scientific
Significant Figures
SI Prefixes
Scales
This is a linear scale - each interval between
integers is the same.
0
1
2
3
4
5
6
7
8
9
Rulers use linear scales to measure distances
(inch, meter, etc), along a line.
But linear scales are used for other measured
quantities besides distance such as frequency
(Hz) or velocity (m/s or mph).
10
Scales
This is a linear scale - each interval between
integers is the same.
0
1
2
3
4
5
6
7
8
9
10
On a linear scale, the intervals between powers
of 10 are not equal; for example,
the interval between 1 and 10 is 9
the interval between 10 and 100 is 90
the interval between 100 and 1000 is 900, etc.
Scales
The scale on which the intervals between powers
of 10 are equal is called a logarithmic scale.
100
101
102
103
104
105
106
107
108
109
1010
But on a log scale, the interval between integers
is not contant.
To find the interval between integers on a log
scale, you must understand exponential notation
and logarithms.
Exponential Notation
Any number can be written in exponential notation.
For example, the number 101.325 can be written by
moving the decimal point to the left
101.325100
0 places
10.1325101
1 place
1.01325102
2 places
0.101325103
3 places
0.0101325104
4 places
etc.
When the decimal moves left, the exponent (power
of 10) increases
Exponential Notation
The same number, 101.325, can be written by moving
the decimal point to the right
0 places
1 place
2 places
3 places
4 places
etc.
101.325100
1013.2510-1
10132.510-2
10132510-3
101325010-4
When the decimal moves right, the exponent
(power of 10) decreases
Exponential Notation
There is one form of exponential notation which is so
special it has a special name:
101.325 = 1.01325102 is called scientific notation.
Any number, no matter how large or how small, can
be written in scientific notation as N10n where
1 ≤ N < 10
- < n < 
Exponent n is an integer. [note that 0 = 1  10-]
N is any decimal number from 1.000... to 9.999...
Scientific Notation
There is one form of exponential notation which is so
special it has a special name:
101.325 = 1.01325102 is called scientific notation.
Any number, no matter how large or how small, can
be written in scientific notation as N10n where
1 ≤ N < 10
- < n < 
In scientific notation, the single digit to the left of the
decimal point and all of the digits written to the right
of the decimal point are called significant digits or
significant figures.
Exponential Notation
Some examples of scientific notation:
Any number, no matter how
large or how small, can
-5
0.000032 = 3.210 (2 sig figs) n
be written in scientific notation
as N10 where
2
147.4 = 1.47410 (4 sig figs)
1 ≤ N < 10
2000 = 2103 (1
-sig
< nfig)
<
2000.0 = 2.0000103 (5 sig figs)
The number of sig figs depends on the precision
of the measurement.
Logarithms
x
1
2
3
4
5
6
7
8
9
log(x) 0.000 0.301 0.477 0.602 0.699 0.778 0.845 0.903 0.954
The common logarithm of a number x is y if
log(x) = y
x = 10y
For example, the logs of the integers from 1 to 9
(to three significant figures) are shown above.
The logs of these numbers can be placed on a log
scale as fractions of the distance between 1 (100)
and 10 (101).
Log Scales
x
1
2
3
4
5
6
7
8
9
log(x) 0.000 0.301 0.477 0.602 0.699 0.778 0.845 0.903 0.954
100
1
101
0.301d
2
3
4
5
6
7
8
9 10
0.477d
d
The logs of these numbers can be placed on a log
scale as fractions of the distance between 1 (100)
and 10 (101).
Log Scales
x
1
2
3
4
5
6
7
8
9
log(x) 0.000 0.301 0.477 0.602 0.699 0.778 0.845 0.903 0.954
10n
110n
10n+1
2
3
4
5
6
7
8
9 1010n
These fractional distances are the same between
any successive powers of 10 on a log scale.
Successive powers of 10 on a log scale are called
decades.
Log Scales
x
1
2
3
4
5
6
7
8
9
log(x) 0.000 0.301 0.477 0.602 0.699 0.778 0.845 0.903 0.954
10n
110n
10n+1
2
3
4
5
6
7
8
9 1010n
To find any number on a log scale
1. Write the number in scientific notation: N  10n
2. Find the appropriate decade (10n) on the scale.
3. Find N (approximately) within that decade.
Log Scales
Example: on the log scale below, find the number
648.22
Solution: write the number in scientific notation
with (at most) two sig figs
6.5102
Here is 6.5102 (approximately)
Here is 6102
Here is 1102
10-3
10-2
10-1
100
101
102
103
Log Scales
Example: on the log scale below, find the number
36010-4
Solution: write the number in scientific notation
with (at most) two sig figs
3.610-2
Here is 110-2
Here is 310-2
Here is 3.610-2 (approximately)
10-3
10-2
10-1
100
101
102
103
Log Scales
Exercise: on the log scale below, write the
approximate value of the number to
which each arrow points
10-3
10-2
2
6
10-1
4
100
1
7
10+1
9
10+2
3
10+3
5
8
Log Scales
Log scales are most useful when the numbers to be
presented have a wide range of magnitudes.
Examples:
• the energies of earthquakes (Richter scale)
• the acidity of solutions (pH scale)
• the size or mass of material objects, from
atomic nuclei to clusters of galaxies
The common (base 10) log scale below shows 24
decades of magnitude, from small (10-15) to large
(109).
-15
-12
-9
-6
-3
0
3
6
9
Log Scales
On this scale, every third exponent is numbered, and
the unit of measurement (e.g., length, m) is denoted
at zero (100 = 1).
All other numbers on this scale are either fractions
(negative exponents) or multiples (positive
exponents) of this unit.
We use SI prefixes in front of the unit to indicate the
decade in which a value is found. The prefixes in
most common use name every third decade.
m
-15
-12
-9
-6
-3
0
3
6
9
Log Scales
femto10-15
kilo103
pico10-12
Mega106
nano10-9
Giga109
We use SI prefixes in front
of the unit to name the
-6
micro- (m) 10
decade in which a value
is found. The prefixes in
-3
milli10
most common use name every third decade.
fm
pm
nm
mm
mm
m
km
Mm
Gm
-15
-12
-9
-6
-3
0
3
6
9