Transcript § 1-1 Functions

Cultural Connection
Puritans and Seadogs
The Expanse of Europe – 1492 –1700.
Student led discussion.
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9 – The Dawn of Modern
Mathematics
The student will learn about
Some European
mathematical giants.
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§9-1 The Seventeenth Century
Student Discussion.
3
§9-2 John Napier
Student Discussion.
4
§9-3 Logarithms
Student Discussion.
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§9-3 Logarithms
More from me later
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§9-4 The Savilian and Lucasian
Professorships
Student Discussion.
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§9-5 Harriot and Oughtred
Student Discussion.
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§9-6 Galileo Galilei
Student Discussion.
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§9-7 Johann Kepler
Student Discussion.
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§9–8 Gérard Desargues
Student Discussion.
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§9–9 Blaise Pascal
Student Discussion.
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§9–9 Blaise Pascal
“Problem of the Points” at end if time.
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Logarithms
Forerunner to logs was Prosthaphaeresis. Werner
used it in astronomy for calculations.
2cosAcosB = cos (A – B) + cos (A + B)
2sinAsinB = cos (A – B) - cos (A + B)
2sinAcosB = sin (A – B) + sin (A + B)
2cosAsinB = sin (A – B) - sin (A + B)
Example from astronomy using
2cosAcosB = cos (A – B) + cos (A + B)
Find sin 7º = tan 7º cos 7º (1/2 chord of 14º )
= 0.1227845 · 0.9925462
This becomes our multiplication problem.
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Logarithms 2
This becomes our multiplication problem.
0.1227845 · 0.9925462
Let’s multiply!
0.9925462
0.1227845
49627310
397018480
7940369600
69478234000
198509240000
1985092400000
9925462000000
0.12186928889390
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Logarithms 3
2cosAcosB = cos (A – B) + cos (A + B)
sin 7º = 0.1227845 · 0.9925462
Let 2cos A = 0.1227845, then A = 86.4802681
cos B = 0.9925462, then B = 7.0000000
A – B = 79.4802681 and A + B = 93.4802681
cos (A – B) =
0.1825741
+ cos (A + B) = - 0.0607048
0.1218693
Which is the sin 7º
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Logarithms 4
Napier used “Geometry of Motion” to try to
understand logarithms.
P
-4a
-3a
-2a
-a
0
a
2a
3a
4a
b1
b2
b3 b4
Q
b –4 b –3
b –2
b –1 1
P moves with constant velocity so it covers any unit
interval in the same time.
Q moves so that it covers each interval in the same
time period. What is implied about the velocity? The
velocity is proportional to the distance from Q.
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Logarithms 5
Please take out a sheet of paper and fold it in half ten
times.
Logarithms were developed in three ways • as an artificial number. Napier/Briggs 1614
• as an area measure of a hyperbola. Newton 1660.
• as an infinite series. Mercator 1670.
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Logarithms Artificial Numbers
The previous discussion on on Prosthaphaeresis and
the geometry of motion were contributing factors.
Historically the comparison of Arithmetic and
Geometric progressions like the one used by Napier
was a contributing factor.
Indeed there is evidence on Babylonian tablets of
this comparison.
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Logarithms Artificial Numbers 2
The Babylonians were close to developing logarithms.
They had developed the following table!
2
1
4
2
8
3
16
4
32
5
64
6
128
7
256
8
512
9
1024
10
...
...
Note the first column is a geometric
progression and the second is
arithmetic.
To multiply 16 times 64 from the left
column they would add 4 and 6 from
the right column to get 10 and look up
the corresponding number 1024 on
the left
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Logarithms Artificial Numbers 3
Logarithms were developed for plane and spherical
trig calculations in astronomy. They were used in
developing a table of log sines using a circle of radius
10,000,000 = 10 7 since the best trig tables had seven
digit accuracy. There was also a need to keep the base
of the log system small to help in interpolation.
Napier chose as his base 0.9999999. To avoid
decimals he multiplied by 10 7.
Hence N = 10 7 (1 – 1/ 10 7)L where N is a number
and L is its Napier Log.
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Logarithms Artificial Numbers 4
Hence N = 10 7 (1 – 1/ 10 7)L where N is a number
and L is its Napier Log and a table follows:

sin · 10 7
Napier’s Log
90
80
60
10000000
9848078
8660254
0
153089
1438410
45
30
10
7
2
1
7071068
5000000
1736482
1218693
348995
174524
3465736
6931472
17507240
885264
33552829
40482777
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Logarithms Artificial Numbers 5
From the previous problem by Prostaphaeresis
sin 7º = tan 7º · cos 7º
tan 7º · 10 7 = 1227846 and the Log = 20973240
cos 7º · 10 7 = 9925462 and the Log =
74818
The sum
= 21048058
And 10 7 (1 – 1/10 7 )21048058 = 0.1218693 = sin 7º
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Logarithms Artificial Numbers 6
Henry Briggs traveled to Scotland to pay his respects
to Napier. They became friends and Briggs convinced
Napier that if log 1 = 0 and log 10 = 1 life would be
better. Hence Briggsian or common logs were born.
The first tables to 14 places were published in 1624.
These tables were used until 1920 when they were
replaced by a set of 20 place tables.
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Logarithms Artificial Numbers 7
Modern method sin 7º = tan 7º · cos 7º
Log tan 7º = - 0.9108562
Log cos 7º = - 0.0032493
The sum
= - 0.9141055
And inverse log sin ( – 0.9141055) = 7º .
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Logarithms as Areas.
Log development as an area measure under a
hyperbola (1660). Natural logs.
Consider y = 1/x
a
1
then 
dx  ln a
1 x
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Logarithms as a Series.
Log development as a series a la Mercator (1670).
Natural logs.
2
3
4
x
x
x
ln ( 1  x )  x 


...
2
3
4
Converges for - 1 < x  1.
Show convergence on a
graphing calculator.
Let x = 1 and calculate the
ln 2.
Show a slide rule calculation.
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§9–9 Problem of Points
“Problem of the Points” at end, if time.
Helen and Tom are playing a game with stakes
of a “Grotto’s” pizza. They flip a fair coin and
every time it comes up heads Helen wins a point
and every time it comes up tails, Tom wins a
point. The first one to get five points wins the
pizza. Helen is ahead 3 to 2 when the bell rings
for Math History class. How do they divide the
pizza fairly?
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§9–9 Problem of Points 2
1/4
H
H
1/8
1/16
H
T
H
T
T
H = ¼ + ¼ + 3/16 = 11/16
H
H
H
T
T
T
T
H
H
T = 1/8 + 3/16 = 5/16
T
T
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Assignment
Papers presented from
Chapters 5 and 6.
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