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14 – The Later 19th Century –
Arithmetization of Analysis
The student will learn about
the contributions to
mathematics and
mathematicians of the
late 19th century.
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§14-1 Sequel to Euclid
Student Discussion.
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§14-1 Sequel to Euclid
“. . . A course in this material is very desirable for
every perspective teacher of high-school geometry.
The material is definitely elementary, but not easy,
and is extremely fascinating.”
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§14-2 Three Famous Problems
Student Discussion.
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§14-2 Construction Limits
1. Can construct only algebraic numbers.
i.e. solutions to polynomial equations with rational
coefficients. Example : x2 – 2 = 0
Note: non-algebraic numbers are transcendental
numbers.
2. Can not construct roots of cubic equations with
rational coefficients but with no rational roots.
Descartes’ rational root test. Example 8x3 – 6x –1 = 0
Possible rational roots are ± 1, ± ½, ± ¼, and ± 1/8.
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§14-2 Quadrature of a Circle 2
Reduces to the equation –
s
s 2 = π r 2 or s = r  π
r
However,  π is not an
algebraic number and hence
cannot be constructed.
s2 = r2
x2 - π = 0
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§14-2 Duplication Problem
Reduces to the equation
x 3 = 2 or x 3 – 2 = 0
But this has no rational
roots and hence is not
possible.
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§14-2 Angle Trisection
Trig Identity cos θ = 4 cos 3 (θ/3) – 3 cos (θ/3)
Let θ = 60º and x = cos (θ/3) then
the identity becomes:
½ = 4 x 3 – 3x or 8x 3 – 6x – 1 = 0
But this has no rational
roots and hence is not
possible.
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§14 -3 Compass or Straightedge
Student Discussion.
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§14 -3 Compass
Lorenzo Mascheroni and Georg Mohr
All Euclidean constructions can be done by compass
alone.
Need only show:
1. Intersection of two lines.
2. Intersection of one line and a circle.
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§14 -3 Straightedge
Jean Victor Poncelet
All Euclidean constructions can be done by straight
edge alone in the presence of one circle with center.
Fully developed by Jacob Steiner later.
Need only show:
1. Intersection of one line and a circle.
2. Intersection of two circles.
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§14 -3 Compass or Straightedge
Abû’l-Wefâ proposed a straightedge and a rusty
compass.
Yet others used a two-edged ruler with sides not
necessarily parallel.
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§14- 4 Projective Geometry
Student Discussion.
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§14- 4 Poncelet
Principle of duality
Two points determine a line.
Two lines determine a point.
Principle of continuity – from a case proven in the real
plane there is a continuation into the imaginary plane.
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14-5 Analytic Geometry
Student Discussion.
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14-5 Julius Plücker
Line Coordinates
• A line is defined by the negative reciprocals of its x
and y intercepts.
• A point now becomes a “linear” equation.
• A line becomes an ordered pair of real numbers.
More later.
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§14 - 6 N-Dimensional Geometry
Student Comment
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§14 - 6 N-Dimensional Geometry
Hyperspace for n dimensions and n > 3.
Emerged from analysis where analytic treatment
could be extended to arbitrary many variables.
n dimensional space has • Points as ordered n-tuples (x 1, x 2, . . . , x n)
• Metric d (x, y) = [(x 1–y1) 2 + (x 2–y2) 2 + . . . +(x n–yn) 2]
• Sphere of radius r and center (a 1, a 2, . . . , a n )
(x 1–a1) 2 + (x 2–a2) 2 + . . . +(x n–an) 2 = r2
• Line through (x 1, x 2, . . . , x n) and (y 1, y 2, . . . , y n)
(k (y 1–x1) 2, k (y 2–x2) 2, . . . , k (y n–xn) 2 ) k  0.
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§14-7 Differential Geometry
Student Discussion.
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§14 – 8 Klein and the
Erlanger Program
Student Discussion.
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§14 – 9 Arithmetization of Analysis
Student Discussion.
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§14–10 Weierstrass and Riemann
Student Discussion.
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§14–11 Cantor, Kronecker, and
Poincaré
Student Discussion.
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§14–12 Kovalevsky, Noether and
Scott
Student Discussion.
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§14–13 Prime Numbers
Student Discussion.
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§14–13 How many Prime Numbers
Is there a formula to calculate the number of primes
less than some given number?
Consider the following:
n
Number of primes < n
10
4
100
14
...
...
10 9
50,847,534
10 10
455,052,511
n
???
Confirm this.
 n / ln n
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§14–13
2 n - 1 generates primes:
n
1
2n - 1
1
2
3
4
3
7
31
5
...
127
521
63
...
39 digit prime
How many digits?
216,091
64,828 digits
2n- 1
Composite
2n
10
20
30
40
...
10k
# digits
4
7
10
13
1 + 3k
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2n
§14–13
2 1
2n
Fermat thought that 2  1 generated only primes:
n
2n
1
2 1
5
2
17
3
257
4
65,537
5
4,294,967,297
Composite
Also composite for n = 145 and
lots of others
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§14–13 Palindromic Primes
11, 131, 151, . . . , 345676543, . . .
There are no four digit palindromic primes. WHY?
11 is the only palindromic primes with an even
number of digits.
There are 5,172 five digit palindromic primes.
Homework – find the smallest five digit prime.
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Functions to Generate Primes
f (n) = n 2 – n + 41 yields primes for n < 41.
n
1
2
3
4
5
6
f (n)
41
43
47
53
61
71
...
f (n) = n 2 – 79 n + 160 yields primes.
Homework – find a polynomial that yields all primes.
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Twin Primes
2, 3 and 5, 7 and 11, 13 and 137, 139 and 1007, 1009
and infinitely many more.
My new phone number is 2 5 · 5 3 · 11 · 191
Note 2, 5, 11 and 191 are the first of twin primes.
Note 2, 5, 11, and 191 are all palindromic primes.
Goldbach’s Conjecture – Every even integer > 2 can
be written as a sum of two prime numbers.
1000 = 3 + 997
Homework – write 2002 as the sum of two primes.
Goldbach Bingo
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Poincaré’s Model
Hyperbolic Geometry
Normal points
m
o
P
n
l
Ideal points
Ultra-ideal points
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Assignment
Papers presented from
Chapters 11 and 12.
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