Numbers are man's work
Gerhard Post, DWMP
Mathematisch Café, 17 juni 2013
Numbers are man's work
The dear God has made the whole numbers,
all the rest is man's work.
Leopold Kronecker (1823 - 1891)
Leopold Kronecker
Two interwoven stories:
• The concept “number”
• The representation of a number.
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Egyptian fractions
®
A Number is a sum of distinct unit fractions,
1
5
1 1
such as 8 = 3 + +
4
24
Rhind papyrus (1650 BC)
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Egyptian fractions: construction
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Egyptian fractions: why ?
A possible reason is easier (physical) division:
5
8
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1
2
= +
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8
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The Greek
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A Number is a ratio of integers
or: a number is a solution to an equation of the form:
c1 x + c0 = 0
(c1 and c0 integers)
Hippasus (5th century BC) is believed
2 is not a number
to have discovered that 2 is not a
number
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The Greek (after Hippasus)
®
A Number is a solution to an equation of the
form:
cn x n + cn-1 x n-1 + … + c1 x + c0 = 0
for integers cn ,…,c0.
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Orloj, Prague (15th century)
Orloj - Astronomical Clock - Prague
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Orloj, Prague (15th century)
Toothed wheels
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Orloj, Prague
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A Number is a ratio of ‘small’ integers
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Orloj, Prague
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A Number is a ratio of ‘small’ integers
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Orloj, Prague
How to construct these small integers ?
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The Italians (Cardano’s “Ars Magna”, 1545)
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A Number is a solution to an equation of the
form:
cn x n + cn-1 x n-1 + … + c1 x + c0 = 0
Girolamo Cardano
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Niccolò Tartaglia
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Lodovico Ferrari
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Solve: x 3 + a x 2 + b x + c = 0
1
1. Replace x by (x  a) (drop the prime) gets rid of x 2 :
3
x3 + b x + c = 0
2. Substitute u - v for x
(u 3  3uv(u  v)  v 3) + b (u  v) + c = 0
3. Take 3uv = b:
u 3  v3 + c = 0
4. Substitute v = 1/3 b/u → quadratic equation in u3.
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Simon Stevin Brugensis (1548 1620)
®
A Number is a
decimal expansion
Simon Stevin
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Beginning of 19th century
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A Number is an algebraic number (since 500 BC)
An algebraic number is a solution to an
equation of the form:
cn x n + cn-1 x n-1 + … + c1 x + c0 = 0
for integers cn ,…,c0.
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Joseph Liouville (1809 - 1882)
f(x) = cn x n + cn-1 x n-1 + … + c1 x + c0 = 0
(integers cn ,…, c0).
If  is an irrational algebraic number satisfying f ()=0
the equation above, then there exists a number A > 0
such that, for all integers p and q with q > 0:
𝑝
𝐴
 − > 𝑛
𝑞
𝑞
𝑝
𝑞
The key observation to prove this is: |f ( )| 
𝑝
𝑞
and 𝑓 𝛼 − 𝑓( ) = 𝑓’ (𝑐) (𝛼 −
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𝑝
𝑞
1
𝑞𝑛
𝑝
𝑞
if f ( ) ≠ 0,
)
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Joseph Liouville (1809 - 1882)
®
A Number is an algebraic or a Liouville number
A Liouville number is a number  with
the property that, for every positive
integer n, there exist integers p and q
with q > 0 and such that
0<  −
𝑝
𝑞
<
1
𝑞𝑛
Joseph Liouville
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Joseph Liouville (1809 - 1882)
Liouville’s constant: 𝛼 =
1
101!
+
1
102!
+
1
103!
+…
= 0.11000100000000000000000100…
Q: How many Liouville numbers are there?
A: As many as all decimal expansions…
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Georg Cantor (1845 –1918)
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A Number is a decimal expansion
Not all infinities
are the same
Georg Cantor
Leopold Kronecker: “I don't know what predominates in Cantor's
theory – philosophy or theology, but I am sure that there is no
mathematics there.”
David Hilbert: “No one will drive us from the paradise which
Cantor created for us.”
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Conclusions
®
A Number is …
Although the numbers are man’s work,
they brought us to paradise…
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