Gl b l F
Global
Focus on K
Knowledgel d-CreatingC ti M
Mathematicsth
tiLecture three

Mathematics ‘‘On Campus’’
Creating words, Originating worlds
2009.10.22
Thee figures,gu es, pphotos
otos aandd moving
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ages witht ‡‡marksa s attac
attached
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to their copyright holders. Reusing or reproducing them is prohibited
unless permission is obtained directly from such copyright holders.

Number System• Natural number 0,0 1
1, 2,2 3,3 44, ….
• Integer0, ±1, ±2, ±3, ±4, ….
• Rational number• Real number, － , 0,0 ….
, e = 2.7182818....,
π=3.141592653....,
3 141592653….
• Complex
pnumber,,cos +sin , ….

at elementary school• Numerical position.... 1st -4
4th grade at elementary school• 0 (any number when multiplied by zero iszero))....
3rd graded• fraction (Integer to rational number)....
4thgrade)• decimal ((rational number to real number)....
4th grade
4th grade• Number line (real number) ....

at and after Junior high school• Negative number(natural number to Integer).... 1st grade
at junior high school• Square root(rational number to real number).... 3rd grade
at junior high school• Prime number(Integer) ....
3rd grade
att junior
j i hihighh schoolh l•
is not a rational number .... 1st grade at high school• Mathematical
M thti l induction(naturali d ti ( t l number)....
b )2nd grade at high school• Imaginary number (real(number to complex number)....
)
2nd grade at high school• theth definitiond fi iti off reall numberb .... 1st1 t graded att university?
iit ?

In the historical order• Fraction .... Ancient Mesopotamia•

iis an iirrationaltil number....Ancientb
A i tG
GreekkPythagoreany gschool ((6c. BC))• The theory of ratio....real number theory at
Ancient Greek
Eudoxos ((4c. BC))
• Numerical position，0 .... India(around 6c)• NegativeNti numberb.... India(aroundI di (d 7c)
7 )Pythagoras(580 BC? - 497 BC?)
Reprinted fromhttp://en.wikipedia.org/wiki/File:Kapitolinischer_Pythagoras_adjusted.jpg(2010/09/03)

In the historical order• decimald i l…. endd off 1616c.
• Imaginary number …. Cardano(1545)• Fundamental theorem of algebra…. Gauss(1799)

In the mathematical order• natural number to integer .... subtraction• integer to rational numbernumber....
division• rational number to real number .... length, area
differentiation/integration
pnumber ....
• real number to complex
solution of equation

real number to complex number• the Cubic Formula(Cardano formula)

the concept of imaginary number isnecessary to get to the solution,even if theh solutions
l i
are real.lCardano (1501.9.24 - 1576.9.21)
Reprintedpfromhttp://en.wikipedia.org/wiki/File:Cardano.jpg(2010/09/03)

Cardano’s
Cardanos formula (1535,1545)(1535 1545)• Cubic equation X3 = 3pX + q
subsidiary 「quadratic」equation「Y3 + Z3 = q, Y Z = p
2
X = Y + Z,Z ωY
Y + ω Z,Z23
ω Y + ωZ.
ωZ (ω = 1)

Cardano’s
Cardanos formula (1535,1545)(1535 1545)• X3 = 15X + 4 = 3pX + q
3
3
Y + Z = q = 4, Y Z = p = 5
Y3, Z3 = 2 ± 11 i
X = Y + Z,= (2 + i) + (2 － i) = 4Real number to Imaginary number• Fundamental theorem of algebra(d’Alembert/Gauss(dAlembert/Gauss theorem)
All equations have complex numbersolution(s)No need to extend the world ofnumber system(the end of story?)

Gauss (1777.4.30 - 1855.2.23)
Reprinted fromhttp://en.wikipedia.org/wiki/File:Carl_Friedrich_Gauss.jpg(2010/09/03)

Complex plane (Gauss plane)• A complex number=a point on the planei+iA complex number=a point on the plane• multiplication of complex numbers：
rotation and extension－1+1+i+ii1

Complex numbers and matrices
multiplied bymultiplied by+i :1 tot －1+ito：+iitoto

Complex numbers to Quaternions• Complex number： a + b i (a, b are real)• quaternion： a + b i + c j + d k(a, b, c, d are real)(Hamilton 1843.10.14)
1 ij=－ji=k• i2 = j2 = k2 = － 1,

• Giving up the commutatibility

Hamilton ((1805.8.4 –1865.9.2))
Reprinted fromhttp://en.wikipedia.org/wiki/File:WilliamRowanHamilton.jpeg(2010/09/03)

In the latter half of the 19th century• the definition of Natural number....
Peano axioms (1891)• 0 is a natural number.
• if n is a natural number, then n + 1 is also anatural number• there is no natural number n such that 0 = n+1• if n + 1 = m + 1 then n = m• The principle of mathematical induction

Peano ((1858.8.27 – 1932.4.20))
Reprinted fromhttp://en.wikipedia.org/wiki/File:Giuseppe_Peano.jpg(2010/09/03)

In the latter half of 19th century• the definition of real number ....
Cantor the founder of set theory (1872)
Dedekind (1858.11.24)(1858 11 24)
ffoundations
d ti
off theth world
ld off numberbsystem
yestablished(the end of story this time?)

Cantor(1845.3.3 –1918.1.6)
Reprinted fromhttp://en.wikipedia.org/wiki/File:Georg_Cantor.jpg(2010/09/03)

mathematically• natural number to integer…subtraction• Integer
I tt rational
toti
l number…divisionbdi i i• Rational number to real
number…length,areadifferentiation/integration• real number to complex number....
solution of equation

Rational number to real number
number....length
lengthThe ratio between the length of the side ofa square and the diagonal
is1

not a rational number有理数から実数へ・・・長さ
正方形の１辺と対角線の長さの比1有理数ではないは

is not a rational number
suppose=(i d ibl ) then(irreducible)
th2m2 = n2m2 = 2n’2

n is even n=2n’
m is evencontradiction

rational number to real number
number....areaarea• The area of a disk
• TheTh numberb off □
times
the area of each □Th smallll □ limit
Theli it

The area of a disk

Rational number to real number・・
differentiation, integration, limit• Exponential function (ex)’= ex, e0=1,e = 2.7182818....
= 1+
1 1+ + += + + + +=1-+–++++ +・・+ + ....

–+ ....

What is real number?
• A point on a line
・ what is a point on a line?....
reall numberba circular argument• the limit of rational number• A real number can be expressed as(i fi it )d i l(infinite)decimal‡Reprinted fromhttp://www.ms.u‐
tokyo.ac.jp/gcoe/summary/documents/pamphlet_Part16.pdf( 2010/09/03)

creating the system of number• The human understanding of numbers has madeprogress as human extended the world of number.
• It begins
b i withith naturalt l numbers,bas we count things, one, two, …
th subtractionthenbt ti leadsl d tto th
the conceptt off integer,i tdivision to fraction, and so forth.
• OnO theh contrary, reall numberb whichhi h iis thought
hh to bbeordinary number, is considered to be a point on thenumber line.
line• This way of thinking was so natural that it was not until
the latter 19th century that the true property of real
numbers was understood.

The theory of ratio by Eudoxos(the theory of quantity)• A method which deals with the ratioof quantity (real number) by integers• A : B = a : b means
with any given natural number m, n > 0mA
A > nBBma > nbbmA < nB
ma < nb

Eudoxos?(400 BC? – 350 BC?)

Dedekind cut (1858.11.24)(1858 11 24)• A method which deals with real number thattreats rational number as elements• Real number x is the set of rational number｛ ｜ is｛r｜ri rational
til anddr<x}• Hence if the real number equation x = y holds,r<xr<ywhere r is any rational number.

Reprinted fromhttp://ja.wikipedia.org/wiki/ファイル:Dedekind stamp.jpg(2010/09/03)

Dedekind(1831.10.6 – 1916.2.12)

1 = 0.999999....
0 999999
becauser<1
r < 0.999999....
0 999999
where r anyyg
given rational numberNumber System (19th century)• Natural number 0,
0 1
1, 2,
2 3,
3 4
4, ….
• Integer0, ±1, ±2, ±3, ±4, ….
• Rational number• Real number, － , 0,0 ….
, e = 2.7182818....,
π=3.141592653....,
3 141592653….
• Complexpnumber,,cos +sin , ….

The discovery of a new numberworld (1897)• Construction of real number based on theconcept of rational number(Dedekind, Cantor)• The discovery of a method to extend the number
world• using the same method,method
the world of rational number was extended in adifferent direction
the great revolution in number theory (Hensel)

Reprinted fromhttp://paginas.fe.up.pt/~sam/homepage/inftheory.htm(2010/09/03)

HHensell (1861.12.29–1941.6.1)

The number system(after 1897)
rational number

2-adicnumber3-adicnumber5-adicnumber7-adicnumber........

real
numbercomplexnumberb

Method to extend the world of numbers• real numbers are limits of rational numbers(Cantor 1872)• corresponding to each prime number
p = 2, 3, 5, 7, ....,There exists
one way to specify limit (topology),(topology)
and one can construct number system(Hensel 1897)
there is no other way (Ostrowski)

The Way to specify limits .... topology• Real number
1 0
1,0.1,
1 0
0.01,
01 00.001,001 .... → 0•77-adicadic number
1, 7, 49, 343, 2401, .... → 0

7 adic number .... multiplied by 7 equals
7-adic• (the set of all integers) multiplied by 7 equals((the set of all multiplespof 7).
)• (the set of all integers) is divided by 7 into equalparts(the set of all multiples of 7)(the set of all integers which leave theremainder of 1 when divided by 7)，........
( the set of all integers which leave theremainder of 6 when divided by 7 ).

The advantages of extending
the number world• Equations become easier to solve
x2 – a = 0 (a( integer)i t)• Concerned with rational number• It cannot be determined until one
decompose a into prime number

The advantages of extending
the number worldx2 – a = 0• Concerned with real number
ifa > 0 solvable
whereas a < 0 not solvable

The advantages of extending
the number worldx2 – a = 0• Concerned with 7
7-adicadic number• solvable if the remainder a leaves when
di id d b
dividedby 7 iis either
ith off th
the ffollowing
ll i
1 = 12, 2 = 32 - 7, or 4 = 22(Hensel)
not solvable if the remainder is 3, 5, or 6

Local global principle (Hasse principle 1921)
Local-global
In the case of quadratic equation• solvable as rational number• solvable as p-adic number for all prime
number p• also solvable as real number

((c)Konrad)
Jacobs

Hasse (1898.8.25–1979.12.26)
Reprinted fromhttp://ja.wikipedia.org/wiki/ファイル:Helmut_Hasse.jpg(2010/09/03)

The number systemy(the first half of the 20th century)
rational number andits extensions
2-adicnumberand its
extensions

3-adicnumberand its
extensions

5-adicnumberand its
extensions

7-adicnumberand its
extensions........

real
numberandcomplexnumber• The fundamental framework of modern number
theory, that enables one to reach even for thesettlement of Fermat’s
Fermat s last theorem, wasoriginated.