Does Abstract Math belong to Elementary Math ? 

The answer is : “Yes” but with some exceptions.

Most pedagogy mistake made in Abstract Algebra teaching is in the wrong order (by historical chronological sequence of discovery):

[X] Group -> Ring -> Field 

It would be better, conceptual wise, to reverse the teaching order as:

Field -> Ring -> Group

or better still as (the author thinks):

Ring -> Field -> Group

  • Reason 1: Ring is the Integers, most familiar to 8~ 10-year-old kids in primary school arithmetic class involving only 3 operations: ” + – x”.
  • Reason 2: Field is the Real numbers familiar in calculators involving 4 operations: ” + – × ÷”, 1 extra division operation than Ring.
  • Reason 3: Group is “Symmetry”, although mistakenly viewed as ONLY 1 operation, but not as easily understandable like Ring and Field, because group operation can be non-numeric such as “rotation” of triangles, “permutation” of roots of equation, “composition” of functions, etc. The only familiar Group is (Z,+), ie Integers under ” +” operation.

Some features which separate Advanced Math from Elementary Math are:

  • Proof [1]
  • Infinity [2]
  • Abstract [3]
  • Non Visual [4]
  •  

    Note [1]: “Proof” is, unfortunately, postponed from high-school Math to university level. This does not include the Euclidean Geometry axiomatic proof or Trigonometry Identity proof, which are still in Secondary school Elementary Math but less emphasized since the 1990s (unfortunately).

    Note [2]: However, some “potential” infinity still in Elementary math, such as 1/3 = 0.3333…only the “Cantor” Infinity of Real number, {\aleph_{0}, \aleph_{1}} etc are excluded.

    Note [3]: Some abstract Algebra like the axioms in Ring and Field  (but not Group) can be in Elementary Math to “prove” (as in [1]): eg. By distributive law
    (a + b).(a - b) = a.(a - b) + b.(a - b)
    (a + b).(a - b) = a^{2}- ab + ba - b^{2}
    By commutative law
    (a + b).(a - b) = a^{2}- ab + ab- b^{2}
    (a + b). (a - b) = a^{2} - b^{2}

    Note [4]: Geometry was a “Visual” Math in Euclidean Geometry since ancient Greek. By 17 CE, Fermat and Descartes introduced Algebra into Geometry as the Analytical Geometry, still visual in (x, y) coordinate graphs. 

    20 CE Klein proposed treating Geometry as Group Transformation of Symmetry.

    Abstract Algebra concept “Vector Space” with inner (aka dot) product is introduced into High School (Baccalaureate) Elementary Math – a fancy name in  “AFFINE GEOMETRY” (仿射几何 , see Video 31). 

    eg. Let vectors
    u = (x,y), v = (a, b)
     
    Translation:
    \boxed {u + v = (x,y) + (a, b) = (x+a, y+b)}
     
    Stretching by a factor { \lambda} (“scalar”):
    \boxed {\lambda.u = \lambda. (x,y) = (\lambda{x},  \lambda{y})}

    Distance (x,y) from origin: |(x,y)|
    \boxed {(x,y).(x,y) =x^{2}+ y^{2} = { |(x,y)|}^{2}}
     

    Angle { \theta} between 2 vectors {(x_{1},y_{1}), (x_{2},y_{2})} :

    \boxed { (x_{1},y_{1}).(x_{2},y_{2}) =| (x_{1},y_{1})|.| (x_{2},y_{2})| \cos \theta}

    Ref: 《Elements of Mathematics – From Euclid to Gödel》by John Stillwell (Princeton University Press, 2016) [NLB # 510.711]

    Advertisements

    Richard Dedekind

    Julius Wilhelmina Richard Dedekind (6 Oct 1831 – 12 Feb 1916)

    – Last student of Gauss at Göttingen
    – Student and closed friend of Dirichlet who influenced his Mathematical education
    – Introduced the word Field (Körper)
    – Gave the first university course on Galois Theory
    – Developed Real Number ‘Dedekind Cut‘ in 1872
    – Accomplished musician
    – Never married, lived with his unmarried sister until death
    “Whatever provable should be proved.”
    – By 1858: still yet established ?
    \sqrt{2}.\sqrt{3} = \sqrt{2.3}
    – Gave strong support to Cantor on Infinite Set.

    http://www-history.mcs.st-and.ac.uk/Biographies/Dedekind.html

    What is Ideal ?

    Anything inside x outside still comes back inside

    => Zero x Anything = Zero

    => Even x Anything = Even

    Mathematically,

    1. nZ is an Ideal, represented by (n)

    Eg. Even subring (2Z) x anything big Ring Z = 2Z = Even

    2. (football) Field F is ‘sooo BIG’ that

    (inside = outside)

    => Field has NO Ideal (except trivial 0 and F)

    Why was Ideal invented ? because of ‘failure” of UNIQUE Primes Factorization” for this case (example):

    6 = 2 x 3
    but also
    6=(1+\sqrt{-5})(1-\sqrt{-5})
    => two factorizations !
    => violates the Fundamental Law of Arithmetic which says UNIQUE Prime Factorization

    Unique Prime factors exist called Ideal Primes: \mbox{gcd = 2} , \mbox{ 3}, (1+\sqrt{-5}), (1-\sqrt{-5})

    Greatest Common Divisor (gcd or H.C.F.):
    For n,m in Z
    gcd (a,b)= ma+nb
    Example: gcd(6,8) = (-1).6+(1).8=2
    (m=-1, n=-1)

    Dedekind’s Ideals (Ij):
    6 =2×3= u.v =I1.I2.I3.I4 ;
    u= (1+\sqrt{-5})
    v=(1-\sqrt{-5})

    Let gcd(2,u) = 2M+N.u
    M,N in form of a+b\sqrt{-5}

    1. Principal Ideals:
    2M = (2) = multiple of 2

    2. Ideals (nonPrincipal) = 2M+N.u

    3. Ideal prime factors: 6=2 x 3=u.v
    Let
    I1= gcd(2, u)
    I2=gcd(2, v)
    I3=gcd(3, u)
    I4=gcd(3, v)
    Easy to verify (by definition):
    I1.I2=(2)
    I3.I4=(3)
    I1.I3=(u)
    I1.I4=(v)
    => Ij are prime & unique factors of 6=I1.I2.I3.I4
    => Fundamental Law of Arithmetic satisfied!

    =>Ij “Ideal“-ly exist! hidden behind ‘compound’ (2,3,u,v) !

    Verify : gcd(2, 1+√-5).gcd(2, 1-√-5)=(2) ?

    Proof by definition:

    [2m+n(1+√-5)][2m’+n'(1-√-5)]
    =[2m+n+n√-5 ][2m’+n’-n’√-5]
    = 4mm’+2mn’+2nm’+6nn’
    = 2(2mm’+mn’+m’n+3nn’)
    = 2M
    = 2 multiples
    = (2) = Principal Ideal

    Field: Galois, Dedekind

    Dedekind
    (1831-1916)

    Dedelind was the 1st person in the world to define Field:
    “Any system of infinitely many real or complex numbers, which in itself is so ‘closed’ and complete, that +, – , *, / of any 2 numbers always produces a number of the same system.”

    Heinrich Weber (1842-1913) gave the abstract definition of Field.

    Field Characteristic

    1. Field classification by Ernst Steinitz @ 1910
    2. Given a Field, we start with the element that acts as 0, and repeatedly add the element that acts as 1.
    3. If after p additions, we obtain 0 again, p must be prime number, and we say that the Field has characteristic p;
    4. If we never get back to 0, the Field has characteristic 0. (e.g. Complex Field)

    Example: GF(2) = {0,1|+} ; prime p = 2
    1st + (start with 0):
    0 + 1 = 1
    2nd (=p) +:
    1 + 1 = 0 => back to 0 again!
    => GF(2) characteristic p= 2

    Galois Field GF(p)

    1. For each prime p, there are infinitely many finite fields of characteristic p, known as Galois fields GF(p).

    2. For each positive power of prime p, there is exactly one field.
    (This is the only IMPORTANT Theorem need to know in Field Theory)
    E.g. GF(2) = {0,1}

    Math Game: Chinese 9-Ring Puzzle  (九连环 Jiu Lian Huan)

    http://www.google.com.sg/imgres?imgurl=http://info.makepolo.com/uploadfile/2012/0723/20120723100653765.jpg&imgrefurl=http://info.makepolo.com/htmls/6/69/2669.html&h=400&w=533&sz=44&tbnid=ExodLfHv3cQjHM:&tbnh=91&tbnw=121&zoom=1&usg=__hsZaBecpPNdvTvguQbaQftCsXgo=&docid=qXMWtmo8A-vXEM&hl=en&sa=X&ei=f2NaUciqKYrOrQeT2YHgDw&sqi=2&ved=0CEsQ9QEwAg&dur=591

    To solve Chinese ancient 9-Ring Puzzle (九连环) needs a “Vector Space V(9,K) over Field K”

    finite Field K = Galois Field GF(2) = {0,1|+,*}
    and 9-dimension Vector Space V(9,K):
    V(0)=(0,0,0,0,0,0,0,0,0) ->
    V(j) =(0,0,… 0,1,..0,0) ->
    V(9)= (0,0,0,0,0,0,0,0,1)

    From start V(0) to ending V(9) = 511 steps.

    Eigenvector & Eigenvalue

    1. Matrix (M): stretch & twist space
    2. Vector (v): a distance along some direction
    3. M.v = v’ stretched & twisted by M

    Some directions are special:-
    a) v stretched but not twisted = Eigenvector;
    b) The amount of stretch = constant = Eigenvalue (λ)

    Let M the matrix, λ its eigenvalue,
    v eigenvector.
    By definition: M.v = λ.v
    v = I.v (I identity matrix)
    M.v = λI.v
    (M – λI).v=0
    As v is non-zero,
    1. Determinant (M- λI) =0 => find λ
    2. M.v = λ.v => find v

    Note1: Why call Eigenvalue ?
    From German: “Die dem Problem eigentuemlichen Werte
    = “The values belonging to this problem
    => eigenWerte = EigenValue
    Eigenvalue also called ‘characteristic values’ or ‘autovalues’.
    Eigen in English = Characteristic (but already used for Field).

    Note2: Schrödinger Quantum equation’s Eigenvalue = Maximum probability of electron presence at the orbit outside nucleus.

    Note3: Excellent further explanation of the eigenvector and eigenvalue:

    http://lpsa.swarthmore.edu/MtrxVibe/EigMat/MatrixEigen.html