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)
    \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]

    群论的哲学 Philosophical Group Theory

    ​在一个群体里, 每个会员互动中存在一种”运作” (binary operation, 记号 *)关系, 并遵守以下4个原则:

    1) 肥水不流外人田: 任何互动的结果要回归 群体。(Closure) = C

    2) 互动不分前后次序 (Associative) = A

    (a*b)*c = a*(b*c)

    3) 群体有个 共同 的 “中立身份 (Neutral / Identity) = N (记号: e)

    4) 和而不同: 每个人的意见都容许存在反面的意见 “逆元” (Inverse) = I (记号: a 的逆元 = a^{-1})

    Agree to disagree = Neutral

    a*a^{-1} = e


    群体的 “美 : “对称” 

    如果没有 (3)&(4): 半群

    如果没有 (4) 反对者: 么半群
    以上是 Group  (群 ) 数学的定义: “CAN I”

    CA = Semi-Group 半群

    CAN = Monoid 么半群

    群是 Evariste Galois 19 岁数学天才 (伽罗瓦)在法国大革命时牢狱中发明的, 解决 300年来 Quintic Equations (5次或以上的 方程式) 没有 “根式” 解 [1] (radical roots)。19世纪的 Modern Math (Abstract Algebra) 从此诞生, 群用来解释自然科学(物理, 化学, 生物)里 “对称”现象。Nobel Physicists (1958) 杨振宁/李政道 用群来证明物理 弱力 (Weak Force) 粒子(Particles) 的不对称 (Assymetry )。

    Note [1]:”根式” = \sqrt[n]{x}

    比如: Quadratic equation (二次方程式) 有 “根式” 解:[最早发现者 : Babylon  和 三国时期的吴国 数学家 赵爽]

    {a.x^{2} + b.x + c = 0}
    \boxed{x= \frac{-b \pm \sqrt{b^{2}-4ac} }{2a}}

    3次方程式 (16世纪 意大利数学家 解根 时发现 i =\sqrt {-1} ) & 4次方程式也有 “根式” 解, 但5次或以上没有 — 这和 根 (roots) 的”对称”有关系。

    1930民国初 一位苏家驹教授 写论文称他证明 5次方程有”根式”解。当时15岁的杂货店员 华罗庚 用19世纪Galois的证明 反驳。此事得到清华大学数学系主任 (留法博士) 熊庆来教授 和 杨武之教授 (杨振宁的父亲, 中国第一位”庚子赔款”奖学金 留美 数学博士)的注意, 破格收”只有中二学历”的华罗庚进入清华读数学。 毕业后, 熊再保送他去 英国剑桥大学深造三年, 跟从当代世界最伟大的数论大师  Prof G.Hardy (Ramanujian 的老师)。华罗庚青出于蓝, 比2位恩师更有成就, 尤其对 1960 ~ 1980的 中学数学教育的贡献, 考察苏联 大师 А. Н. Колмогоров 的数学教育改革, 编撰中国中学数学课本, 影响中国和星/马华校学子。

    Group Theory in Rubik’s Cube & Music

    Group is the “mathematical language” of Symmetry — the beauty which pleases human’s eyes and animal’s eyes: We are attracted by a symmetrical face (五官端正), bees are attracted by a symmetrical flower…

    Group was discovered by a 19-year-old French genius Evariste Galois (sound: \ga-lua, 1811-1832), who was attempting to explain why any quintic equations (polynomial with degree 5 or more) could not have a solution formula (like quadratic, cubic, equations) using +, -, ×, /, nth root radicals. This 300-year problem since 16th century defeated even then the world’s greatest Mathematician Gauss. Before a fatal duel which killed him, Galois wrote down his discovery of ‘Group Theory’ which was understood only 14 years later by Professor Louisville of the Ecole Polytechnique — ironically the same university which failed Galois twice in admission exams (Concours, aka French imitation of Chinese 科举).

    Group has 4 properties: “CAN I ?”

    C: Closure
    A: Associative
    N: Neutral element (or Identity)
    I: Inverse

    Rubik’s cube is group. It is a fun game.

    Music is group. It is pleasing to ears if the music is nice, or “symmetric”.

    Cayley graphs and the geometry of groups

    Terence Tao’s wrote this easy-to-understand Group Theory. It is rare for mathematicians to be also a good writer in explaining difficult math in layman’s terms. Highly recommended!

    To be continued (Part 2):

    What's new

    In most undergraduate courses, groups are first introduced as a primarily algebraic concept – a set equipped with a number of algebraic operations (group multiplication, multiplicative inverse, and multiplicative identity) and obeying a number of rules of algebra (most notably the associative law). It is only somewhat later that one learns that groups are not solely an algebraic object, but can also be equipped with the structure of a manifold (giving rise to Lie groups) or a topological space (giving rise to topological groups). (See also this post for a number of other ways to think about groups.)

    Another important way to enrich the structure of a group $latex {G}&fg=000000$ is to give it some geometry. A fundamental way to provide such a geometric structure is to specify a list of generators $latex {S}&fg=000000$ of the group $latex {G}&fg=000000$. Let us call such a pair $latex {(G,S)}&fg=000000$…

    View original post 2,096 more words

    Great Math Popular Books


    日本数学科普作家写的好书。深入浅出, 适合中学生读最高深数学。

    結城 浩 Hiroshi Yuki (1963 -) is a Japanese Math Popular Book Writer for Secondary and High School students. In the “Galois Theory” (Chapter 10) he boldly attempted to explain to them such complicated concepts: Quotient Group, Field Extension, Group Order, Normal Sub-Group, Solvable Group …

    Evariste Galois‘s genius is he built a “bridge” between Field (域/体) and Group (群) – both new concepts invented by him. The “bridge” is called Galois Group, or by Emile Artin the “Group Automorphism”. He transferred the difficult problem of solving complicated 4 -ops (+-×/) Field (coefficients) to the single-op (permutation of roots) Group.

    Galois Group is the ultimate TRUTH of all Math — Fermat’s Last Theorem, and any advanced Math, will use Galois Group or Field, to solve. Prof C.N. Yang 杨振宁 Nobel-Prize Physics discovery was based on Group Theory.

    Evariste Galois was a French Math genius, died at 21 in a duel during French Revolution. He is the ‘Father’ of Modern Algebra. Failed 2 years in Ecole Polytechnique CONCOURS Entrance Exams, then kicked out by Ecole Normale Supérieure, his Math was not understood by all the 19th century World’s greatest Math Masters : Gauss, Fourier, Poisson, Cauchy…

    “Galois Theory” — the ultimate Math “葵花宝典” (a.k.a. “Kongfu Bible“) — is only taught in the Math Honors Undergraduate or Masters degree Course.

    自己学习《Galois Theory》(Page 365):
    — “高中会教这种困难的数学吗 ?”
    — “…我觉得比起給高中老师教, 不如自己好好学吧。”
    — ” 重要的是自己学习。”

    Modern Algebra (Abstract Algebra) Made Easy

    UReddit Courses:

    Modern Algebra (Abstract Algebra) Made Easy

    This video series is really well done ! short and sharp, yet cover the entire syllabus in the Group Theory.

    Strongly recommended for those Math-inclined students from upper secondary schools (Secondary 3 to JC2). Although the Singapore school Math syllabus based on Cambridge ‘O’ and ‘A’ level do not cover modern math – which is a serious weakness for being biaised on computational applied math, an outdated pedagogy for the last 40 years with no major changes – we miss the latest Math development since 19 century, the so called ‘Modern Math’ but already 300 years old.

    Group Theory is the stepping stone to open the door of interesting advanced Math, physics, chemistry, bio-science and engineering. It should not be limited only to the Math-major undergraduates in university. (Note: Why France and China make Modern Algebra compulsory for all science and engineering students )

    Part 0: Binary Operations

    Part 1: Group

    Note: Why ‘e’ for Identity, ‘Z’ integers

    Part 2: Subgroup

    Part 3: Cyclic Group & its Generator

    Part 4: Permutations

    Part 5: Orbits & Cycles

    Part 6 : Cosets & Lagrange’s Theorem

    Part 7 : Direct Products / Finitely generated Abelian groups

    Part 8: Group Homomorphism

    Part 9: Quotient Groups

    Part 10: Rings & Fields

    Part 11: Integral Domains

    Reference: further studies in deeper and advanced Abstract Algebra at:

    Harvard Online Free Course by Prof Benedict Gross