直观 数学 Intuition in Abstract Math

Can Abstract Math be intuitive, ie understood with concrete examples from daily life objects and phenomena ?

Yes! and Abstract Math should be taught by intuitive way!

1. 直观 线性空间 : Intuition in Linear Space

(Part I & II) 矩阵 (Matrix), 线性变换 (Linear Transformation)


(Part III)


Animation: English (Chinese subtitles)


2. 直观 群论 (Intuition in Group Theory)



Symmetry, Algebra and the “Monster”

Very good introduction of Modern Math concept “Group” to secondary school math students by an American high school teacher.



  • Symmetry of a Square
  • Isometry (*) or Rigid Motion (刚体运动) = no change in shape and size after a transformation
  • What is a Group (群 “CAN I” ) ? = Closure Associative Neutral Inverse
  • Monster Group = God ?
  • String Theory: Higgs boson (玻色子) aka “God Particles”

Note (*): “保距映射” (Isometry),是指在度量空间 (metric space) 之中保持距离不变的”同构“关系 (Isomorphism) 。几何学中的对应概念是 “全等变换”

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。