Our Daily Story #8: The Rigorous Mathematician with epsilon-delta

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http://en.m.wikipedia.org/wiki/Augustin-Louis_Cauchy

We mentioned Augustin Louis Cauchy in the tragic stories of Galois and Abel. Had Cauchy been more generous and kind enough to submit the two young mathematicians’ papers to the French Academy of Sciences, their fates would have been different and they would not have died so young.

Cauchy was excellent in language. He was the 2nd most prolific writer (of Math papers) after Euler in history. When he was a math prodigy, his neighbor — the great French mathematician and scientist Pierre-Simon Laplace — advised Cauchy’s father to focus the boy on language before touching mathematics. (Teachers / Parents take note of the importance of language in Math education.)

Cauchy’s language education made him very rigorous in micro-details. This was the man who developed the most rigorous epsilon-delta Advanced Calculus (called Analysis) after Newton / Lebniz had invented the non-rigorous Calculus (why?).

Rigorous epsilon-delta Analysis:

https://tomcircle.wordpress.com/2013/03/31/432/

The modern epsilon-delta definition by Cauchy was arithmetized by Weierstrass as below:

\displaystyle\lim_{x\to a}f(x) = L \iff
\forall  \varepsilon >0, \exists \delta >0 such that
\boxed{0<|x-a|<\delta \implies |f(x)-L|< \varepsilon}

Note:
When I first encountered this epsilon-delta ‘horor’ in the 1st year Math in France, I saw my French classmates reciting at ease the definition learned in their Baccalaureate (but not taught in our British Cambridge G.C.E. A-Level Math):
Quelque soit ε > 0, il existe δ > 0, tel que ...”

If you can’t understand the ‘ε-δ’ definition, just memorize it like a poem – at least better than saying it wrongly:
For all ε > 0, there exists δ > 0, such that …

Watch the excellent Khan Academy Lecture on Limit with ‘ε-δ’ :

(To continue in YouTube on the same webpage of this video)

References:

1. Series Convergence with ‘ε-δ’
https://tomcircle.wordpress.com/2013/05/10/analysis-by-timothy-gowers/

2. Limit with ‘ε-δ’
https://tomcircle.wordpress.com/2013/04/06/limit-%ce%b5-%ce%b4-analysis/

3. Sequence with ‘ε-δ’
https://tomcircle.wordpress.com/2013/05/30/sequence-limit/

4. Limit and Continuity: ‘ε-δ’
https://tomcircle.wordpress.com/2013/04/25/%ce%b5%ce%b4-limit-continuity/

Group Theorems: Lagrange, Sylow, Cauchy

1. Lagrange Theorem:
Order of subgroup H divides order of Group G

Converse false:
having h | g does not imply there exists a subgroup H of order h.
Example: Z3 = {0,1,2} is not subgroup of Z6
although o(Z3)= 3 which divides o(Z6)= 6

However,
if h = p (prime number),
=>
2. Cauchy Theorem: if p | g
then G contains an element x (so a subgroup) of order p.
ie.
x^{p} = e ∀x∈ G

3. Sylow Theorem :
for p prime,
if p^n | g
=> G has a subgroup H of order p^n:
h= p^{n}

Conclusion: h | g
Lagrange (h) => Sylow (h=p^n) => Cauchy (h= p, n=1)

Trick to Remember:

g = kh (god =kind holy)
=> h | g
g : order of group G
h : order of subgroup H of G
k : index

Note:
Prime order Group is cyclic
(Z/pZ, +) order p is cyclic & commutative.

Order 4: Z4 not isomorphic to Z2xZ2

Order 6: only Z6 isomorphic Z2xZ3.
Z6 non-commutative

S3 = {1 2 3} ≈ D3 Not Abelian
(1 2)(1 3) = (1 3 2)

(1 3)(1 2) =(1 2 3)

Lagrange: |G|=6
=> order of subgroups in G = 1,2,3,6
6= 2×3
Cauchy : 2|6, 3|6 (2,3 prime)
=> order of elements in G
= 2, 3

Rigorous Calculus: ε-δ Analysis

Rigorous Analysis epsilon-delta (ε-δ)
Cauchy gave epsilon-delta the rigor to Analysis, Weierstrass ‘arithmatized‘ it to become the standard language of modern analysis.

1) Limit was first defined by Cauchy in “Analyse Algébrique” (1821)

2) Cauchy repeatedly used ‘Limit’ in the book Chapter 3 “Résumé des Leçons sur le Calcul infinitésimal” (1823) for ‘derivative’ of f as the limit of

\frac{f(x+i)-f(x)}{i}  when i -> 0

3) He introduced ε-δ in Chapter 7 to prove ‘Mean Value Theorem‘: Denote by (ε , δ) 2 small numbers, such that 0< i ≤ δ , and for all x between (x+i) and x,

f ‘(x)- ε < \frac{f(x+i)-f(x)}{i} < f'(x)+ ε

4) These ε-δ Cauchy’s proof method became the standard definition of Limit of Function in Analysis.

5) They are notorious for causing widespread discomfort among future math students. In fact, when it was first introduced by Cauchy in the Ecole Polytechnique Lecture, the French Napoleon top students booed at him and Cauchy received warning from the school.

Note 1: From the textbook ‘Calculus‘ (1980, USA):
If can’t understand the ‘ε-δ’ definition, just memorize it like a poem – at least better than saying it wrongly.
E.g. “For all ε>0, there exists δ>0, …

Note 2: George Polya: “The students are not trained in ‘ε-δ’, teaching them Calculus is like dropping these rules from the sky...”