Concrete and Abstract in Modern Math

华罗庚 《数论导引 》序言
Preface on “Introduction to Number Theory” by Hua Luogeng (1956).

“Math evolved from concrete to abstract, the former is the source of inspiration of the latter. One cannot just study the abstract definitions and theorems without going back to the source of concrete examples, which has proven valuable applications in Physics and other sciences.”

“Mathematics, in essence, is about the study of Shapes and Numbers. From Shapes give rise to the Geometrical Intuition, from Numbers give the Relationship and Concepts


Our Daily Story #10: A Shop Assistant Math Professor

In the previous story (#9) we mentioned Ramanujan having the luck of being spotted by Prof G.H. Hardy as the treasure of mathematics, another Chinese Hua Luogeng 华罗庚, 20 years younger than Ramanujan,  was also coached by Prof Hardy, although Prof Hardy did not realize Hua’s potential later to the modern mathematics in China.

Hua dropped out of secondary school due to poverty, he worked in his father’s little grocery shop as the shop assistant. His talent was spotted by the French-educated mathematician Prof Xiong Qinlai ( 熊庆来) in Tsinghua University 清华大学 , from a paper the young boy published – on Quintic Equation Solvability error made by a Math Professor Su. Hua was admitted to Tsinghua University as assistant math lecturer on exception. Later he was sent to Cambridge on 庚子赔款 Boxer Indemnity scholarship.

When Prof Hardy met Hua, he let Hua choose between:
1) Work on a PhD degree on one research topic; or
2) Work on any topic without a PhD degree.
Hua chose the 2nd option to spend his most productive 2 years in Cambridge, making good friends among the world’s top mathematicians (Paul Erdös, André Weil, etc), and published more than 10 breakthroughs in mathematics.

Hua influenced the math education in Chinese secondary schools and universities. He adopted the best from Europe, the USA and the Russian syllabus, modified and translated them into Chinese textbooks after WW II.

In 1980s after the disastrous 10-year Mao’s Cultural Revolution when China was closed to the outside world, Hua convinced Deng Xiaoping to allow Chinese students participate in the International Mathematics Olympiad (IMO). He and his team designed training programmes and promoted math competitions throughout China. In the following 20 years Chinese IMO teams (CHN) have been dominating IMO Championships and winning Gold medals.

Hua died of heart failure at 74, during a lecture in Japan at the podium.


Patriotic Mathematician

华罗庚 Hua Luogeng, while was on research scholarship in the USA, decided to leave for home immediately to help build the new China in 1949.

He wrote this famous letter of appeal to his Chinese compatriot scientists enjoying a comfortable life in the western countries:

Friends, the paradise (Liang Yuan) is good but not for long stay. For the truth, for our country and our people, let’s go home …”

He reached China via Hong Kong, just in time before the USA FBI retained other Chinese scientists from leaving the USA, notably Qian Xuesen 钱学森 who later championed the China Spaceship Programme by sending Chinese astronauts to space.


Self-Study Math Master

Hua Luogeng (华罗庚) urged using the daily 10-20 mins intervals while waiting for buses, queues, idle times, make it at least 1 hour a day to read Math books which you carry along with you.
Hua advised on speedy self-learning Math :
1) Choose the Best book on the Topic written by the Master (say, Abstract Algebra), read completely and do the exercises.
2) Read other reference books. Read only those new topics not covered in 1).
If not much new things, return them to bookshelf. This way speed up reading many books in short time.
3) Then read International renown Math Journals.
Beware 90% are copy-cats or rubbish by University lecturers to meet their yearly publishing quota. Only < 10% are masterpieces.
4) Pick one topic to do your independent research.
5) Discuss with friends with better knowledge in the field.
This way you can be a Master in the topic in 4 to 5 years.

Open Cubic Root

Mental Trick

It was discovered by the Martial Art writer Liang Yusheng 武侠小说家 梁羽生 (《白发魔女传》作者), who met Hua Luogeng (华罗庚) @1979 in England:
2³= [8]
8³= 51[2]
3³= 2[7]
7³= 34[3]

The last digit pairs :
[2 <->8] , [3 <-> 7]
Others unchanged.


\sqrt[3]{658503} = N
Last three digits 503 <-> …[7]
First three digits 658:
 (8³ =512)< 658 < (729 = 9³)
=>  8
Answer : \sqrt[3]{658503} = N= 87
Note: Similar trick for opening \sqrt[23] {200 digits} by an indian lady Ms Shakuntala (83) dubbed “Human computer”.

Indian Lady Shakuntala

Human computer