The Importance of Mathematics in one's Education

By


Wu Siu Yan


24 June 2003

In primary school, we learn addition, subtraction, multiplication, division, fraction, ...

With these, we can do many "commercial arithmetics".

...... COMPANY LIMITED Description Qty Price Total xxxx 10 9.5 xxxx 25 12.9 xxxx 120 6.4 -------- ========

Could you imagine a person with no such maths knowledge running a business ? He may not be able to calculate a simple invoice !

In secondary school, one learns more maths -- geometry, algebra, trigonometry, .....

With these, one may calculate, say, the distance from shore of the ship he is sailing,

if he knows the height of the lighthouse from chart, and the angle.

Who do you think would be able to sail a ship better, one who knows trigonometry/geometry and one who doesn't ?

When one begins to learn Calculus, one's ability to calculate increases further, e.g. he is able to calculate the range of a projectile,

and knows that if he aims at 45^o elevation, the projectile will reach the farthest.

With elementary calculus, one can also calculate the "escape velocity", the minimum velocity that a rocket must reach in order to escape from Earth's gravitational field and reach outward.

If one furthers one's education in Maths, and learns, say,

• Advanced Calculus
  
  



• Complex Number and Complex Analysis
  
  



• Linear Algebra (which includes elements of abstract algebra such as "group, ring, field".)
  
  



• Ordinary Differential Equation
  
  



• Vector Analysis
  
  



• Statistics
  
  



• Classical Differential Geometry
  
  



• Classical Mechanics
  
  



• Numerical Analysis
  
  



• Calculus of Variation
  
  



• Partial Differentiation Equation
  
  



• Numerical solution of Partial Differentiation Equation, such as "Finite Element Method", "Boundary Element Method",...
  
  



• Number Theory
  
  



• ....

then one's ability to "compute" increases tremendously.

Build yourself a solid foundation in Maths, then all branches of engineering/physics will be opened to you

Do you wish to know how to compute the "stresses" in a bridge, in a high-rise building, in machine parts, in the wing of an aircraft, in the hull of a ship ?

Do you wish to know why statistics works ? that one can know the statistics of the whole population just by sampling a few, and can say, "The mean value is ..., and error is no more than 3% (say), and this should be correct 95 cases out of 100." ?

Do you wish to understand "Quantum Mechanics" ? How one single partial differential equation can give us immense insight into the Periodic Table -- the formation of all elements on Earth ?

If you have built yourself a solid foundation in Maths, then for all branches of engineering/physics such as Ship-building, aeronautical engineering, civil engineering, electric and electronic engineering, ...., you only need to study things pertaining to that but not its Maths. Your knowledge of Maths will help you a lot in mastering such knowledge, and you can learn much faster and more effective than one whose Maths foundation is shaky.

Not only in Engineering, but your knowledge of "Ordinary differential equations", or "Dynamical Systems", which gives you insight into system dynamics, will enable you to know what the "economists" are talking about, and form better judgements.

Who do you think will have a better world view, one who understands Maths and one who doesn't ?

Your deficiency in Maths will make you feel unsure all the times

On the other hand, if you have not built yourself a good foundation in Maths, your study of engineering will only be half-rewarded. YOU WILL ALWAYS BE UNSURE, YOU WILL ALWAYS HAVE THE FEELING OF UNCERTAINTY, and you can never step beyond what you have been taught.

I have seen many engineers whose efforts are hindered by their lack of mastery of Math.

Even in computer programming -- if one has studied Maths and has carried out proofs rigorously and logically, one can write much better computer programs than one who hasn't.

You cannot learn "good" maths at today's universities; they are too occupied with "abstraction"

I have first-hand experience of that. When I entered University, I studied all Maths subjects (8 papers Maths, British University style), in the hope that Maths would enable me to calculate a lot. However, I found the emphasis was on abstraction; and I graduated from University with a feeling that I no longer wanted to study Maths anymore.

Later, because I had to write computer programs for engineers, I learned "Applied Maths" by myself; and I began to find immense joy and satisfaction in those, because I CAN USE MATHS TO SOLVE CONCRETE PROBLEMS.

However, many modern text-books on Maths are not enlightening, because of their too much emphasis on abstraction, and not the connection of Maths with the real world.

Serge Lang, a Mathematician who has written many excellent Maths textbooks, said in the Preface of his book "Complex Analysis",

Serge Lang : "In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc.) and I would recommend to anyone to look through them. More recent texts have emphasized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex analysis : the power series expansion, the uniqueness of analytic continuation, and the calculus of residues. The systematic elementary development of formal and convergent power series were standard fare on the German texts, but only Cartan, in the more recent books, includes this material, which I think is quite essential, e.g. for differential equations. ..."

Therefore, if you wish to learn more Maths so that you can compute more, I would advise you to read old text-books. I have included quite some references in the preface to my Maths ebook.

The present state of knowledge of the world is not perfect, and will increase if knowledge of Maths increases

The concept of "time, space" in Einstein's Relativity is not perfect, Maxwell's Equations on electro-magnetism to describe the electromagnetic world is not perfect, Schrodinger's equation in describing the Quantum world is not perfect, ...

Personally, I think the answer lies in "Quaternion". If we know more about "Quaternion Calculus, Quaternion power series, Quaternion Calculus of Variation, Quaternion equation, ....", and be able to formulate physical laws in terms of such, we can arrive at a better description of the physical world.

Start learning now

Logic is most important in Maths, but you can learn logic quickly. I have written a logic tutorial consisting of only 10 pages (5 pages exercises and 5 pages answers). If you work through all the exercises in it, you would have good knowledge of logic. GUARANTEED. GUARANTEE VOID IF YOU MISS ONE SINGLE EXERCISE, YOU MUST WORK THROUGH ALL THE EXERCISES.

Do not hope to attain mastery of Maths quickly. Complete mastery of maths -- including numerical solution of Partial Differential Equations -- will take some years. But be patient. Learn a little every day, apply your Maths knowledge constantly, and pray to GOD to give you talents in this, and one day, when you have ventured into many branches of engineering and physics with your Maths, and see the beauty of GOD's creation, you will find extreme satisfaction.