I heard this story from one of my ex-colleagues in Cathay Pacific, and I took the liberty of doing some editing and giving it a title.
There were four Catholic men and one Catholic woman having coffee. The first Catholic man stood up and said proudly, "My son is a priest. When he walks into a room, everybody calls him
'Father'."
The second Catholic man stood up and said proudly in a loud voice, "My son is a Bishop. When he walks into a room, everybody says
'Your Grace'."
The third Catholic man stood up and said proudly in an even louder voice, "My son is a Cardinal. When he walks into a room, everybody says
'Your Eminence'."
The fourth Catholic man stood up and said proudly in the loudest voice, "My son is the Pope. When he walks into a room, everybody says
'Your Holiness'."
The Catholic woman was sipping her coffee in silence. The four men looked down at her and gave her a subtle "Well ...... ?". The woman then raised her head and said shyly, "I have a daughter. She's
I taught private tuition in Mathematics to students junior than me while I was studying in secondary school. Over time, I came to become aware of some skills in handling some special situations during the lessons. Here are things you may say in class under certain circumstances:
. You don't want to go through all the possible cases. You just show one, and let the student figure out the rest,
"Without loss of generality, .......... "
. You know it's true, but you've lost your notes and you don't know how to prove it,
"It is obvious that .......... "
. You are struggling on how to explain, and your student suddenly reminds you the critical point,
"It now becomes clear that .......... "
. You feel tired after a long day's study yourself, and you don't want to explain in details,
When mathematician (or anyone working with mathematics) finishes a problem, he or she tends to tell the whole world that the work was quite easily done (Q.E.D.).
More often than not, they have already worked on it for days, if not weeks!
Talent is something you can do easily while others find it difficult to do.
Genius is something you can do easily while others find it impossible to do.
When Gauss (1777-1855) was a child, he attended the primary school in his local town. One day, the teacher found the class was too unsettled. In order to keep the class quiet for a while, he asked the children to sum the numbers from 1 to 100 before they could be dismissed. The class did stay silent while everyone was busy sketching their calculations on papers. Gauss didn't move. He stared at the blackboard for a few seconds, raised his hand and said, " The answer is 5050." The teacher looked at Gauss in disbelief, thinking he was just keen to leave the classroom. Gauss then explained step by step as follows:
"We call the sum of the numbers 1 to 100 S. Then,
S = 1 + 2 + .................. + 99 + 100
If we reverse the order of the numbers and add, the sum is still the same S.
S = 100 + 99 + .................. + 2 + 1
If we take the two together (i.e. add them vertically), we have two times the sum 2S, but the terms become 101 all the way for the 100 terms.
2S = 101 + 101 + .......... + 101 + 101
Since the numbers are the same, we don't need to add 100 times but we can multiply by 100. Simpler still, multiplying by 100 is just to add two zeros at the end of the number.
2S = 101 x 100 = 10100
To get back to the sum S, we just divide by 2.
S = 10100/2 = 5050."
Q.E.D.
Following on Gauss idea, we can develop further to generalize to sum the numbers from 1 to N (no matter what N is):
1 + 2 + ............... + N = N(N+1)/2
or, to sum from the number M to number N:
M + (M+1) + .......... + (N-1) + N = (N-M+1)(N+M)/2
If we still want to go further, we can research more to sum their squares:
1 + 22 + 32 + ………. + N2
or their cubes,
1 + 23 + 33 + ………. + N3
or, to push to the extreme,
1 + 2m + 3m + ………. + Nm
(no matter what m is).
But these require further mathematical techniques like recursive formula and Bernouilli numbers. I can touch on more as time permits.
