(This is my first essay and I have put a great deal of work in it. It encompasses the ideas ranging from the subjects of maths, physics to chess. To write an essay on definitions was not an easy task. There are a lot of questions which were never answered. I was lucky to have discovered some of the answers myself. In this process, I have learnt that you cannot wholly rely on teachers for all your questions. Sometimes you have to make a leap, take that risk and go on for adventures. The very topic might sound boring to some of you but it is, in some sense, a collection of my good and bad experiences. The two instances illustrated are purely fictional. This essay is more of a conversation and I too need explanations on some topics.
Any grammatical mistake that might have cropped up still represents incompetence on my part. Healthy suggestion about improving my way of writing is what I need the most. Those of you who can clear my fog of ignorance are warmly welcomed. )
On Definitions
“If you understand a mathematical term but cannot express it in words or define it precisely, you are in the dark world of illusion. This illusion of knowledge is a real threat to students. The good students susceptible to this threat can never become better students.”
– Anonymous
An old man enters the classroom. We can hardly call him bald since the remains of hair are still noticeable on some parts of his head. A pair of thick glasses clearly tells us that his head is a deposit box of information. The way he picks up a marker and scribbles on the white board provides us with sufficient hints to guess his personality. What do we call a man like that? Oh yes, a teacher of mathematics.
The students try to focus more on his bad art which gradually turns out to be an equation. The equation runs as follows:
ϴc=S/r
The teacher clearly explains everything regarding the topic and then demonstrates how the equation can be applied in various problems. The students are then put to an immediate test. Krishna, the topper of the class, solves all the problem with little difficulty. Fairly content with himself he says, “Sir, I have solved all the problems.”
“Good! Can you define a radian?”
“It’s a circular measure of an angle.”
“Well that was not a definition, I guess.”
“Why do I need to define it? I know it.”
“Your inability to define it shows your incompetence, whereas your unwillingness shows your ignorance.”
One may be tempted to compare this particular case with the famous scene of the bollywood movie 3 Idiots.
Ranchhod Das Chhanchad, an exceptionally talented student, is asked to define a machine. Instead of giving the textbook answer, he defines a machine as:
“Anything that simplifies a man’s work is a machine.”
The teacher is quite disappointed by such an answer. He, then, asks the same question to Chatur Ramalingam. Chatur comes to his teacher’s rescue and defines a machine in no less than, I guess, 50 words.
There follows a very good scene in which Ranchhod teaches all of us the importance of economy of words.
The reader might jump at the conclusion that definitions do nothing but only come in the way of true understanding of concepts. As I see, the case is just the opposite.
Being unable to grasp the subtle ideas embodied in definitions, teachers often commit a public offence which is then transferred from one student to another unless, of course, there is someone more vigilant.
For instance, once some students were taking lectures on circular motion. The teacher had to explain how a body moving in a circular path can have acceleration despite having a constant speed. The very first sentence of the teacher was horrifying for a student. The teacher (instead of saying constant speed in a circular path) said that a body moving with a constant velocity in a circular path can have acceleration. The student retorted back by saying that a body cannot have uniform velocity in a circular path. The teacher should have realized his mistake but he had his own dignity to defend. He chose the safer way by saying that both the terms were acceptable. No wonder, this mistake then turned into a public mistake.
Of course, the aforementioned case can easily be forgotten. However, there are some cases to be dealt with much more carefulness. Hundreds of students have encountered problems in trigonometry in which it is required to compute the sine of angle 720 . Nobody even bothers to ask why 361◦ has no meaning in plane geometry but 1 million degree does make sense in trigonometry. It may be even more surprising to know that Euclid, the author of Elements , never measured an angle in degrees, grades, radians or any such units. Without measuring an angle like that how could he lay the foundation of all the geometry that we learnt in high schools? The answer is in his definition of an angle. His definition of right angle is:
“If one straight line stands on another straight line, it makes two adjacent angles. If both the angles are equal, either of them is called a right angle.”
The common definition of a right angle is:
“An angle in measurement 90◦ is called a right angle.
So basically, it is the right angle which comes first in history and then the degree measurement. Did it surprise you? Well, try proving two angles to be equal without using a protractor. How would you say that one angle equals the other without using specific measurements like 30◦? If you can do it, you will probably admire the beauty of geometry.
As a chess player I always prefer to teach a beginner the knight’s move by using a board rather than explaining him in words. It is because even a man with average talent can learn it by actually watching other players move. If he tries to consult a chess book then he finds himself in the middle of nowhere. Following are the standard definitions of a knight’s move:
- A knight moves in an L shape. It moves two squares straight and then one square perpendicular to the former ones.
- A knight kept on a square moves to the opposite corner of a 2×3 rectangle.
- A knight kept on a square moves to any one of the closest squares not on its diagonal, rank or file.
The first is a layman’s definition. The second is a refined definition and the third is the most beautiful of all. Why beautiful? Put a queen in the central square of a 5×5 square. Mark off the squares not controlled by a queen. Now remove the queen and put a knight on the same square and observe. You are bound to find the third definition superior to all others!
In physics and often in real life, it is practical and even sensible to drop out definition altogether. Sir Isaac Newton once said,” I won’t define Time. Everybody knows what time is.”
Hardly had he imagined that about two and a half centuries later some man would astound the whole world with his perception of Time. We all know his name: Albert Einstein
He is the one to state that time and spaces are related.
If you are a physics student, try to recall what you understand by a charge. Can you define it? Benjamin Franklin discovered the existence of two opposite types of charges. Did he prove one kind to be positive and the other to be negative? No. He named the two charges positive and negative. The rest is history. Now there’s an interesting point. If he had named the two charges in the reverse way, the world would have been a lot easier. The electrons would have positive charge then. Besides, the definition of electric potential would surely make some sense. Electromagnetism would be so beautiful then. There wouldn’t be any doubt about the sign conventions.
There’s another interesting example. The definition of one metre has been defined three times by the International Bureau of Weights and Measures. Why were the definitions changed? The current definition is: the distance travelled by light in 1÷299792458 second in vacuum. For God’s sake, why to use that complex number for defining one metre when following definition would be a lot simpler:
Distance travelled in 1 millionth of a second
Or,
Distance travelled in 100 thousand of a second
There’s a very good explanation for it but I leave my readers to discover it themselves.
Well, the most irritating definition in physics for me is that of energy. They say that energy is the capacity of doing work. Now tell me how you wish me to have a physical insight about something defined in such an abstract way. Einstein added more to my misery. His famous equation e=mc2 states the relationship between mass and energy. Mass is something having a physical existence and which we can perceive by one or more of our sense organs. Now how can something called mass be related with energy which is capacity of doing work. The word capacity is abstract. How are these things related? I wish someone could answer it for me.
It needs experience and an open-minded attitude to relish the taste of definitions and the concepts hidden within. I don’t want to be dogmatic and hence, would not give my own opinion. I leave my readers to decide whether the bald headed teacher was right or not.
shyamchessplayer 