PoPP3- Pythagoras theorem [Ved1]
Hallo! Time for some math. Fundamentals of mathematics are very interesting. This is because of the way these ideas came to life. Take for example the ‘Pythagoras theorem’. When you look at it at first glance, you see only a formula. A formula made up of letters with the number 2 on top of each of them. But, the idea of the theorem is something more deep. Most of the basic mathematics that you will learn in school stems from calculations on a farming field. Pythagoras theorem is no exception. Look at these notes I’ve written below.
Pythagoras theorem originated as a relation between the areas of the squares on the triangle. Take any Pythagorean triple and you will observe this phenomenon of the areas. You could go out on a beach and try drawing squares such that you have a right-angled triangle forming on the inside. Measure the sides of the squares and calculate the areas by hand. You will see that the theorem holds true. But, what you did on the beach is an empirical proof. Theorems in mathematics are not empirical, rather they have a greater meaning. Its called deduction. I can’t explain it in words. You will have to feel it.
The proof of the pythagoras theorem I performed above is a geometrical proof. The proof stems from the fact that a square can be made by joining 2 right angled triangles together.
Far more powerful is the algebraic proof that I wrote above. Now comes the most important part of today’s lesson. Remember when I told you that the Pythagoras theorem started-off as a relation between the areas of the squares? The algebraic proof shows you that the Pythagoras theorem can also be used to define the length of a line between 2 points.
Pythaogoreans were a sect existing during the time of Greek mathematics. To put time into perspective, the Indus valley civilization existed some 2000 years Before Christ was born. The Babylonians also existed some 2000 B.C. Greek civilization was around 700 B.C. Pythagoras lived at around 500 B.C. When the Pythagoreans found that the length of the side opposite the right angle(with 1 and 1 as its shorter sides) cannot be measured accurately, they did some horrible things. Thus, began a new puzzle in mathematics called the Irrational numbers.
How do you prove that a number is irrational?