User:Kishu29196

Introduction
For most of people, visual memory is more powerful than linear memory of steps in a proof.The various relationships embedded on a good diagram represent real mathematics awaiting recognition and verbalization. To help students learn and remember mathematics, proofs without words are often more accurate than proofs with words. Visual proofs enjoy a long history, going from some of the earliest recorded mathematical writings through to the proofs without words column that first appeared in Mathematics Magazine in The College Mathematics Journal. Proofs without words have enjoyed broad appreciation as a pedagogical devices and recreational puzzles.

Definition
In Mathematics, a Proof Without Words(PWW) is a proof of an identity or mathematical statement which can be demonstrated as self evident by a diagram without any accompanying explanation.

Characteristics
These proofs being more self evident in nature, they are considered to be more elegant than formal and mathematically rigorous proofs. When the diagram demonstrates a particular case of general statement, to be a proof, it must be generalizable. Every proof of a Mathematical Statement can not be given by a diagram but even if a PWW struggles to express clear statements of consequence and inferential ordering, it can illustrate the key idea or main insight that a more completely written up proof could exploit.

There are many examples for proof without words for various mathematical statements. Among them few of them are really very rigorous and helps students to understand in a very easy manner. Among such examples few of them are discussed below.

1. Sum of odd numbers
The statement that the sum of all positive odd numbers up to ( 2n-1 ) is a perfect square, more specifically , the perfect square n2 which can be demonstrated by a proof without words as shown in the below figure.



The first square is formed by 1 block; 1 is the first square. The next strip, made of white squares, shows how adding 3 more blocks makes another square :four. The next strip, made of black squares, shows how adding 5 more blocks makes next square. This process continues which can be generalized to attain the required result.

2. Pythagoras Theorem
Pythagoras theorem plays an important role in many fields of mathematics. For example it is a basis for an important branch of Mathematics which is trigonometry and in its arithmetic form it connects Geometry and Algebra. The statement of this theorem is as follows

"In a Right angled triangle,the square of the hypotenuse is equal to the sum of squares of other two sides."

If the hypotenuse is 'c', adjacent side is 'a' and opposite side is 'b' , then by the Pythagoras theorem they are related as   c2 = a2 + b2



Both the figures are having equal areas ie (a+b)2. Now in the figure one its clear that the area of a square having side 'c' is given by

C2 = (a+b)2  - 4 (1/2) ab = a2 + b2 + 2ab – 2ab = a2 + b2

Therefore its clear from the figure that  c2 = a2 + b2

Though it may not give exact proof, by the diagram and some fundamental identities , the proof will be understood and visualized.

Thus proof without words, though its not in more of practical use and not considerable much for the point of examination view. There play a great role in understanding concepts and visualize them.