Identities, Transformations & Cylic Order____ A Knol.

______Identities,Transformations & Cyclic order_____________

This is a very useful information for the school students studying algebra at the lower school level.This is my personal research based topic and I would be happy to receive any useful feedback for the information to be upgraded.

Bhatia, Manohar. Identities,Transformations & Cyclic order [Internet]. Version 2. Knol. 2009 Jan 2. Available from: http://knol.google.com/k/manohar-bhatia/identities-transformations-cyclic-order/1g86tdy1pj9kg/5.CitationEmailPrintFavoriteCollect this page
_______ Conceptualized & written________
By
Manohar Bhatia
Contents:__________
This is a very important chapter for school students learning algebra.I found many students getting unnecessarily confused solving problems on cyclic expressions.I have discovered a very simple and effective way to tackle such problems.

This involves factorisations of cyclic expressions.Here,it is very important for the student to understand the important term 'cyclic'This word 'cyclic' is derived from the word 'cycle' or 'bicycle' which we ride daily.A bicycle is a 2-wheeled mechanical vehicle,where a person sits on a seat provided and paddles the vehicle with his 2 feet.The perfect synchronised movement that is obtained by paddling allows the bicycle to move forward without toppling.The 2 wheels of the bicycle are moving round and round and this is called 'cyclic movement'

Now, this same principle is involved in solving the expressions of identities and transformations that are cyclic in nature.An identity is an algebraic expression or a statement which is true for all values of letters involved in it.For example, take an expression.........a raised to power 3 + b raised to power 3 = (a+b)(a square - ab + b square) If you put any values in a and b, you will observe that Left Hand Side= Right Hand Side.Therefore, an identity always asserts that expressions are always equal,provided if the law is applied properly to it. And....................
WHAT IS THE CYCLIC LAW?
Imagine a wheel of a bicycle going round and round.Now, put 3 points, a,b,c, on the wheel at a short distance from each other and watch the wheel.You will observe that a is followed by b, b by c, and c by a.This pattern can never be changed because of the movement of the wheel.
Now, multiply a by b, b by c and c by a and you get an expression...........
ab + bc + ca...................(1)
Now, interchange the alphabates and you get...................
bc + ca + ab...................(2)
From the above expressions, we can comclude that (1) = (2)

Therefore,
ab + bc + ca = bc + ca + ab
Such expressions are called identities that are cyclic in order,because these expressions follow the usuual CYCLIC LAW and ordered movements of a bicycle involving points a,b,c.
The 4 operations of a cyclic movement are.......................
(1) Addition .................... (a + b), (b + c), (c + a)
(2) Subtraction....................(a - b), (b - c), (c - a)
(3) Multiplication..................ab, bc, ca
(4) Division......................... a/b, b/c, c/a

However complicated your problem may be, you can easily solve these problems by applying the above cyclic law to these expressions. Just remember the order:------ (a+b), (b+c), (c+a) or (a-b),(b-c),(c-a) or ab, bc, ca or a/b, b/a, c/a.
Always remember to bring the complicated espressions to the above forms by making minor adjustements of plus,minus,multiply or division signs and then proceed.Just as a movement of a bicyle cannot change, so also these identities will always be true on both left hand and right hand sides of an expression.
Manohar Bhatia.
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Manohar Bhatia
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