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#M-5.   Approximations

(M-4) Identities

In some algebraic manipulations entire expressions get multiplied. For instance, one can write

(a + b)c = ac + bc

This is not an equation but an identity, an expression true for any three numbers (a,b,c). For instance if a = 3, b = 7, c = 5, then

(3 + 7)(5) = (3)(5) + (7)(5) = 15 + 35 = 50

If the addition is performed first

(3 + 7)(5) = (10)(5) = 50

Identities do not add any information about the quantities which they contain, because they are true for any values those quantities may take. They are however useful in reshuffling equations to new, cleaner forms. The identity written on top is actually one of the basic properties of numbers ("the distributive law"). From it one gets more generally

(a + b)(c + d) = (a + b)c + (a + b)d

which can be further broken up and which holds for any values of (a,b,c,d). In particular

(a + b)2 = (a + b)(a + b) = (a + b)a + (a + b)b

= a2 + ba + ab + b2

= a2 + 2ab + b2

which is quite useful (you can try it out with some specific values for a and b). Similarly

(a – b) 2 = (a – b)(a – b) = (a – b)(a) + (a – b)( –b)

= a2 – ba – ab + b2

= a2 – 2ab + b2

Again, the two last identities

(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2

hold for any values of a and b, and as will be seen, are very useful in proving Pythagoras' Theorem.

--------------------------
  Another useful identity is obtained by multiplying (a–b) times (a+b). You need remember that the minus sign goes with (–b), because you might just as well have written
(a + (–b))            in place of             (a – b)
So
(a – b) (a + b) = (a + (–b))(a + b )

= a2 +
(–b) a + ab + (–b)b   =   a2 – ba + ab – b2   =   a2 – b2


Rewriting just the beginning and the end
(a – b)(a + b) = a2 – b2


This too will be useful in deriving the theorem of Pythagoras, in a different way.

Finally, two more identities which you may occasionally encounter:

(a – b)(a2 + ab + b2) = a3 – b3

(a + b)(a2 – ab + b2) = a3 + b3
)

Next Stop:   #M-5    Deriving Approximate Results

Author and Curator:   Dr. David P. Stern
     Mail to Dr.Stern:   stargaze("at" symbol)phy6.org .
Last updated 25 November 2001