Unlike the previous posts, my proof is algebra-free.

Just stare at this picture long enough and the rule

will appear to you "stereogram-style".

Arrange 4 identical rectangles, each of dimension a x

b, into a square, like this:

* * * * * * * * * * * * * *

* * *

* * *

* * *

* * * * * * * * * * *

* * * *

* * * *

* * * *

* * * *

* * * * * * * * * * *

* * *

* * *

* * *

* * * * * * * * * * * * * *

(The above picture might get messed up when I send

this, but I think it's fairly obvious what is meant)

If you can't see the "quarter squares rule" in the

picture then these observations might help:

1. Area of large square = (a+b)^2

2. Area of square "hole" in centre = (a-b)^2

3. Area of large square = combined area of the four

rectangles plus area of the square hole at centre

Hence, (a+b)^2 = 4ab + (a-b)^2

Divide through by 4 gives the rule.

RobF

--- peterschreyfogl <

peterschreyfogl@...> wrote:

> Thank's for your reply, Scott.

>

> I see. It is the proof that the Quarter Squares-Rule

> produces the

> result ab.

>

> Do you see also how one can derive the identity by

> transforming the

> Binomial formulas? Maybe it is not correct like it

> is said on

> Wikipedia. Nonetheless, through some logic there are

> ways how one can

> get to the Quarter Squares-Rule. -- The question is

> HOW.

>

> t.y., Michael

>

>

> --- In MentalCalculation@yahoogroups.com, "P Scott

> Horne"

> <patrick.horne@...> wrote:

> >

> > ((a + b)/2)^2 = (a^2 + 2ab + b^2)/4

> >

> > ((a - b)/2)^2 = (a^2 - 2ab + b^2)/4

> >

> > Subtract the latter from the former, and you get

> >

> > (2ab + 2ab)/4,

> >

> > which is

> >

> > ab.

> >

> > QED.

> >

> > P Scott Horne

> >

> >

> >

> >

> > _____

> >

> > From: MentalCalculation@yahoogroups.com

> > [mailto:MentalCalculation@yahoogroups.com] On

> Behalf Of

> peterschreyfogl

> > Sent: 30 janvier 2007 15:03

> > To: MentalCalculation@yahoogroups.com

> > Subject: [Mental Calculation] Deriving the Quarter

> Squares-Rule.

> >

> >

> >

> >

> > Hello!

> >

> > At http://de.wikipedia

> >

>

<

http://de.wikipedia.org/wiki/Kopfrechnen#Rechentricks>

> > .org/wiki/Kopfrechnen#Rechentricks it is stated

> > that one can derive the Quarter Squares-Rule by

> transforming the

> > Binomials 1. (a + b)^2 = a^2 + 2ab + b^2 and 2. (a

> - b)^2 = a^2 -

> 2ab

> > + b^2.

> >

> > Well, I get to the point where: ab = ((a + b)^2 -

> a^2 - b^2)/2, but

> > how do I get to ab = ((a + b)/2)^2 - ((a -

> b)/2)^2?

> >

> > y.s.,

> > Michael (peter.schreyfogl)

> >

> >

> >

> >

> >

> >

> >

> > [Non-text portions of this message have been

> removed]

> >

>

>

>

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