### PWW: How Geometry Helps Algebra

*Proofs Without Words* is a great educational device that helps students understand and teachers convey mathematical facts. Professor Roger Nelson of Lewis & Clark College has a special knack for the PWW; a rare issue of Mathematics Magazine comes out without one if his creations. The latest (June 2011) is no exception.

What do you see?

The diagram looks like a square, with opposite side marked: one ; the other . We realize that the intention is to convey a relationship between two numbers (no doubt positive, as both designate segment lengths.) The relationship at hand is naturally .

The square is split into 4 rectangles with a small square hole left over. Each of the rectangles has area while the area of the square is . It is now not hard to surmise the embedded message: . The hole disappears when . In that case the inequality becomes equality . In all other cases the hole is present, making the inequality strict.

Since , we may rewrite the inequality as which begs a slight modification which is achieved by dividing both sides by : . This is the inequality the exercise was intended to convey. The inequality holds, provided . There is another inequality that holds under the same condition:

What does this one tell you?

There is again a square, this time with the side . So the area of the square is at least . The square is covered with an overlap by two squares with side and two squares with side , implying

.

This is almost immediate. Next, the appearance of double squares on the left may remind you of a simple algebraic identity (and subsequent inequality)

.

Using this leads to

.

And from here we get an algebraic confirmation of the insight suggested by the diagram :

So we get an inequality to remember:

,

provided .

### References

- C. Alsina and R. B. Nelsen, Charming Proofs: A Journey Into Elegant Mathematics, MAA, 2010
- C. Alsina and R. B. Nelsen,
__Proof Without Words: Inequalities for Two Numbers Whose Sum Is One__,*Math. Mag.***84**(2011) 228 - R. B. Nelson,
*Proofs Without Words*, MAA, 1993 - R. B. Nelson,
*Proofs Without Words II*, MAA, 2000

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June 7th, 2011 at 10:06 pmProofs Without Words is a great educational device indeed. I followed your recommendation and had a look at it. Can already see how that shall help. Appreciate your input, this is a great site!

August 16th, 2011 at 6:44 am