### 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 p + q; the other 1. 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 p + q = 1.

The square is split into 4 p\times q rectangles with a small square hole left over. Each of the rectangles has area pq while the area of the square is 1\times 1=1. It is now not hard to surmise the embedded message: 1\ge 4pq. The hole disappears when p = q = \frac{1}{2}. In that case the inequality becomes equality 1=4\cdot \frac{1}{2}\cdot \frac{1}{2}. In all other cases the hole is present, making the inequality strict.

Since p+q=1, we may rewrite the inequality as p+q\ge 4pq which begs a slight modification which is achieved by dividing both sides by pq: \frac{1}{p}+\frac{1}{q}\ge 4. This is the inequality the exercise was intended to convey. The inequality holds, provided p+q=1. 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 (p+\frac{1}{p})+(q+\frac{1}{q})\ge 1+4=5. So the area of the square is at least 25. The square is covered with an overlap by two squares with side p+\frac{1}{p} and two squares with side q+\frac{1}{q}, implying

2(p+\frac{1}{p})^2+2(q+\frac{1}{q})^2\ge 25.

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

2a^2+2b^2=(a+b)^2+(a-b)^2\ge (a+b)^2.

Using this leads to

2(p+\frac{1}{p})^2+2(q+\frac{1}{q})^2\ge 25.

And from here we get an algebraic confirmation of the insight suggested by the diagram 2(p+\frac{1}{p})^2+2(q+\frac{1}{q})^2\ge 25:

2(p+\frac{1}{p})^2+2(q+\frac{1}{q})^2\ge (p+\frac{1}{p}+q\frac{1}{q})^2\ge 5^2= 25.

So we get an inequality to remember:

(p+\frac{1}{p})^2+(q+\frac{1}{q})^2\ge \frac{25}{2},

provided p+q=1.

### 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