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:
- 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