CTK Insights

04 Jun

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

  1. C. Alsina and R. B. Nelsen, Charming Proofs: A Journey Into Elegant Mathematics, MAA, 2010
  2. C. Alsina and R. B. Nelsen, Proof Without Words: Inequalities for Two Numbers Whose Sum Is One, Math. Mag. 84 (2011) 228
  3. R. B. Nelson, Proofs Without Words, MAA, 1993
  4. R. B. Nelson, Proofs Without Words II, MAA, 2000

2 Responses to “PWW: How Geometry Helps Algebra”

  1. 1
    Weekly Picks « Mathblogging.org — the Blog Says:

    [...] Come Saturday The Geomblog gave advice on applying for academic jobs this fall (in the US), A Neighborhood of Infinity takes a Quora question on diffraction to a whole new level and CTK Insights studies proofs without words. [...]

  2. 2
    Neon Rider Enthusiast Says:

    Proofs 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!

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