Steiner's Problem

(thing) by futurebird (2.3 y) (print) ? (I like it!) 2 C!s Sat Mar 18 2006 at 4:20:24

The square root of a number, say sqrt(36) is the number that when multiplied with itself yields 36.

6*6 = 36 or
6² = 36

so

sqrt(36) = 6

But, most natural numbers do not have natural numbers as their square roots. For example the square root of 2 starts out 1.414213... and it just keeps going forever without any repeating pattern. It is an irrational number. Most square roots turn out to be irrational numbers. If you multiply 1.414213*1.414213 = 1.99999841 not exactly 2. But it is close enough for most practical purposes, such as creating a square garden with an area of two acres. Since square roots involve finding the product of two of the same number, we could also call them "2^nd roots." There are also 3^rd roots ( called cube roots) 4^th roots and 5^th roots and 6^th roots ... etc.

The 4^th root of 2 is about 1.1892

1.1892 * 1.1892 * 1.1892 * 1.1892 = 1.1892⁴ = 1.99995214 or almost 2

Just like square roots, cube roots, 4^th roots, 5^th root etc. are also mostly irrational. That is, we can't write the majority of them as a ratio of natural numbers. We can't even write them in their entirety in our base ten decimal system either since the decimal system is just short hand for a type of fraction (or ratio) composed of natural numbers where the denominator is a power of ten. Irrational numbers are not that strange-- most of the real numbers are irrational. This leads us to irrational roots... You see, not only do we have 5^th root and 6^th roots, but we also have 2.5^th roots and even 0.5 roots. You can even take root using an irrational number like pi or the square root of 2.

Consider the pi root of 5, that is a number that when multiplied with itself pi times (or a little over three times) equals 5. The pi root of 5 is about 1.66915. Now here is the big question: What happens when you take the x ^th root of x when x is any real number?

Let's see (all values are approximate)
square root of 2 is about 1.41421356
cube root of 3 is about 1.44224957
4^th root of 4 is about 1.41421356
5^th root of 5 is about 1.37972966
100^th root of 100 is about 1.04712855

If you plot this as a function ( f(x) = x^(1/x) remember x^(1/n) = the nth root of x ) you will see that it rises up rapidly at first then between 2 and 3 it starts to fall getting closer and closer to 1 for high values of x. So what is the maximum value of this function? And what value of x will maximize the function?

The value of x that will maximize the function is e

the e^th root of e is about 1.4446678

And that is as good as it gets. All other roots of the form "the xth root of x" are smaller. The problem of maximizing the xth root of x is known as Jakob Steiner's (He was largely self-taught and was professor of geometry at the Univ. of Berlin from 1834.) problem. I didn't solve it on my own. This just happens to be one of my favorite definitions of e.

printable version
chaos

Jakob Steiner

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