__Calculating e__

The "magic" number e is fundamentally defined by the formula:

which means the same as:

In other words, as m approaches infinity, the result approaches e. The value of e can be calculated from this formula by a computer program like for pi, but convergence is very slow. We can also write:

Now if we expand

by the Binomial Theorem and take n to the limit we get:

which is the series expansion for e^{x}. If x is made equal to 1, the series will give the value of e. The series converges rapidly, so not too many terms are needed for reasonable accuracy.

__The Moebius Strip__

A strip of paper bent into a circle and stapled has 2 surfaces, the outside and the inside. Such a hoop has 2 edges also, with a measurable distance between them. When one end of the strip is twisted a half turn before stapling, it is called a moebius strip. How many surfaces does it have? How many edges? The books say one each. Some people say two each like the hoop. Clearly there is still a measurable width as before. Is this not the distance between 2 edges? One can look at a section of the paper and turn it over to look on the 'other' side. If it has 2 sides, then it has 2 surfaces they say. Logic demands it.

You can draw a center line around the outside of the hoop until it joins itself at the starting point. The inside of the hoop has no line. If you do this with the moebius strip, the line ends up on 'both' sides of any section of the paper. This proves that it has only one surface, some say. The same thing happens if you color 'one' edge. 'Both' edges get the color. It must be the same edge, but how can there be a distance between 1 edge? The human mind is certainly twisted along with the paper!

Maybe mathematics can solve the debate. When you cut the hoop on the center line, 2 pieces result. These 2 pieces have 4 surfaces. With the moebius strip you get only 1 piece. The piece that you get has 2 full twists! The piece has 2 surfaces. Let's take this to stage two. We cut each of the two hoop pieces down the center as before. Now we have 4 pieces with 8 surfaces. After cutting the moebius piece we find that we have 2 distinct pieces that are entangled. Each piece has 1 full twist, so there are 4 surfaces. When the pieces are laid down on a table, they can be arranged into two figure 8's that are woven like a lawn chair.

Now let's write some formulas for the hoop, and apply them to the moebius strip. Let the number of surfaces before the first cut be designated Sb. Let the number of surfaces after the cut be designated Sa. Let the number of pieces after the cut be designated Pa. For stage one we have the two formulas: Sb = Pa, and Sb = Sa/2. For stage two we have the two formulas: Sb = Pa/2, and Sb = Sa/4. Applying these formulas to the moebius strip, we have:

Sb = Pa = 1

Sb = Sa/2 = 2/2 = 1

Sb = Pa/2 = 2/2 = 1

Sb = Sa/4 = 4/4 = 1

All four formulas agree that the moebius strip had 1 surface before the cut! Algebra has come to the rescue.

1## Re: Number Theory - Calculating e and The Moebius Strip