Cut a piece of wire 10 centimeters long and place a mark every centimeter for a scale. Bend the wire into a perfect circle which has circumference 10 cm. Pick a mark to be the start of our trip. Travel 3 cm along the wire in a counterclockwise direction and connect back to the starting point with a straight segment. Continue to move and connect back, and along the circle, you will have traveled
3, 6, 9, 12, and some more, and then finally 30 cm
where the trip ends because 30 is a multiple of 10 as well as 3, and ten steps gets you to land back at the start. We have just constructed the star pattern (10, 3), one of UB's favorites, pictured above. An attractive star, but not an infinite trip as promised.
The infinite trip is accomplished by taking advantage of two facts: increments of the square root of two (√2) can be marked as a scale on the wire, and
√2 is irrational. Now we repeat the process and create a (10, √2) star starting at point A, the first several segments of which are shown below.
The completed (hah!) star is an infinite trip around this circle. Why? Because if we ever returned to the starting mark, then a whole number n times √2 would equal a whole multiple of 10. In algebrese it would look like
n √2 = 10m
and that implies √2 = 10m/n which makes √2 a rational number, which it isn't.
We're taught in geometry that a circle has an infinite number of points, and so an infinite trip is possible, BUT here's a kicker, this infinite trip will never visit all of the circle's points and we can prove it.
We will show that, beginning at a common point A, a (10, √3) star and a (10, √6) star, though visiting an infinite number of points on the circle, will never arrive at another common point. [Figure below] If they did so, then a whole number of steps (k) times √3 would equal another number of steps (m) times √6. That's impossible because if
k√3 = m√6,
then k/m, a rational, equals √6/√3, but that equals √2, an irrational. QED as they say.
In the figure two star patterns are initiated at point A. Both take an infinite journey within the circle and they never share another point! For both trips the points visited are said to be "countably infinite," since they are in an order – first, second, third, and so on.
Cantor found that an infinite set which appeared to be merely part of another infinite set could count up the same as the "larger" set. He also found that adding onto, or even doubling the items in an infinite set would not change the count. The mathematical establishment of the late nineteenth century was not equipped to deal with these incomprehensibilities. Many reacted scornfully to Cantor's new theories of sets. He took it all too personally and suffered from depression and, eventually, mental breakdown. Rather than gloss over these gritty bits of numerical bad news though, Cantor released his big bombshell: he had found sets which could not be matched with the infinite counting numbers. They were even larger! For example, the number of points in our circle is uncountable.
Infinite Ponderables for the Reader
Ponderable 1. The rationals all can be expressed as infinite decimals. One-half could be written as 0.5000000... and one-third 0.333333.... Let's pretend to make an infinite decimal. We will choose randomly each digit after the decimal point, one at a time, by rolling a ten-faced die, numbered zero through nine. Trust me that such dice exist. What are our chances of randomly creating an infinite decimal which is rational?
Ponderable 2. You may have doubts about whether those endlessly repeating decimals are all fractions (rationals). There is a method for converting repeaters to fractions, but I'll tell you the chocolate story instead. I found this example in Playing with Infinity by Rozsa Peter.
A certain candy manufacturer wraps a coupon inside every Biggie Bar, and they sell for a dollar. Collect ten coupons and get a free Biggie. Can you give the exact value of one Biggie Bar with its coupon? You may say it's worth a dollar because that's what you paid for it. Remember though, that the coupon represents one-tenth of a free bar. This might cause you to think that the package is worth $1.10 because ten coupons are worth a dollar (if you like chocolate). Alas, you are forgetting that one-tenth of another free bar comes, in effect, with one-tenth of yet another coupon, and that each fraction of a coupon represents tiny fractions of coupons to come. Complete redemption comes only with the exact answer – a rational number.
1. It would be impossible to create an infinite rational decimal by generating the digits randomly. A rational must either terminate with an infinite string of zeros, or repeat a pattern of digits ad infinitum. This makes yet another argument that the irrationals is a larger set than the countably infinite rationals.
2. What is a Biggie Bar with its coupon worth? Assume the bar is worth a dollar. The coupon is worth one-tenth a second (free) bar plus a tenth of that bar's coupon, which is one-tenth of one-tenth of a third bar plus one-hundredth of its coupon, which is one-tenth of one-tenth of one-tenth of another free bar, and so on. In dollars and cents the total is $1.1111111..., which is the fraction 10/9.
An easier way to figure this result is to find a kindly storekeeper, buy nine bars for nine dollars, and ask him to give you the tenth bar for free. You can then unwrap all ten bars and hand over the requisite ten coupons. You have 10 bars costing $9, and so each Biggie Bar is worth 1and 1/9 dollars.
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