HS #79 2022.2.10
Three Mysterious Numbers
When I teach precalculus, I bring rope of various lengths to class and invite students to pair up. One holds an end stationery, and the partner – holding the other end and keeping the rope taught – counts paces around the circumference of the resulting circle, and then across its diameter. When I did it, I counted 50 paces around, and 16 across. The ratio: 50/16 = 3.125 is an approximation of pi. The Greek letter pi represents the ratio of the length of circumference to the length of diameter.
This number, so easily understood, is mysterious. If you try to write it out as a decimal, it never stops and never starts repeating. Is there a place in the decimal expansion where there are 1000 sevens in a row? Most certainly.
Pi shows up other places as well. If you continue the pattern: 1 - 1/3 + 1/5 - 1/7 + 1/9 – 1/11 + . . and multiply the result by 4, you’ll get pi.
Or try this: Snip a piece of paper clip to the length of the distance between the lines of ruled paper. Then randomly spin and drop the clip onto the paper T times, counting the number of times (C) the clip crosses one of the lines. Then pi = 2T/C. How cool is that!
Perhaps the most surprising result involving pi is also the simplest. A 25,000 mile length of string will lay around the equator of the earth. How much longer does the string need to be if the string is raised one foot above the earth along the entire equator? I guessed 10,000 miles. Answer: If r is the length of the radius of the earth in feet, then the equator is 2(pi)r feet. The circumference of the larger circle is 2(pi)(r+1) feet. The difference in lengths is 2(pi) feet – just over 6 feet longer! Blows my mind.
Here’s another mysterious number: If you have a calculator, multiply (1+1/2) by itself twice. Then multiply (1+1/3) by itself three times. Then multiply (1+1/4) by itself four times, etc. What happens to this sequence of numbers? As the sequence continues, the portion inside the parenthesis decreases to 1, but since it is multiplied by itself more times, the resulting number grows larger. So there is tension – a struggle. Some students think the resulting numbers will get closer to 1. Others think they will get arbitrarily large. Turns out that neither happens. The struggle ends in compromise. The numbers get closer and closer to e = 2.718 . . It’s another number that never stops.
The number e is at the heart of the “normal distribution” or “bell shaped curve” which is used extensively in statistics. So, like pi, it is super important in understanding the world.
One other mysterious – beautiful – number: Consider the sequence: 1, 1, 2, 3, 5, 8, 13, 21, . . where each successive number is gained by adding the previous two. This is the famous Fibonacci Sequence. These numbers are found throughout nature – count the petals on a flower or the seeds in an arch of a sunflower, or the knobs in a row of a pineapple and you will often get a Fibonacci number. If you divide a Fibonacci number by its predecessor, the result will be about phi = 1.618 . . Yet another number that goes on forever. This number is called the Golden Ratio. If you construct a rectangle so that the length of the longer side is phi times the length of the shorter side, you get a Golden Rectangle. Remove a square from a Golden Rectangle, and the remainder is another Golden Rectangle. When I ask 5thgraders how many times this can be done, they shout “Infinite!” Their response gives me goosebumps; how interesting that even children imagine infinity. (See Ecclesiastes 3:11.) Golden Rectangles are found throughout art and architecture. da Vinci used them extensively. Google it and see for yourself.
On the other side of the interesting-spectrum of numbers are 0 and 1 - the most basic and ordinary numbers around. Adding 0 doesn’t change a number, and multiplying by 1 doesn’t change it. 1 has a counterpart, i, which is its imaginary cousin. How eerie then that 0, 1, i, e, and pi can all be combined in one simple equation – called Euler’s Equation, perhaps the Holy Grail of Mathematics. Yet one more surprise: “Euler” is pronounced, “oiler.” Pronouncing it correctly will impress any mathematician.
