Alex's Adventures in Numberland - Alex Bellos [117]
Customizing the harmonic series gets more intriguing than this, though. The decision to extract 9s was arbitrary. If I had extracted all the terms containing 8 from the harmonic series, the remaining terms would also converge to a finite number. As it would if I extracted only the terms with a 7, or indeed with any single digit. In fact, we do not even need to limit ourselves to single digits. Remove all terms including any number, and the thinned-out harmonic series is convergent. This works with, say, 9 or 42 or 666 or 314159, and the same reasoning applies.
I’ll use the example of 666. Between 1 and 1000 the number 666 occurs once. Between 1 and 10,000 it occurs 20 times, and between 1 and 100,000 it occurs 300 times. In other words, the percentage occurrence of 666 is 0.1 percent in the first 1000 numbers, 0.2 percent in the first 10,000 and 0.3 percent in the first 100,000. As you consider larger and larger numbers, the string of digits 666 is proportionately more and more common. It will eventually be the case that almost all numbers have a 666 in them. So, almost all terms in the harmonic series will eventually have a 666 in them. Exclude them from the harmonic series and the depleted series converges.
In 2008 Thomas Schmelzer and Robert Baillie calculated that the harmonic series without any term containing the number 314159 adds up to a little over 2.3 million. It is a large number, but a long, long way from infinity.
A corollary of this result is that the harmonic series with only the terms including 314159 must add up to infinity. In other words, the series:
adds up to infinity. Even though it starts with a tiny number and the terms get only tinier, the sum of the terms will eventually surpass any number you want. The reason, again, is because once numbers get very large, almost every number has a 314159 in it. Almost all unit fractions contain a 314159.
Let’s take a look at one last infinite series, one that brings us back to the mysteries of the primes. The prime harmonic series is the series of unit fractions where the denominators are the prime numbers:
The primes get scarcer as numbers get higher, so one might expect that this series doesn’t have the momentum to add up to infinity. Yet, unbelievably, it does. Counter-intuitive and spectacular, the result makes us realize the power and importance of the primes. They can be seen not only as the building blocks of natural numbers, but also as the building blocks of infinity.
CHAPTER EIGHT
Gold Finger
Sitting with me in nunge at home, Eddy Levin handed me a sheet of white paper and asked me to write out my name in capital letters. Levin, who is 75 years old and has a donnish face with grey stubble and a long forehead, used to be a dentist. He lives in East Finchley, north London, on a street that is the epitome of prosperous and conservative suburban Britain. Expensive cars sat in the driveways of between-the-wars brick houses with freshly trimmed hedges and bright green lawns. I took the paper and wrote: ALEX BELLOS.
Levin then picked up a stainless-steel instrument that looked like a small claw, with three prongs. With a steady hand he held it up to the paper and started to analyse my script. He lined up the instrument to the E in my first name with the concentration of a rabbi preparing a circumcision.
‘Pretty good,’ he said.
Levin’s claw is his own invention. The three prongs are positioned in such a way that the tips of the prongs stay on the same line and in the same ratio to each other when the claw opens out. He designed the instrument so that the distance between the middle prong and the prong above it is always 1.618 times the distance between the middle prong and the prong below it. Because this number is better known as the golden mean he calls his tool the Golden Mean Gauge. (Other synonyms for 1.618 include the golden ratio, the divine proportion and , or phi.) Levin put the gauge on my letter E so that the tip of one claw was on the top horizontal bar of the E, the