Among the new things I’ve been trying – podcasts, mathematics, computer programming, etc. – is trying to get a little bit better on the piano. I’ve set the guitar aside (for the most part) and focused more on piano-based songs.
For someone that doesn’t read music, this can be somewhat of a challenge. I play guitar chords on the piano. That is to say, when I play a “D” chord on the guitar, I know what those notes are, and I can then play the corresponding notes on the piano.
But to make a song sound like the actual song, it helps to be able to play at least a part of the actual song, be it a particular rhythmic part, or the introduction, or some distinctive part of a given song.
What follows are the intros to eleven songs and the “guts” of a twelfth. How many of these twelve songs do you recognize?
There’s a bit of an eclectic mix here:
-Three songs from the 70’s. Six from the 80’s. Three from the 2000’s.
-Three artists are represented twice – two are bands who had their hits in the 80’s, one band who is pretty big right now.
-Only one of the artists is not longer with us – the remainder are still making music, albeit some with different lineups.
I also mention the Parker Square at the beginning of the video. Matt Parker, British mathematician/comedian, discussed his aptly named Parker Square in a video for Numberphile a number of years ago, characterizing it as the “mascot of giving it a go” – in other words, don’t be afraid to try something new.
So, give this a go – how many of the songs can you recognize and name?
This is the story of my greatest mathematical “discovery”
to-date.
In his entertaining video on leap years (link below),
Numberphile contributor Matt Parker mentions towards the end of the video that
the next “power year” will be in 2048.
That is to say, the next year that is a power of 2 will be
2048: 211 = 2048. The last
one was the year 1024 (210) and the next one won’t be until way in
the future – the year 4096 (212), to be exact.
A little quick math revealed that I will turn 84 in April of
2048 (assuming I’m still around to annoy people), and I thought it would be a
mildly interesting blog post if I could associate 84 and 2048 somehow,
preferably mathematically.
I spent about 15 minutes tackling this from different
angles. They’re both divisible by 2, and their greatest common factor is (only)
4, which is surprising considering they both have 12 factors (including themselves):
Then I remembered Matt’s video on compatible pairs (also
linked below), and I applied that system to these two numbers. Nothing special
about 84, other than it is an abundant number, which is a number whose proper divisors
(there’s another term for this, which I didn’t know about, and which we’ll get
to shortly) sum to a number greater than itself. In this case, 84’s abundant
(or excessive) number is 140 (1+2+3+4+6+7+12+14+21+28+42).
2048 was a different story. It immediately struck me that
the sum of the proper divisors of 2048 came out to be 2047 – one less than the
number itself. “How odd!”, I thought to myself, followed by “I
wonder how prevalent this is?”.
I quickly threw together a spreadsheet listing the first 100
natural numbers along, with all of their factors. After the quick calculation
of adding up all factors except the last one (the number itself), a pattern
emerged. This appeared to be happening with every power of 2!
(At this point, I have to say that, had I been paying
attention, what was happening would have been readily apparent. My number
manipulation also revealed another pattern, one where the sum of the proper
divisors of a number equaled the number itself. There are two numbers that this
happens with in the set of natural numbers to 100…6 and 28. It honestly
didn’t register with me what was happening…)
Here is the pattern that I discovered:
Sure enough, the powers of 2 less than 100, i.e. 20 through 26, followed this pattern, and they were the only numbers that were doing so! I did the same calculation for 27 through 212, and:
…the pattern held! I was super excited, thinking I’d found
some hitherto unknown, or at least unexplored, feature of the powers of 2.
So excited, in fact, that I sent Matt Parker and email about
it, detailing my “discovery” much as I’ve done above.
At this point, I closed everything down and went about my
day. I went to listen to a friend play guitar at a consignment market, grabbed
something to eat, washed my car. Went home and began surfing the interwebs a
little bit.
That’s when I came across the Wikipedia page for Powers of 2
(yes, there is one, unsurprisingly – I’ve linked it below as well).
In short order, I found that:
-Summing the proper divisors has a technical name, the
aliquot sum. I couldn’t find a proper etymology of the term, but this function
has been known for thousands of years. Well, of course it has.
-What I “discovered” is known as the set of
“almost perfect numbers” – almost perfect because their aliquot sum
is one less than the number itself.
Which brings us to 6 and 28, who’s aliquot sum equals the
number itself. As many videos I’ve watched regarding Fermat and Mersenne prime
numbers, I should have recognized that these are the first two perfect numbers,
so named because of the very quality I “discovered” in my spreadsheet
above.
After sending Matt a follow up “never mind!”
email, and texting everyone else that I was, in fact, not about to become
math-famous, I shut down my computer for the day and enjoyed the unseasonable
beautiful weather we were experiencing.
I still haven’t found a blog-worthy, interesting connection between
84 and 2048 – I’m sure they’re out there, I just need to dig a little deeper.
Like I should have done initially when I made this amazing
“discovery!”
Thoughts 