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!”
File this one under “duh”: I’ve always assumed that the terms ‘rational’ and ‘irrational’ when relating to numbers was an indication of their behavior. 1/4, for instance, is a rational number, because it equates to 0.25 – it behaves in a rational way.
1/3, on the other hand, is 0.333333…, never ending – it behaves in an irrational manner.
Not only is the above NOT true mathematically, that is NOT where the terms come from.
Rational numbers are numbers that can be expressed as the ratio
of two integers – i.e., a fraction:
1/4 = 0.25
1/3 = 0.33333…
98/100 = 0.98
On the other hand, numbers that CANNOT be expressed as a ratio between two integers – for example, π, e, or the square root of 2 – are all irrational numbers:
π = 3.1415926535…
e = 2.7182818…
√2 = 1.4142135…
The Golden Ratio (1.6180339…) is another example of an
irrational number. The difference between the numbers in the Fibonacci
Sequence, to use one example, follow the Golden Ratio as the sequence
increases.
And speaking of cool numbers, if you divide 987654321 by 123456789 on your phone calculator, you may get an answer of 8. I’ve always enjoyed this, as I’ve always considered 8 to be my lucky number, based solely upon a fortune cookie I got when I was a kid. (This is also why I say my favorite color is red – the fortune cookie told me so – when it’s actually blue.)
However, you’re more likely to get an answer like 8.00000007, which seems much more accurate. But still close to 8, right?
But turn your phone calculator on its side to engage the
scientific calculator functions, and the answer becomes 8.000000072900001. Much
more accurate still!
But it doesn’t end there. The full remainder of this operation is another example of digits that repeat to infinity:
So, how many irrational numbers are there? That’s an
interesting question, and involves countable versus uncountable infinity.
We already know that there are an infinite number of
counting (or whole) numbers. “Countable” in this context doesn’t
necessarily mean we can count them, just that it is possible to count
them in theory. We can say n=1, for instance, and then apply the formula n=(n+1)
to generate the series of natural numbers that increment by one.
Real numbers, on the other hand, are uncountable. In fact, there are more real numbers between 0 and 1 then there are natural numbers between 0 and infinity.
Think of it this way: where would you start counting the
real numbers between 0 and 1? You can start with, say, 0.1. Well, there’s at
least one number between 0 and 0.1 – it’s 0.01. And there’s at least one number
between 0 and 0.01 – 0.001. And you can do this literally forever, and still
not determine where to start counting the real numbers between 0 and 1.
Irrational numbers are similar – they are part of the set of
uncountable infinite numbers. So in a sense, even though both the sets of rational
numbers and irrational numbers are infinite, there are more irrational numbers
than there are rational numbers, just as there are more real numbers than there
are natural numbers, even though both sets are infinite.
My brain is now a smoldering heap, so that will be it for today. But tomorrow I have a nifty little thing to share regarding an interesting characteristic of numbers that are a power of 2, so until then – thanks for reading!
I preface this by saying zero original research went into
this other than my interest in the subject and the God-given ability to Google
stuff.
You probably already know that we use a base-10 number
system to count stuff – 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. But there are other number
bases – some of which you already know about, even if you didn’t realize it.
The language of computers, binary, is nothing more than a
base-2 system, where every digit can be represented by zeroes and ones. Thus, when
converting from base-2 to base-10, you’d count 0, 1, 10, 11, 100, 101, 110,
111, 1000, 1001, 1010…and so on. Here is what these base-2 numbers equate to:
0 = 0
1 = 1
10 = 3
11 = 4
100 = 5
101 = 6
110 = 7
and so on, so that (for example):
111001 = 57
(Links are at the bottom of the page if you want to explore this, as well as the other things discussed below)
But there are other bases – an infinite number of bases, actually. The Babylonians used a base-60(!) number system, and Mayans used a base-20 system. Hexadecimal (base-16) systems are relatively common, used in everything from computer programming (as a way to condense binary) to our everyday usage of pounds and ounces.
In fact, you use an octovigesimal system every day. Better known as base-28, this is what our current Gregorian calendar is (loosely) based upon.
So, what if I wanted to see if my name is represented in the digits of π? How would this even be possible? What would be the best way to convert the base-10 digits of π to letters?
Well, we’d want to use base-27 for this. Base-27 is
represented thus:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J,
K, L, M, N, O and P
To convert this to alpha, we’d assign 1=A, 2=B, 3=C and so
on, letting zero represent a space, i.e. 0=””.
Doing this let’s us then convert π in base-27 from 3.3MQ53… to C.CVEZC. And we’re off and running!
(“Why not use base-26?” I hear you thinking. Well, we want to be able to represent each letter of the alphabet with a letter so that A=1, B=2, etc., and we need a 27th place to capture our zero value, so base-27 it is!)
There is a conversion tool online (also linked below) that
allows you to enter your name (or any word, really) to see if it is represented
in this converted base-27 π. My name, STEVE, appears starting with the 8,857,158th
digit of π:
Image courtesy of Dr Mike’s Math Games
Some fun facts about the occurrence of words in (this
version of) π-27:
The first spelled-out number appears at the 4,259th
position: SIX
The first ordinal number appears at the 222,386th position,
and of course it’s FIRST
JEDI appears at position 1,126,698, but the SITH overpowers
them at the earlier position of 804,693.
Within the first 30,000,000 digits of our converted π-27,
the word MATHEMATICS does not appear. Hopefully it will show up in the second
group of thirty million characters…
And where does PI appear? Not surprisingly (given that it’s only two letters long), it pops up for the first time starting at position 18.
So, click on the link below (or right here if you’re impatient) and see where your name first appears!
One caveat, though – this database only includes the first ~30 million converted digits of π, and your chances of finding your name decreases substantially as the length of your name grows longer. If your name is three (or fewer) letters long, you have virtually a 100% chance of finding it. Five letter names (like mine) carry a 56% chance of success, but seven letter names have virtually zero chance of being represented.
Good luck, and let me know in the comments if you’ve found your name in the first thirty million digits of π!
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