Sometimes we need to commit a number to short-term memory, such as an street address or parking garage berth. More important numbers (for phones or credit cards, PINs or birthdays) call for long-term memorization although smart phones increasingly fill that gap. But even with a smart phone it is pleasantly useful to have the option to use one’s own brain for such things. To do this, I rely on something I call the Sweet Numerology Keyfob. Before I explain that outlandish phrase, here is some background...
Digital data is hard to memorize because digits are abstract, while our human memory works best on concrete data such as images or meaningful phrases. There are various systems to encode digits in this way, so as to make them “stick” better in the memory. The simplest systems to learn replace each digit by an word, and produce a sentence corresponding to the original sequence of digits. Here is an example where each digit N is represented by a word with N letters:
How I wish I could recollect pi!
3-- 1 4--- 1 5---- 9-------- 2-
But one digit per word is a waste of words. Here’s why, in terms of basic information theory: To encode a random stream of decimal digits, we will need a minimum of about 3.3 bits per digit. (Are the digits of pi really random? Probably, but nobody knows for sure.) Meanwhile, as shown by various statistical studies and human tests, English words carry 6-12 bits of information. This suggests that each word should be able to carry not one but two or three digits of “payload”. In fact, it is possible to do just that. With a small amount of practice, it is easy (and fun, in a nerdy way) to learn how.
(Here is more on the information content of English: Shannon’s pioneering study measured an upper bound of 2.3 bits per letter and speculated on an actual value near 1 bit. Experiments with human prediction have refined the upper bound to 1.2 (). The theory is fascinating. A small study of Latin and Voynichese texts shows those languages exhibit statistics consistent with 9-10 bits per word, similar to Shannon’s upper bound. Note that information content is not constant but depends on factors like style and genre; striking, artificially constructed mnemonic phrases will probably have higher entropy than narrative text.)
For representing digits by words, I use a system I learned from Kenneth Higbee’s wonderful book, Your Memory: How It Works and How to Improve It. I call it the Sweet Numerology Keyfob, but Higbee calls it the Phonetic System; it is also referred to (for no apparent reason) as the Major System. The technique is about 300 years old.
The idea is to make sounds (not letters or words) correspond to digits. In order to give just the right amount of creative freedom, vowel sounds are ignored; only consonants are significant. The following words all have the same sequence of consonant sounds, and therefore represent the same number: meteor, meter, metro, mitre, motor, mutter, amateur, emitter. The consonants are MTR, and the number happens to be 314. Because this code is perceived phonetically, spellings with a double T still contribute a single digit, because to the ear there is only one T sound. Depending on how you pronounce it, a word like hattrick or rattail probably contributes a pair of 1 digits, one for each T. My point is that your ear, not your eye, will be the judge.
Here is a table relating the digits to some of the consonants:
|0||S||sea, see, sigh, so, sue, ass, ice|
|1||T||tea, tee, tie, toe, to, at, oat|
|2||N||neigh, knee, nigh, no, new, an, in|
|3||M||may, me, my, mow, moo, am, ohm|
|4||R||ray, raw, awry, row, rue, are, oar|
|5||L||lay, lee, ally, low, Louie, ale, Ollie|
|6||SH||she, shy, show, shoe, ash, ashy|
|7||K||key, Kay, echo, coo, icky, auk|
|8||F||fee, fie, foe, foo, if, off|
|9||P||pea, pie, Poe, ape|
|(none)||AEIOU||ah, eh?, I, oh!, you|
As you can see from these examples, silent letters are ignored, like K in knee and GH in night.
Note that the letter combination SH is a single sound, and is used to encode a digit (6).
The point of giving so many examples in the right-hand column is to suggest the range of choices available for each digit. As with the “number of letters per word” code used for pi above, for any given number there will be many words to choose from when deciding on a mnemonic phrase. Longer words with more consonants can be used to encode multiple digits, giving even more choices.
Using phonetics instead of letters makes it possible to use the system without attending to the written form of a word. In practice, this makes encoding and decoding easier and more reliable.
There are only a few other rules to know. First, three sounds are ignored even though they can be classified as consonants. They are W, H, and Y. (You may well ask, “why”?) Thus, the phrase happy hippo whip counts as 999. What about the other consonants? They are grouped with the ones already mentioned above, according to similarity of sound. Any consonant which vibrates the vocal cords is treated like the corresponding consonant which does not, so a hard G goes with its fraternal twin K, etc. Also, the various sounds spelled TH are grouped with T, which differs from the treatment of SH and CH.
(Technically, a voiced consonant is treated as if it were unvoiced, and a fricative consonant is treated as if it were a plain stop, if it does not already have a role in the encoding.)
Here is an updated table. It is helpful to read the examples aloud. (But don’t let your family overhear.)
|0||S/Z||sue, zoo, ease, wheeze|
|1||T/D/TH||two, due, the, aid, weighed, mayday|
|2||N||no, enough, whine, anyway, Hawaiian|
|3||M||mow, yam, home, Omaha|
|4||R||row, war(*), airway(*)|
|5||L||low, wool, hollow|
|6||SH/CH/ZH/J/(G)||show, Joe, chew, huge, awash|
|7||K/G(hard)||coat, goat, wacky, wig|
|8||F/V||few, view, half, heavy|
|9||P/B||pie, buy, hope, oboe|
|(none)||AEIOU||ah, eh?, I, oh!, you|
|(none)||WHY||why, how, highway|
Although the tables above may seem daunting, their core information is simple and easy to learn, if you use your ear instead of your eye. There are many ways to remember the basic sequence of ten letters, but I like to tie it up in a phrase which both describes and encodes the technique:
SweeT Nu-MeR-oLoGy Key-FoB
0---1 2--3-4--5-6- 7---8-9
Now let’s use this system to encode pi, using about the same size phrase as above:
How I wish I could recollect pi!
3-- 1 4--- 1 5---- 9-------- 2-
Moderately pinch Lemuel phobic.
3-1-4-1-5- 9-26- 5-3--5 8--9-7
Meteorite lupine, chew a lame wolfback.
3-1--4-1- 5-9-2- 6--- - 5-3- --589-7-
Motherhood, help any shallow male phobic.
3-1--4---1 5-9- -2- 6--5--- 3-5- 8--9-7.
My tired lip now shall my laugh pique.
3- 1-4-1 5-9 2-- 6--5- 3- 5--8- 9-7--
It can be difficult to get “home run” matches like moderately for 31415, but with a little practice, phrases which combine only two or three digits per word are easy to build quickly without mechanical assistance. Memorizing a few digits using short words is the most common way I use this technique. Less commonly, I work hard to get a good vivid encoding for a number I care about, such as a family birthday or phone number.
You will notice that there is no clever tie-in to pi itself in these phrases. This is a natural effect of the high compression level of the encoding. There is not much “slack” for an author to choose words that are relevant to the subject. Although this might seem to be a defect of the phonetic system, it is easy to compensate for, by mentally adding a tie-in. If you are imagining Lemuel getting pinched, give him a pie in the face also, and you are done. This extra mental step is more than compensated by the doubled efficiency of the encoding itself.
Once I got started it was hard to stop. Eventually I got the following quasi-story which encodes the first 51 digits of pi in 20 words:
Immoderately, punchily, my wolfpack bowmen may free Ginger,
--3-1-4-1-5- 9-26--5- 3--5 8--9-7 9--3-2 3-- 84-- 6-26-4
maim via ammo hangable, sniff heavy rutabaga;
3--3 8-- -3-- --27-95- 02-8- ---8- 4-1-9-7-
the sheep imbibe my accolades.
1-- 6---9 -39-9- 3- -7--5-1-0
That is enough pi for now. The occasion of this post (which is getting long) was a different and much simpler constant. It is used to define the Kelvin and Celsius scales with reference to the triple point of water. The magic number is 273.16, which is (by definition, and exactly) the Kelvin-scale temperature of water in its triple point. This number, along with its close neighbor 273.15, defines both the Kelvin and Celsius scales. In particular, the temperature we call “absolute zero” is -273.15 Celsius. Because the freezing point of water varies slightly depending on pressure, the Kelvin and Celsius scales are defined relative to the triple point, which is very slightly warmer than normal freezing. This hundredth of a degree only makes a difference to scientists, but it is nice to know.
Although I find that number number 273.16 interesting, it tends to slip my mind. Today I made a mnemonic to nail it in there:
And there is a bonus here, a convenient coincidence. (Not quite a mathematical coincidence.) The digits 273 also present the temperature of the cosmos to three significant figures. More specifically, the temperature of the cosmic microwave background radiation has been measured as 2.726 degrees Kelvin. Rounding gives 2.73K, or 1% of the freezing point of water (273K).
The actual ratio is not 1/100 but rather more like 10/1002. Still, it is interesting (to me at least) to note that the cold of space is two orders of magnitude colder than a cold day in most parts of the Earth. It is also interesting to think that this ratio (10/1002) is probably the same everywhere in cosmos. It is slightly more impressive, however, to people who use base 10.
Although I expect to use 273 for both purposes, the more precise value of the CMBR is of course worth remembering. It is, after all, one of the most important physical discoveries of late twentieth century. The CMBR is (to me) like the sound of the sea, unvarying across aeons. Folklore says (erroneously and/or poetically) that you can hear the sea in a shell, for example, in a conch shell. I suggest that some part of the CMBR can also be found there:
In a conch.
-2 - 7-26-