Counting on one’s fingers and thumb (or digits) is a convenient way to physically keep track of numbers – and most everyone can do it; at least up to 5. It seems that the way we are most used to in Canada, is to unfurl our digits, one at a time, until all are unfurled. Each digit represents one number – totaling 5. This was the only method I knew of before I taught English in Korea. There, I discovered that it was possible to count to 10 on one hand. Their method is to start with digits unfurled, retract them one-by-one until all are retracted (at which point they have counted to “Dah-Seoht”, if not “Oh”, or 5). Then they unfurl their digits one-by-one, until all are unfurled again. They have then counted to “Yeohl”, if not “Sheehb”, or “10”. (They have two different sets of terms for the various numbers).

    I have also, since then, discovered a slightly different method for counting to 10 on one hand, which the Chinese use. Their system is to unfurl their fingers, like us, until they have counted to 5 (“Wu”). At this point, they form five distinct hand gestures, signifying 6 (“Liu”) through 10 (“Shi”), respectively.

    Of course, there is always the possibility of using both hands, as well. If we can count to “5” on one hand, we can certainly count to “5+5 = 10” on two. Furthermore, being able to count to “10” on one hand, like the Koreans, Chinese, and etc. actually makes it convenient to count to “100” (or even 110) on two. You can do this by using your second-hand (e.g. left hand) digits to signify “10, 20, 30 … 100”. (I know, you may be asking what the point of all this is. Well, as I have written before, “Knowledge is power!” – so…: prepare to be empowered!)

    Recently, I did some serious thinking about what other possibilities there were. What I discovered was that the surface of the “counting potential” had only been scratched! What do I mean? Well, let’s start by extending all of our digits (i.e. fingers and thumb) on our first hand (e.g. right hand). Let’s call this “00000” (one zero for each digit), equaling 0. Now, just so we know where we’re going, pull all your digits in to make a “fist”. Let’s call that “11111”. What does it equal? Let’s wait and see!... So, an extended digit means “0”, and pulled in, or retracted, means binary “1” - Ready?

    Now, with all fingers extended, pull your thumb in against your palm, and call it: “1”. Now, extend your thumb, and retract your index finger. Let’s say that’s “2”. Well, if your index finger retracted equals “2”, and your thumb retracted equals “1”, why not put them together! Pull your thumb in and use it to help hold down your index finger and you have “2 + 1” which equals “3”!

    (By the way, you may think it makes more sense to assume that an extended finger signifies a “1”, and, retracted, “0” – if you do, I used to be a part of that club (otherwise, I’d invite you to join it)! When, however, I discovered the technique I am describing, I changed my mind. Why? Because I find this technique is actually a lot easier and intuitive to use.) If, however, you can find a better technique - “Good on ya!” (Australian expression)

    Well, if we can count to “3” on two digits, I’ll be jiggered if we can’t count to “7” on three! Extend your thumb and index finger, and retract the middle finger. Let’s call that “4” (by the way, I find that “scrunching” a lone retracted finger against the palm of my hand, when my thumb is not retracted to help with the job, works very well). . Now, retract your thumb and use it to help hold down your “4”, or middle, finger. “4+1” equals “5”. Extend your thumb and retract your index finger (of course, the “4”, or middle, finger next to it should still be retracted). What’s “4+2”? You’ve got it! “6”! Last but not least, retract your thumb, use it to help hold down your index finger again, and you’ve got yourself “4+2+1 = 7”! Are we on a roll, or are we on a roll?

    Alright, its time for: “Count to 15 on four fingers”! All fingers extended except the ring finger (second from little) – retract that one. Say hello to “8”. Keep ring finger retracted and retract thumb (using it to help hold down ring finger). “8+1 = 9”! Extend thumb and retract index (together with ring finger, still retracted). “8+2 = 10”. Pull thumb in again (and use it to hold down index, so that the ring and index fingers, and thumb, are retracted). “8+2+1 = 11”. Now, extend the thumb and index fingers, and retract the middle finger (along with ring finger, still retracted). “8+4 = 12”. Retract thumb (and use it to help hold down middle finger). “8+4+1 = 13”. Extend thumb, and retract index finger (along with middle and ring, still retracted). “8+4+2 = 14”. And, to top it all off, retract your thumb, and place it on your retracted index finger. When you do, your thumb and all of your fingers, except the little finger, should be retracted. “8+4+2+1 = 15”.

    Well, anyone that can count to 15 on four digits can count to 31 on five digits, right? That’s right. In fact, let’s just extend all digits, retract our little finger, call that “16”, and ride the gravy train… With our little finger retracted through all of the following steps!…

    Pull in your thumb, using it to help hold down your little finger: “16+1 = 17”. Extend your thumb and retract your index finger. “16+2 = 18”. Pull in your thumb and place it on your retracted index finger and you have “16+2+1” which equals “19”!

    Extend your thumb and index finger, and retract the middle finger. Let’s call that “16+4=20”. Now, pull your thumb in again (using it to help keep your middle finger retracted). “16+4+1” equals “21”. Extend your thumb and retract your index finger (of course, the middle finger next to it should still be retracted). What’s “16+4+2”. You’ve got it again! “22”! Retract your thumb and place it on retracted index finger, and now you’ve got “16+4+2+1”, i.e., “23”! (your ring finger should be the only extended digit at this point). OK. If we were on a “roll” before, now we must be on a “‘rrrrrrrrrroll’ (“rolling” tongue while pronouncing the word)!”

    Extend thumb, index and middle fingers, and retract ring finger (and little finger, which, or course, should have been retracted the whole time since we passed “16”). This time around, say hello to “16+8=24”. Keep little and ring fingers retracted and pull thumb in (using it to help hold down ring finger). “16+8+1=25”. Extend thumb and retract index (together with little and ring finger). “16+8+2=26” (please point your hand down at this point – just to make sure no-one gets the wrong idea from your sole, extended middle finger (or, at the very least, curl it back so that it is only partially extended)!) Pull thumb in again (holding down index, so that it, together with little and ring fingers, are retracted) “16+8+2+1=27” (now the middle finger alone is extended and the potential for misunderstanding is even greater, so, again, please point your hand down (or curl your middle finger back part of the way)!) Now, extend the thumb and index fingers, and retract the middle finger (of course, with the little and ring fingers still retracted). “16+8+4=28”. Retract thumb and use it to hold down middle finger. “16+8+4+1=29”. Extend thumb, and retract index finger (all four fingers should now be retracted). “16+8+4+2=30”. And to finish off, pull in your thumb and place it on your retracted index finger. When you do, all of your fingers (and thumb) should be retracted into a fist: “16+8+4+2+1=31”.

    Congratulations! 31 on one hand! Perhaps you thought you could only count to 5! If so, I hope you realize there’s always more than meets the eye. Do you realize you just did what a computer does! Give it one “on/off” switch, and it can count to “1”. Give it two switches, and it can count to “11” or, to us (now that we can count this high on two fingers), two retracted digits, or “2+1=3”. Give it five switches, and it can count to “11111” (to us, all retracted digits), or “16+8+4+2+1”=31!" Here, take a look at the following spreadsheet, showing the parallel between binary numbers and what we just did on our fingers.

    Note the parallel between the binary representation of each number, and the graphic representation of retracted or extended fingers. For example, for “7”, the binary number is “111”, or a retracted middle finger, index finger, and thumb. For eight, the binary number becomes, “1000”, or a retracted ring finger, and an extended middle finger, index finger, and thumb. For “16”, the binary number is “10000”, or a retracted little finger, with all other digits extended. And for 31, the binary number is “11111” - or all digits retracted. Finally, we have “32”, or “1 00000”, or all digits extended on first hand (for example, your right hand, unless you are left handed), and your thumb retracted on your second hand (e.g. left hand).

View Table #1

    Now, earlier, I alluded to the fact that one hand is not all we have – and that, therefore, we are not limited to what we can count to on one hand. Friends, the half has definitely not been told when it comes to what can be done on two hands! Let’s explore! Again, retract the thumb on your second (e.g. left) hand, call it “32”, and count to “31” on your first hand all over again (just like you did above). Only, this time, “1” is “32+1” which equals “33”. “2” is “32+2=34”. “4” is “32+4=36”. “8” is “32+8=40”. “16” is “32+16”=48. The destination of this “train” is where the thumb on your second hand and all the digits on your first hand are retracted; and the tally is: “32+16+8+4+2+1” which equals “63”. 63 is already more than double what we were able to do on one hand – and we’ve only used the thumb from our second hand! Here’s a spreadsheet that illustrates counting from 32 to 63:

View Table #2

    Well, friends, we are entering the “64-zone”. On second hand (or, for most, the left hand), extend thumb and retract index finger. What then? You guessed it! Count to 31 on your first hand yet again! Only now “1, 2, and 3…” are “65, 66, and 67…”, respectively. “4, 8, and 16” are “68, 72, and 80” (of course, all we are doing is adding 64 to them). And the second digit of our second hand plus all the digits of our first hand total: “64+16+8+4+1” which equals 95. Here’s another spreadsheet to help with understanding:

View Table #3

    Retract the thumb and use it to help hold down the index finger of your second hand, and you’re looking at “96”. Hold that position on your second hand, count to 31 on your first hand, adding 96 to every number, and before you know it, you’re the proud owner of your first seven retracted digits, adding up to:

    “64+32+16+8+4+2+1”; which equals “127”! The third digit on your second hand (e.g. middle finger), retracted alone signifies “128”… The fourth digit: “256” …- and the fifth: “512”… As a matter of fact, before you know it, you realize (if your experience is that of mine) that you are doing the same thing on your left hand that you did on your right – just in multiples of 32:

1 x 32 = 32,          2 x 32 = 64,           3 x 32 = 96,           4 x 32 = 128,…
8 x 32 = 256,…     16 x 32 = 512,…      31 x 32 =…            …992

    When all digits on your second hand are retracted, you have “(16+8+4+2+1) x 32 = (31) x 32 = (512 + 256 + 128 + 64 + 32) = 992”. When all ten digits are retracted, we are looking at: “(512+256+128+64+32) + (16+8+4+2+1)” which equals “(992) + (31) = 1023”. Here’s a table showing the progressive sums from “512” to “1” (and vice versa):

    That’s it! That is 1 to 1023 (or binary “11111 11111”) on ten fingers! All you have to do at that point is extend all your fingers (give your hands a good shake to loosen them up), say, “1024 (visualizing “1 00000 00000”)!” and bask in the moment! By the way, I’ll have you realize that you just counted one binary “k”! Remember the article, “Byting Off More Than You Can Chew?” Well, you just “chewed” it! People in the Computer Industry converse in up to “1 00000 00000” binary (or 1024) bytes – and then they start all over! So, for example, a binary “kilobyte” (if not “kibibyte”) is “1 00000 00000” (or 1024) bytes. From there, as I explained in the aforementioned article, we have...

- “1 00000 00000” (1024) kilobytes, or a binary megabyte (or “mebibyte”)
- “1 00000 00000” (1024) megabytes equals a binary gigabyte (“gibibyte”)
- “1 00000 00000” (1024) gigabytes makes a binary terabyte (“tebibyte”)

So, if we can count to binary “1 00000 00000”, or 1024, on our fingers - and we just did it (or 1023, which is the next best thing) - we’re set! (Well, OK, maybe there are a few more things to learn, but don’t tell me we didn’t learn anything today)!

    …And, by the way, if anyone ever says, “As many as you can count on one hand”, now you can ask them, “Do you mean ‘5’ or ‘31’?”


Appendix: “Duotrigesimal”(!)

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