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Perhaps counting on your fingers conjures up images of a little kid in preschool using his fingers to count to 10? But, why couldn't such a little kid count to 1023 with those same fingers?

Treat each digit (err, finger) as a bit. If the finger is sticking up, it's a one. Else, it's a zero. Unless your parents were cousins or you lost an argument with a tablesaw, you should have 10 bits with which to count.

The range of unsigned integer values that is representable with 10 bits is 0 to 1023.

I recently thought of teaching my own future children to count to 1023 in binary using this same technique. Unfortunately, I also realized the difficulty in doing so: counting in binary requires either knowledge of multiplication and addition in order to compute the numerical value of a given arrangement of fingers, or the ability to recognize on sight every possible arrangement of fingers (although with just one hand, this is limited to 31 possible numbers instead of 1023, so it's a bit more managable).

There's also the conceptual issue. A preschooler learns to count from one to ten on his/her fingers because they can conceptually recognize four fingers as "four", the same way they identify four apples or four blocks as "four". Identifying "four" as middle-finger-up, all-other-fingers-down does nothing to help the preschooler to recognize "four" of any other things in real life.

However, it does have potential to simplify basic math with older children. Counting in binary on your fingers means that you can calculate sums and differences larger than ten, multiply quickly by two and powers of two, and of course have a leg up on the rest of the class if and when you begin a career in circuit design.

While admittedly the binary system gives you 1024 (ahem, mblase) numbers, and one hand gives you 32 (Ahem, indeed), one can magnify the number grossly by making the base 3 -- trinary. By having the finger position be in the set of Down, Halfway, and Up (rather than just Down and Up) you essentially increase your maximum to 59049 (and 243 on one hand). Taking this concept to a fuller potential, we decide that a person is able to identify 7 degrees of erection for their fingers (thereby creating a septenary system for counting. This would then allow for more than 282 million numbers (16807 one hand). Machine aided, oboy, look out.

Of course, one could use a machine much more effectively by mapping out a high resolution image map of the hand, and store each pixel as an integer, accessing these when a given place on the hand is indicated (hopefully by a very fine tool).

And to answer the question, I can count as high as ten, or even as high as two.

If I was asked to climb a short wall, I could count on my fingers to hold me, but if asked to scale more than 12 feet, with slippery footing, I would prefer to count on a rope held by a trained belayer.

No one has yet explained why, when counting to 1023 is possible, we get through most of our lives only getting to 10 and never get any feeling of "waste". So then... why?

It's all due to redundancy. How many ways are there of representing 1 in the usual system? Ten. And two? Forty-five. Five can be made in no less than 15120 ways.

The ideal system would have only one way of representing each number, to make best use of the possible combinations. The "finger up, finger down" system meets this criterion, since it is effectively a model of binary which we know gives any integer a unique representation.

However, representing the number n in the preschool way is possible in 10Cn different ways; this also shows why 10 is the highest number we can possibly make with this system.

Hence, the preschool system has very high redundancy by duplicity, whereas the previously suggested binary system has none.

There's also the supergroovy Base6 two-line abacus method!

Begin with both hands closed.
For the value 1 (one) raise a single finger on the right hand. (It could be the left if you really wanted, just try to be consistent about it, okay? okay.)
For the value of 2 (two) raise another finger on the right hand to accompany the first.
Three fingers for the number 3, 4 fingers for the number 4, and I'll leave it as an exercise for the reader to hypothesize how many fingers should be extended to represent the number 5.
Ah, but then it gets tricky! You see, to represent 6 you might think you pop up a finger on the left hand (or the right hand, if you decided you had to be different back there at the beginning) and you'd be right. What's different about it is that you close the fingers on the right hand. I.E. one finger up on the left (or right) and no fingers up on the right (or left) hand represents the number 6.
I'magine there are a few folks out there scratching their heads wondering how to go about getting the number 7 to appear. Well, wonder no longer for I shall enlighten you: Pop up a finger on your right hand. For 8 through 11 just keep adding fingers to your right hand 'til you run out of fingers to add. At 12 you raise another finger on your left hand and close the right. Lather, rinse, repeat.
With a little quick and easy math you'll notice you can count to 35 with this nifty method, assuming you have the full suggested compliment of the required parts.

A slight modification to the "base 6 abacus" method allows counting up to 99 without complicated finger contortionism.

Basically, each finger on your dominant hand counts for one. The thumb on your dominant hand is five, the fingers on your off hand are 10, and the thumb on your off hand is 50.

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