Excursions: Math

Note

This is the first of a series of excursions in mathematics. I will make it a practice to set a goal at the beginning of each year that will require me to use what I have been playing with all year. I want to make it an adventure, so, as a goal for my excursions in mathematics, I’m going to use several low tech surveying techniques to measure the distance across the Sipsey River in North Alabama during the Spring SEHowl and then I will verify the result (or, at least, see how close I came). Boy! That creek is cold!

In the meantime, these excursions are merely records of my own adventures in the universe. Most of them will be equipping me to achieve my goal but I will take occasional side trips. You can try out my adventures yourself or you might go a little further with adventures of your own. Above all, enjoy yourself.

Counting

Mathematics is about counting. Addition, subtraction, multiplication, roots, permutations, averages and standard deviations – all are short cuts for counting. Differentiation counts rates – units per units. Integration counts units of area and volume under curves and planes. Topology counts holes and connections.

So it’s not hard to imagine how it all started. Hunters didn’t need to count; they just bagged enough game to fill the larder and quit. But shepherds had to keep track of their flocks. Hunters owned animals only as they killed them, Shepherds and other herdsmen owned the entire flock and had to keep up with gains and loses.

They might have counted on their fingers to start with, but that method had a serious limitation – 10. A flock with only ten sheep is not a very big flock and, sheep being what they are, if there are rams and ewes, they tend to exceed that digital limit rather quickly.

So as they herded a flock into an area, they dropped a pebble into a bag for each animal. When they moved to another area, they transferred the pebbles to another bag. If any pebbles were left over, they had an animal to find.

The development of numeration systems and writing made the whole process much easier and reliable. Most of the earliest documents in history were business documents including inventories.

But are fingers really that limited? How far can you count on your fingers? I can count to 100 on the fingers of both hands – wait make that 1024. Can you figure out how I do that?

You would think that, for two hands of five fingers each, a system with 10 digits would be ideal and that may explain the popularity of our decimal system, but there were a variety or early numeration systems including the wildly popular sexagesimal system which was based on the number 60.

But before we explore more efficient means of finger counting, let’s look at how the decimal system works.

There are 10 digits – one for each finger. In the following diagrams, an X indicates that a finger is down; a O represents a cleared (up) finger. So you would count:

O O O O____O O O O
_______O O________ Starting at zero

O O O O____O O O X
_______O O________ One

O O O O ____O O X X
________O O_______ Two, and on to

O O O O ____X X X X
________O X_______ Five, and

X X X X____X X X X
_______X X_______ Nine, and one more makes

O O O O ____ O O O O
________O O ________ Ten,

but where do you go from there? Well if you want to sacrifice detail – that first digit – you can count by tens using the same scheme – 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. Sacrifice both the two first positions and you can count to a thousand and so on. Each counting sacrifices some detail (except the first – ignore fractions for now). But if you could write the results of all the countings down , you would not have to sacrifice any detail. The right most position represents ones, the next position to the left represents tens, then hundreds, and thousands, and so on. Notice that each position to the left represents another power of ten (The first power of ten is ten multiplied by one, the second is ten multiplied by itself, the third is ten multiplied by that result, the fourth is ten multiplied by itself four times, and so on. The zeroth power of any number is one.)

Can you do better with the decimal system. One thing you would want is a procedure that uses all your fingers and you need a procedure that would represent all the numbers. It would also be convenient to use different hands to represent different place values (powers of ten) and the thumbs might do something different also.

So, as you count with your right hand fingers you count 1, 2, 3, 4 and then there’s a thumb. If you let that represent Five then you can free the fingers up to count again –

O O O O____O O O O
_______O X________ Five

O O O O ____ X O O O
________O X ________ Six

O O O O ____X X O O
_______ O X _______ Seven

O O O O ____ X X X O
_______ O X ________ Eight

O O O O ____ X X X X
_______ O X _______ Nine

and what now? One more is Ten, you could represent that by a finger on the left hand and then you would have all the fingers on the right hand clear again to keep counting, thus:

O O O X ____ O O O O
_______ O O ________ Ten

O O O X ____ X O O O
_______ O O ________ Eleven, and on to

O O O X ____ O O O O
_______ O X ________ Fifteen, and

O O X X ____ O O O O
_______ O O _______ Twenty,

but what happens when you get to

X X X X ____ X X X X
_______ O X _______ Forty-nine?

Well, the left thumb is still clear, so it must represent

O O O O ____ O O O O
_______ X O ________ Fifty.

If you continue counting in the same manner you will pass

O O O O ____ O O O X
________ X O ________ Fifty-one

O O O O ___ O O O O
_______ X X ________ Fifty-five

O O O X ____ O O O O
_______ X O ________ Sixty

and ultimately

X X X X ____ X X X X
______ X X ________ Ninety-nine,

and clear all the fingers to get

O O O O ____ O O O O
_______ O O ________ One hundred.

So, in this way, you can count to one hundred on your fingers. This process is the basis of the Oriental counting and calculation method called chisenbop. In the United States, it’s often called Fingermath.

There are popular ways of extending this process. An early one was called the counting board.

What if you had more hands? If each hand could handle ten digits – 0 to 9 – then ten hands will allow you to count to ten to the tenth power or 10,000,000,000. Ten billion should take care of just about anything we want to count.

Let’s go back to the pebbles the shepherd used to count sheep but, instead of pebbles, use something a little more ornate and constant in size – say, beads. and instead of bags, use a board with grooves cut in it to hold the beads. On each groove, you should have four beads at the bottom and two at the top with different colors. The four at the bottom represent ones and the two at the top represent fives. Having two “thumbs” has some advantages when you start doing calculations later.

This counting board was probably a very early invention but it has one serious drawback. It’s too easy to spill the beads off the board. I have a magnetic chess/checkers board that I can use as a counting board but there aren’t enough pieces for 10 “hands”. There is a more practical answer, though.

String the beads on strings or dowels. If you provide a crossbar or string between the “finger” and “thumb” beads, you can keep them separated. I’ve just described the ever popular abacus.

My first abacus was constructed from a cigar box, fishing line, and clear plastic beads. I still have it around somewhere. Later I got a brass abacus with brass beads on brass rods mounted on a jade base. That one is too pretty to use. My latest is a wooden one. The wooden beads are just snug enough on the rods to provide a nice, slightly “stiff” feel so that the beads will stay in place if there’s no reason fr them to move around.

Abaci are still used by bookkeepers and storekeepers in the Orient to calculate. When you get good at it, you can work calculations almost as quickly as you can with a four function calculator. And the best thing about an abacus is that it doesn’t need batteries.

But we’re still counting and I said that I would tell you how to count to 1024 on your fingers. That number has probably already rung a bell with some of you computer programmer types.

But let’s go back to the decimal system first. Remember that it’s based on powers of ten. Decimal math may be a natural extension of having ten fingers but you can only work with two place values – ones and tens – if you only have two hands.

There is another very natural number system for fingers and, using it, each finger represents a place value. The only way that could happen is if you have two numerals to work with. In other words, if a finger is up, it represents zero; if it’s down, it represents one. And what are the place values?

Well in the decimal system (based on ten), the first eleven place values are

10⁰ = 1
10¹ = 10
10² = 100
10³ = 1000
10⁴ = 10,000
10⁵ = 100,000
10⁶ = 1,000,000
10⁷ = 10,000,000
10⁸ = 100,000,000
10⁹ = 1,000,000,000
10¹⁰ = 10,000,000,000

So in the binary system (based on two), the first eleven place values are:

2⁰ = 1
2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
2⁶ = 64
2⁷ = 128
2⁸ = 256
2⁹ = 512
2¹⁰ = 1024

And there’s that number – 1024

And here’s how you count in binary. Starting from the right, with all fingers clear, each time you press one finger, you clear all fingers to the right and repeat all counting motions up to that point with the new finger held down. It will look like this:

O O O O ___O O O O
_______O O_______ 0 (binary) = 0 (decimal)

O O O O___O O O X
_______O O_______ 1 (binary) = 1 (decimal)

O O O O ___O O X O
_______O O_______ 10 (binary) = 2 (decimal)

O O O O___O O X X
_______O O_______ 11 (binary) = 3 (decimal)

O O O O___O X O O
_______O O _______ 100 (binary) = 4 (decimal)

O O O O___O X O X
_______O O_______ 101 (binary) = 5 (decimal)

O O O O___O X X O
_______O O_______ 110 (binary) = 6 (decimal)

O O O O___O X X X
_______O O______ 111 (binary) = 7 (decimal)

O O O O___X O O O
_______O O_______ 1000 (binary) = 8 (decimal)

and on to

O O O O___O O O O
______O X________ 10000 (binary) = 16 (decimal)

and

X X X X___X X X X
______X X_______ 1111111111 (binary) = 1023 (decimal)

and, finally

O O O O___O O O O
_______O O_______ 1000000000 (binary) = 1024 (decimal)

The real utility of this is not counting but calculating but there are two things you have to get used to to really use the method. One is counting – getting used to the pattern of counting in the binary system. The other is the place values, and after a short time, that becomes natural. There is a memory system that I will cover later that will help until you get used to thinking of your first finger on the right as ones, the second as twos, the third as fours, the fourth as eights, the right thumb as sixteens, and so on.

A lot of you have probably already recognized this as the binary system that computers use. Binary fingermath gives you a unique opportunity to explore how computers think.

For instance, each binary digit is called a bit (Binary digIT). In our fingermath system, each finger represents a bit. Numbers are stored in computer memory in clusters of 8 bits called “bytes”. If you don’t use your thumbs, the fingers on both hands represent a byte. Programmers often use a shorthand by representing the eight bits of a byte as two digit numbers in the hexadecimal system (based on 16). For instance 101101 (binary) = 45 (decimal) = 2D (hexadecimal). (In the hexadecimal system, the numerals are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F).

But for our binary fingermath to really be useful, we need to be able to change back and forth between binary and decimal values. First, let’s see how to go from binary to decimal – that’s the easiest change. All you do is add the place values of the fingers that are down (and we’ll go over mental arithmetic later if that’s a problem). So, when your fingers are in the following position

O O X O___X O O X
______O X_______

you add 128 + 16, which is 144, plus 8 which makes 152, and finally 1 which makes 153. I find it easier to start with the largest place value and work back toward the right. As the sum gets larger, it’s easier to keep up as the additions get smaller.

Now, I’m getting a little ahead of myself because I’ll be playing with addition later, but I’m assuming that most of you are pretty well exposed to it and some of the things I’m going to be doing with binary fingermath needs some prior experience.

Going from the decimal system to the binary system is a little more involved but is easy to do once you get the hang of it. First, figure out the finger that has the largest place value that is less than the decimal number. Say you want to represent 73 as a binary number. The finger that has the largest place value less than 73 is the 64 finger (the left index finger).

O O O X___O O O O
______O O________

Now subtract that number from the original number, 73 – 64 = 9. Next, find the finger with the largest place value less than the result. That wold be the 8 finger and the result looks like this:

O O O X___X O O O
______O O_______

You had 9 left; now you subtract the last place value (8) from that and you are left with 1, which, of course, is represented by the little finger on the right hand. So the final result is:

O O O X___X O O X
______O O_______

and the binary equivalent of 73 (decimal) is 1001001.

There is another way to keep track of counts. You can bend a paper clip so that it will slide along the edge of a ruler. Normally, a metric ruler will be best. That way, the major marks can be 10s while the smaller marks are 1s. Or the major marks can be 100s while the smaller marks are 10s. Or the major marks can be 1s while the smaller marks are tenths. You see, you can choose what the marks represent to fit your problem. As you will see later, this is the basis of a very powerful calculation instrument.

Now, one more consideration before we go on to other things. Say you have counted 15 sheep but you are expecting a neighboring shepherd to give you more later in the day. You will want to add the new sheep to your current total but you don’t want to go back to carrying around bags of pebbles. Is there an effective way to remember the current sum (and, by the way, partial sums and products and other intermediate results in complicated calculations)?

Yes! There is an excellent method available (if you want the details, get a copy of The Memory Book by Harry Lorayne and Jerry Lucas). The concept is simple and easy. There are ten groups of related consonants in the English alphabet. Within each group, the sounds are made similarly. How convenient that there are also ten digits in the decimal number system! And more so, how convenient it is that it is so easy to associate each digit with a consonant. The following table shows you how to do it:

Digit	Consonants	How to remember it
0	z, s, sh	“Zero” begins with a “z”.
1	t, d, th	“t” has one (1) downstroke.
2	n	“n” has to (2) downstrokes.
3	m	“m” has three (3) downstrokes.
4	r	“Four” (4) ends in “r”.
5	l	When you spread the 5 fingers of the right hand, the thumb and ring finger form an “L”.
6	j, g(soft), ch, sh	Capital “J” is almost a backward 6.
7	k, g (hard)	Two 7s can be used to make a “K”.
8	f, ph, v	A cursive “f” is almost an 8.
9	p, b	Capital “P” is almost a 9.

Since vowels have no value, words can be made from numbers by combining consonants with vowels. If you have more than one word number to remember, you can make them more amenable to memory by associating them in some ridiculous fashion.

So, say you have arrived as a shepherd. You now have 273,106 sheep and you want to remember that number until you get more. The letters corresponding to the digits are nkmtsj. You can use the words “knock”, “mit”, “sash”. Visualize yourself walking up to a door but, since you don’t want to break anything, you pull out a catcher’s mitt from your pocket and put it on to knock. Despite your care, you knock so hard that the window sash on the window next to the door falls in. The best scenarios to remember, though, are your own.

Once you get use to the system, you will find yourself memorizing license plates on the cars around you as you drive. It can get irritating.

Oh yeah. One last thing before going on to the next excursion. I mentioned that there was a way to remember the place values of your fingers in binary fingermath. This, of course is it. The fingers of the left hand are too easy to worry about mnemonics (memory aids, that is). From the far right, the finger values are 1, 2, 4, 8, and 16. Possible mnemonics for the left hand are shoe – man (6th finger – 32 – picture yourself slipping your foot into a slice open man – yuck, but I bet you remember it!). Key – chair (7th finger – 64 – picture yourself locking yourself out of your house and trying to use a chair as a key). Ivy – toe knife (8th finger -128 picture yourself using your sharp big toe to trim your ivy). Pie – new latch (9th finger – 256 – a pie made out of shiny new door latches. Mmmmmm, what could be better.) Toes – Latin (10th finger – 512 – picture yourself getting made at your toes for talking to each other in Latin where you can’t understand them (unless you’re fluent in Latin, that is.) And finally, tot – dozin’ rye (All fingers cleared – 1024. Picture a small child trying to wake up a sleeping loaf of rye bread.)

So, now, where’s the practicality of counting with your fingers? I actually use it in one of the tests I give which requires me to count both the number of moves a person makes placing a pin in holes and pressing a button, and the number of errors they make. People normally don’t appreciate how separate their senses are unless they have some head injury that forcibly brings the point home. But, fact is, the senses work pretty much independently to a considerable depth of processing. So, to a large extent you can use, say, your vision and sense of touch separately. What I do is count the number of moves my client makes mentally and count the number of errors they make on my hands. Since they rarely make more than 40 moves in the time required and can’t make more errors than moves, I’m safe using the “count to 100” system. So you can accurately count on your fingers while paying attention to other things.

Now, that should give you nightmares for a month. And with that, we bid the counting excursion adieu. Next – Standards – you gotta have ’em.