Counting and Number Systems

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We all learned how to count as young children: "one, two, three". Iflike me you watched a lot of Sesame Street when you were small, thenyou also learned the Spanish words for those same concepts: "uno, dos,tres". And that pretty much sums up counting, right?

Well, not quite. Today, I would like to try something a littledifferent. Instead of discussing a specific application of math tobusiness, science, or art, I want to go back to the very beginning andsay a few words about "counting" that illustrate the relationshipbetween mathematical abstractions and the real world.

How did early humans first develop mathematics? How do/should childrenlearn about math? Certainly one concept at the heart of mathematicalthinking is counting. Hunters count animals, gatherers countfruits, children count marbles, merchants count inventory (both goodsand coins): pretty much everyone needs to count.

Pretend you are just learning how to count, and you have to make it upas you go along, without cultural knowledge to speed up theprocess. What do you do?

Perhaps you start by making marks in one-to-one correspondence withthe objects you are counting. Thus if you have one apple, you make onemark, as in Figure A. If you have two apples, you make two marks, asin Figure B. If you have six apples, you make six marks, as inFigure F.

This is a good start: it is verifiable (you can visually line up themarks and the apples and confirm that they are in a one-to-onecorrespondence), and it suggests the abstract concept of "number",because "five marks" could mean "five apples" or "five rabbits" orwhatever.

However, it is not that great as a system for organizing yourbusiness. It is a bit like making a map of your town at a scale of oneinch equals one inch: the map has to be as large as your town.What you really want is data compression: a map that fits inyour pocket so you have it when you go out walking, yet summarizesenough about the town to be useful in helping you find your way.

Can you visually distinguish between 5 and 6 marks (figures E and Fabove)? What about between 36 and 37 marks (not shown): short ofre-counting them, it is tough for the human visual system to processthat many marks: the Monotonous Repetition is Unsatisfyingeffect comes in, and our minds get bored.

So we need a better system. The first step we learned as childrenis to use a different symbol to represent groups of five. Forinstance, we can use every fifth "stick" to "tie up a bundle". Underthis system we can more easily distinguish numbers like 4, 5, 6, and 7:

However, it is still difficult to distinguish 101 from 106, because westill have monotonous repetition, just now it is bundles of five thatrepeat.

Nonetheless, we can generalize this approach. Every time we have 5 bundles(of five), let's replace them by yet another new symbol, which standsfor 25 sticks. Since drawing little stick figures is tedious, I amalso going to start writing them using letters rather than pictures,but that's just for my typing convenience. In fact, to make clear thatthese are still "pictures", I will use rather uncommon letters: S willmean a single "stick", B will mean a "bundle of five", and "G" will meana group of five bundles, i.e. 25 sticks. With this system, I can nowcount fairly high without making the symbols too monotonouslyrepetitive. Here are the symbols for a variety of differentnumbers. I have labeled them with their conventional English word names soyou can check your ability to decode them, but remember that at thispoint, we have no English words, no decimal system, only these GBSsymbols.

Symbolic Pictograph	English Interpretation
S	one
SS	two
SSS	three
SSSS	four
B	five
BS	six
BSS	seven
BB	ten
BBS	eleven
BBB	fifteen
BBBB	twenty
BBBBSSSS	twenty four
G	twenty five
GS	twenty six
GBS	thirty one
GGBBSS	sixty two
GGGGS	one hundred one

At this point, we have more or less replicated "Roman Numerals". Theactual Roman Numeral system is a bit more arcane than this, but sinceit is a mathematical dead-end, I am not going to bore you by walkingthrough it. The fatal flaw in both Roman Numerals and our little "GBS"system is that arithmetic is extremely difficult. We have succeeded ingiving (somewhat) short names to (small positive whole) numbers, in asystematic fashion. This makes it relatively easy to count. Butonce we can count, we want to do other things. Most urgently, we wantto add, so that if you have 27 apples and I have 32, we learnthat together we have accumulated a total of 59.

How would addition work in the GBS system? We could try to concatenatethe numbers for 27 and 32, namely "GSS" and "GBSS", which gives (aftersorting), "GGBSSSS", or 59. This is actually correct! Unfortunately,this simple rule fails in lots of other cases. For instance, if we add27 and 33, we get "GGBSSSSS", which is not even a valid number in oursystem. We see the problem: we need to "promote" the five S charactersinto a B, forming "GGBB" or 60.

What about adding 24 and 24?We get "BBBBBBBBSSSSSSSS", which is a huge monotonous mess. Now wehave to group five S characters into a B, forming "BBBBBBBBBSSS". Nowwe group five B characters into a G, forming "GBBBBSSS", or 48.Is this really how you want to do addition? Imagine tryingsubtraction, multiplication, or division!!

Also, if this is your system for writing numbers, how will you devise amechanical "adding machine" to help you? Notice that adding twonumbers sometimes creates a result that is shorter than either of thetwo inputs (e.g. add ten and fifteen). That is going to seriouslycomplicate any kind of system of gears and levers you might try to build.

So now we are in a position to better state what we want a numbersystem to achieve. Not only must it provide a systematic wayto count (i.e. it must give short names to small numbers, andbe able to systematically name any number), but it must alsosupport a simple, systematic way to do addition.It would be great if it also made it equally easy to multiply and divide, butthat may be asking too much - that's why people invented logarithmsand slide rules to supplement adding machines.

It turns out that there are many ways to accomplish thisgoal. Our familiar "decimal system" is one way, but since it is sofamiliar, I am going to present two others instead that are also quitehandy to know about in this age of computers: the "binary system" andthe "hexadecimal system". There are even more exotic systems. Forinstance, Knuth(The Art ofComputer Programming, Volume 2) discusses a perfectly reasonableQuater-imaginarybase number system.

The decimal, binary, hexadecimal and Quater-imaginary systemsall share a key feature: positional notation. This is anotherconceptual leap forward. Although I sorted the letters in my "GBS"system to put the "important" (big) letters first, that was purely atypographical convention which had no real impact: "GBS" and "SBG"both represent the number that we ordinarily call 31, because both arereally just short hand for drawing 31 sticks.

By contrast, in positional number systems, the order inwhich we write the symbols matters. This means that we can achievehigher amounts of data compression, because position can carryinformation implicitly, without us having to spell itout. However, data compression reduces redundancy.Unfortunately, this makes math hard for many people: less redundancymeans greater attention to detail is required to get it right. Aperson whose brain has trouble distinguishing between "GBS" and "GSB"is likely to have trouble with arithmetic.

Ordinary English text contains a great deal ofredundancy. Psychologists have done experiments where they show peoplethe top half of the letters in a sentence, or the bottom half, orevery third letter, and people can often figure out the meaning. If Iwrite "rss r rd, vlts r bl", you might be able to guess that I meant"roses are red, violets are blue". But in math, missing characters arefatal: each one carries its own crucial meaning - there is no analogyto "deleting vowels". There is no question that the numbers 5027 and507 are completely different, and both are contextuallyplausible, or equally "grammatical". You cannot drop (or swap) symbolsin math.

The binary system uses just two fundamental symbols,conventionally written 0 and 1. "Bi" means "two", like in"bicycle". In contrast, "deci" means ten and "hexideci"means 16, so those systems use more fundamental symbols ("digits").The decimal system uses the ten symbols "0123456789". Thehexadecimal system uses these and six more, namely the letters "abcdef".

Since binary uses just two symbols, it is simple to learn. Thesimplicity also facilitates building "adding machines" (nowadays knownas "computers").

Binary is a positional system. When we count, we start with 1. For thenext number, two, we do not have a '2' symbol, so we write 10instead. At this point you are probably saying, wait a minute, 10means "ten", not "two", and you feel hopelessly confused.

So, for purposes of explanation, let's not use 0 and 1 as the binarysymbols: let's use A and B. When we write a binary number such asBABA, the positions matter. The last (rightmost) symbolrepresents individual sticks/apples/rabbits, with A meaning zero and Bmeaning one. The second to last symbol represents bundles oftwo sticks: A still means zero, but B now means one bundle,i.e. two sticks. That's why "BA" stands for"two". The third to last symbol represents groupsof four sticks: A still means zero, but B now means one group,i.e. four sticks. The fourth to last symbol representsgroups of eight sticks, and so forth. Thus "BABA" means "ten", becauseit represents eight plus two.

Here is the same table of numbers we used to describe the GBS system,now converted to binary.

Binary Symbol	English Interpretation
B	one
BA	two
BB	three
BAA	four
BAB	five
BBA	six
BBB	seven
BABA	ten
BABB	eleven
BBBB	fifteen
BABAA	twenty
BBAAA	twenty four
BBAAB	twenty five
BBABA	twenty six
BBBBB	thirty one
BBBBBA	sixty two
BBAABAB	one hundred one

For example, the "one hundred one" case works like this.

There are many good things about this table: larger numbers (inmagnitude) are also longer (in binary form) than smaller ones, so thepattern is more systematic, and easier to mechanize, than for the GBSsystem. Moreover, it is easy to do addition: just apply the samealgorithm you learned in elementary school! Starting from theright-most column, you add, and if the result is too big to fit, you"carry" a "B" ("one") to the column to the left.

The nice part is you don't have to memorize such a big table ofaddition facts! School children have to memorize all possiblesingle-digit addition facts, such as "9+6 = 15". But in binary, theONLY non-trivial single-bit addition fact is "B+B = BA", i.e. the rulethat if both "bits" are "B", you write "A" for the sum and you "carry"the "B".

If you already had two B's in a column, and you received a third oneas a carry, you use the fact that

B+B+B = (B+B)+B = BA+B = BB,

so you would put B as the answerand carry another B to the left.

For instance, to add 10 and 6 to get 16, we work in binary as shownbelow. Move from right to left, noticing that A+A = A, A+B = B+A = B,and when you have B+B, you put A and "carry" the B by following thecurved line.

This is essentially the same thing that happens in ordinarybase-ten (decimal system) addition, for example when we add 9595 and405 to get 10,000.

It also illustrates how we can count arbitrarily high. All we have todo to implement "counting" is to know how to add one. And we can:given any binary number, no matter how long, we can add "B" to it byfollowing the rule described above. When needed, the rule lengthensthe number by one binary digit (bit), as when we go from 15 (BBBB) to16 (BAAAA) - just like in decimal when we go from 99 to 100. We mayor may not have invented English words for numbers withhundreds of digits in them, but they still make sense, and we know howto operate on them, just by operating on their digits.

The only drawback to binary for human (rather than computer)use is the visual monotony that stems from having only 2 symbols. Thehuman visual system can easily distinguish a much wider range ofsymbols (e.g. in English we have 26 letters, each with 2 variations:Upper and lower case). We can take advantage of this to get shorterrepresentations for our numbers. In the decimal system, we use tensymbols, conventionally written 0,1,2,3,4,5,6,7,8,9. As a result, thepositional values for the columns are powers of ten, insteadof Powersof Two. Thus for example, the number 123 in decimal means onehundred (1*10*10) plus twenty (2*10) plus three (3*1). We are so used tothis (once we have graduated from elementary school) that we think of"123" as "being" the number one hundred twenty three. Indeed ourEnglish word text parallels the decimal system very closely (with afew exceptions, like "eleven", that are common enough to get their ownword). That's why saying "10" means "two" in binary confusespeople. Hopefully now you do not mind saying that "BA" means "two" inbinary.

With the decimal system, we have more addition facts to memorize(9+6=15), but we get shorter strings of digits, which are visuallyeasier to look at. With the hexadecimal system, we carry thisidea even further: by using 16 symbols instead of 10, we can make thedigit strings even shorter, but we have yet more addition facts tomemorize. The digits 0..9 retain their ordinary meaning (when used inthe right-most column), but now we add "a" for ten, "b" for eleven,"c" for twelve, "d" for thirteen, "e" for fourteen, and "f" forfifteen. Once you get these memorized, we can do our positionalnotation trick again, this time using powers of 16. The secondcolumn (from the right) represents bundles of 16 sticks. The thirdcolumn represents groups of 16 bundles of 16 sticks, so it reallymeans 256 sticks. And so on.

For example, in hexadecimal, the symbol "a7" means "one hundred sixtyseven", or 167 in our decimal notation, because the "a" in the secondcolumn from the right tells us to take ten copies of sixteen, or 160,and the "7" in the rightmost column means to add seven more, giving167.

Because positional notation involves adding, it is notsurprising that it is easy to perform addition in any positionalsystem. That same elementary school algorithm continues to work inhexadecimal: we just need to modify the addition facts. For instance,in hexadecimal, "5+5=a", and "6+7=d", and "a+c=16", where you have tokeep in mind that in hexadecimal, "16" means 16+6, or 22. To avoidconfusion, computer programmers (who do sometimes actually need to usehexadecimal) write a "0x" in front of hexadecimal numbers todistinguish them from decimal numbers. Thus, we might write "0xa + 0xc= 0x16", and now there is no confusion: this is simply the hexadecimalway of saying that "ten plus twelve equals twenty two".

Here is the same table of numbers we used to describe the GBS system,now converted to hexadecimal.

Hexadecimal Symbol	English Interpretation
0x1	one
0x2	two
0x3	three
0x4	four
0x5	five
0x6	six
0x7	seven
0xa	ten
0xb	eleven
0xf	fifteen
0x14	twenty
0x18	twenty four
0x19	twenty five
0x1a	twenty six
0x1f	thirty one
0x3e	sixty two
0x65	one hundred one

As one last example, 0x123 means, in decimal: 1*256 + 2*16 + 3 = 291.Conversely, if you want to translate 291 back into hexadecimal, youneed to do a somewhat more complex process involving division: 291/256is one with a remainder of 35. Then 35/16 is two, with a remainder ofthree. So 0x123 is the answer. Isn't is nice that we have computersthat can do these kinds of mechanical translation steps for us!

I want to stress that all these multiplications and divisionsare only needed for converting between number systems. If wehad grown up learning hexadecimal instead of decimal, it would beintuitive to us, and we would not need to "convert" back to decimalin order to understand it.

Regardless of which number system we choose for writing, there is anenormously important fact about the real world lurking in thebackground: counting works. If you have five apples lined up ona table, and you count them (carefully, and without eating or droppingone), then you will always get a count of 5, regardless ofwhich end of the line you start from. If you then shuffle the order ofthe apples and recount them, you will still get a count of 5. This mayseem like a trivial observation, but it is not. There is no particularreason why the real world needs to behave in a rational, regular,systematic manner - but it does.

Said differently: as a theoretical mathematician, I can definea number system - take your pick of binary, decimal, hexadecimal, orany other - and thereby create a computational system. Ican define the operation "next", which implements counting; Ican define "addition", and all the other usual arithmeticoperations. The resulting system will be internally consistent, and Ican do all kinds of interesting things with it, BUT, none of thisnecessarily has anything to do with the real world. It is an amazingthing about the world we live in that in fact, when mathematicians dodefine new mathematical systems, they frequently turn out to haveactual predictive value in the real world. We will see an example ofthis next week when I talk about Imaginary Numbers.

At the heart of it, this is why Science works: different people, withdifferent mental models and different political agendas, cannonetheless agree that there are 5 apples on the table: not more, notless.

The real world exhibits an astonishing amount of regularityand repeatability. Not only can we count apples, but we can measuremass, position, velocity and acceleration, and summarize them via"Newtonian Mechanics", from which we can calculate how to launchrockets that can put a human on the Moon and bring them safely backhome. We can measure light, electricity and magnetism, and summarizethem via "Maxwell's Equations", from which we can calculate how tomake radios and electric lights. We can measure radioactivity and thephotoelectric effect and summarize them using "Quantum Mechanics",from which we can calculate how to make the transistors and computerchips that are central to cell phones and the Internet. In each ofthese cases, humans have been able to model the observed regularityand consistency of the real world using mathematics, then turn aroundand use that same mathematics to make predictions about thereal world, predictions that actually come true, predictions that areat the heart of all our modern technologies.

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