Place value and number bases

In brief

Try to write the number of seconds in a day using tally marks, one stroke per second. You would need eighty six thousand four hundred strokes, and reading the result back would be harder than making it. Now write "86400." Five symbols, drawn from a set of only ten, encode a quantity that would otherwise take a wall of tally marks to represent. The difference is not a bigger alphabet of symbols. It is a trick: the same ten symbols mean different amounts depending on where they sit. This entry is about that trick, how position, not new symbols, buys unlimited range from a fixed, small vocabulary.

The full treatment

First look: what a digit's neighbors do to its value

Look at the number 222. It is written with the same digit three times, yet the three copies do not mean the same amount. The rightmost 2 means two units. The middle 2 means two groups of ten, or twenty. The leftmost 2 means two groups of a hundred, or two hundred. Add them and you recover two hundred twenty two. Nothing about the symbol 2 changed; what changed is the job assigned to its position. An odometer makes the same point mechanically: the rightmost wheel turns once per mile, and only when it completes a full turn does the next wheel move by one notch, silently standing for ten times as much per notch as the wheel to its right.

Building the idea: grouping solves the running-out-of-symbols problem

Suppose you only have ten distinct symbols, 0 through 9, and you need to name quantities larger than nine. The constraint is severe: you cannot invent a new symbol for ten, then another for eleven, forever, because a system with infinitely many symbols is unusable, nobody could learn or write it. The solution is grouping. Bundle ten units into one group of ten. When you have ten of those groups, bundle them into one group of a hundred. Keep bundling by tens, forever. Now every quantity can be described by saying how many of each size of bundle you have, and since a group can never hold ten or more of the next size down without triggering another bundle, the count of any single size of bundle is always a single digit, zero through nine. Position is simply a record of which bundle size a digit is counting.

The formal model: base b positional notation

Generalize the bundle size from ten to any whole number b greater than one, called the base. A positional numeral with digits, read right to left, d0, d1, d2, and so on, represents the quantity

N = d0 times b to the power 0, plus d1 times b to the power 1, plus d2 times b to the power 2, and so on,

where each digit di is a whole number from 0 up to b minus 1, never larger, because reaching b of anything triggers a bundle into the next position. The power of b attached to a position is called its place value. Base ten, the everyday case, sets b equal to ten and allows digits 0 through 9. Base two, binary, sets b equal to two and allows only the digits 0 and 1; the positions then carry weights 1, 2, 4, 8, 16, and so on, each one double the last, which is why binary digits map so directly onto physical switches that are simply on or off.

Converting between bases: the mechanism, not just the answer

To convert a quantity already given in one base into another base, you need a procedure, not a lookup table, because the quantity itself does not change, only its dress. Two mechanisms do the job, and each follows directly from the definition above. To find a number's digits in base b, repeatedly divide the quantity by b and record the remainder at each step; the remainder is always less than b by definition of remainder, so it is a valid digit, and it names how many units of the current bundle size are left over once you have removed as many of the next bundle size as possible. Reading the remainders from last to first gives the digits from most significant to least. The reverse direction, turning digits back into a quantity, is just the sum in the formula above, and can be computed efficiently by starting from the leftmost digit, repeatedly multiplying the running total by b, and adding the next digit, a method sometimes called Horner's method, which avoids computing large powers of b separately.

Choosing the base: convention, not necessity

Ten is not privileged by mathematics; it is privileged by anatomy, most counters in history had ten fingers. Babylonian scribes used base sixty, a choice that survives today in sixty minutes to an hour and three hundred sixty degrees in a circle, because sixty divides evenly by many small numbers, which is convenient for fractions. The Maya used base twenty, matching fingers and toes. Modern computing uses base two because a base with only two digits maps cleanly onto a physical device that is reliably either off or on, and uses base sixteen as a compact shorthand for groups of four binary digits at a time. None of these choices change what a quantity is. They change only how many symbols it takes to write it down and how convenient the arithmetic is in that dress.

Lineage

Positional notation was invented independently at least three times. Babylonian scribes developed a base sixty positional system by the early second millennium BCE, using cuneiform marks whose meaning depended on position within a numeral, though for centuries they left an ambiguous blank for an empty position rather than a true zero digit. Indian mathematicians developed the base ten positional system with a genuine zero digit by the middle of the first millennium CE, a system that spread through Al-Khwarizmi's arithmetic treatise into the Islamic world and reached Europe largely through Fibonacci's Liber Abaci in 1202. The Maya, separately, built a base twenty positional system with its own placeholder zero for their calendar calculations. Georges Ifrah's history documents all three inventions and the very different notational choices each culture made while solving the identical underlying problem.

The strongest case for it

Positional notation is why a phone number, a bank balance, and the distance to a star can all be written with a handful of characters regardless of how large the underlying quantity is. It also makes arithmetic mechanical: the column methods for addition, subtraction, multiplication, and long division taught in school work precisely because each column represents a fixed place value, so operations on one column interact with neighboring columns only through carrying or borrowing, a completely regular procedure that scales to numbers of any size. Every digital computer is a direct application of the base two case, and every hexadecimal color code or memory address is an application of base sixteen, so the same one idea, position multiplies meaning, underlies both a child's arithmetic homework and the machine reading this sentence.

The strongest case against it

The convenience has costs. Positional systems require a symbol for an empty position, without which 45 and 405 could look identical, so a placeholder zero is not optional but load bearing, a dependency taken up fully in a separate entry. A single misplaced digit, one wrong position, changes a quantity by an entire power of the base, so positional notation is unusually fragile to transcription error compared to, say, tally marks, where a missing stroke only shifts the count by one. A frequent misconception is comparing numbers written in different bases digit by digit as though the digits alone carried meaning, for instance assuming the base two numeral 111 is smaller than the base ten numeral 12 because it "looks like a smaller number," when 111 in base two equals seven, larger than the two digit base ten numeral. Position value depends entirely on the declared base, and comparing raw digit strings across bases without converting first is simply invalid.

Where it stands now

That positional notation with a fixed base can represent any whole number using a bounded set of digits is settled mathematics, unchanged since its several independent inventions. The choice of base is convention rather than discovery, and different fields keep different conventions for good practical reasons, base ten for everyday use, base two and base sixteen for computing, base sixty surviving in timekeeping and angle measurement. None of this is in dispute; what continues to develop is only engineering practice around which bases suit which machines.

Test yourself

Take the decimal quantity two hundred thirty seven. Convert it to binary using the repeated division by two method, recording each remainder and showing why each one must be a valid digit for that base. Then invent your own base, choose any whole number greater than one that is not two, ten, or sixteen, and convert two hundred thirty seven into that base using the same method. Finally, take your invented base representation and reconstruct the original decimal quantity from it using the multiply and add method, and explain in your own words why the two conversion procedures are exact inverses of each other rather than two independent tricks.