II.1 From Numbers to Number Systems

Fernando Q. Gouvêa

People have been writing numbers down for as long as they have been writing. In every civilization that has developed a way of recording information, we also find a way of recording numbers. Some scholars even argue that numbers came first.

It is fairly clear that numbers first arose as adjectives: they specified how many or how much of something there was. Thus, it was possible to talk about three apricots, say, long before it was possible to talk about the number 3. But once the concept of “threeness” is on the table, so that the same adjective specifies three fish and three horses, and once a written symbol such as “3” is developed that can be used in all of those instances, the conditions exist for 3 itself to emerge as an independent entity. Once it does, we are doing mathematics.

This process seems to have repeated itself many times when new kinds of numbers have been introduced: first a number is used, then it is represented symbolically, and finally it comes to be conceived as a thing in itself and as part of a system of similar entities.

1 Numbers in Early Mathematics

The earliest mathematical documents we know about go back to the civilizations of the ancient Middle East, in Egypt and in Mesopotamia. In both cultures, a scribal class developed. Scribes were responsible for keeping records, which often required them to do arithmetic and solve simple mathematical problems. Most of the mathematical documents we have from those cultures seem to have been created for the use of young scribes learning their craft. Many of them are collections of problems, provided with either answers or brief solutions: twenty-five problems about digging trenches in one tablet, twelve problems requiring the solution of a linear equation in another, problems about squares and their sides in a third.

Numbers were used both for counting and for measuring, so a need for fractional numbers must have come up fairly early. Fractions are complicated to write down, and computing with them can be difficult. Hence, the problem of “broken numbers” may well have been the first really challenging mathematical problem. How does one write down fractions? The Egyptians and the Mesopotamians came up with strikingly different answers, both of which are also quite different from the way we write them today.

In Egypt (and later in Greece and much of the Mediterranean world), the fundamental notion was “the nth part,” as in “the third part of six is two.” In this language, one would express the idea of dividing 7 by 3 as, “What is the third part of seven?” The answer is, “Two and the third.” The process was complicated by an additional restriction: one never recorded a final result using more than one of the same kind of part. Thus, the number we would want to express as “two fifth parts” would have to be given as “the third and the fifteenth.”

In Mesopotamia, we find a very different idea, which may have arisen to allow easy conversion between different kinds of units. First of all, the Babylonians had a way to generate symbols for all the numbers from 1 to 59. For larger numbers, they used a positional system much like the one we use today, but based on 60 rather than 10. So something like 1, 20 means one sixty and twenty units, that is, 1 × 60+ 20 = 80. The same system was then extended to fractions, so that one half was represented as thirty sixtieths. It is convenient to mark the beginning of the fractional part with a semicolon, though this and the comma are a modern convention that has no counterpart in the original texts. Then, for example, 1;24,36 means 1 + + , which we would more usually write as or 1.41. The Mesopotamian way of writing numbers is called a sexagesimal place-value system by analogy with the system we use today, which is, of course, a decimal place-value system.

Neither of these systems is really equipped to deal well with complicated numbers. In Mesopotamia, for example, only finite sexagesimal expressions were employed, so the scribes were not able to write down an exact value for the reciprocal of 7 because there is no finite sexagesimal expression for . In practice, this meant that to divide by 7 required finding an approximate answer. The Egyptian “parts” system, on the other hand, can represent any positive rational number, but doing so may require a sequence of denominators that to our eyes looks very complicated. One of the surviving papyri includes problems that look designed to produce just such complicated answers. One of these answers is “14, the 4th, the 56th, the 97th, the 194th, the 388th, the 679th, the 776th,” which in modern notation is the fraction . It seems that the joy of computation for its own sake became well-established very early in the development of mathematics.

Mediterranean civilizations preserved both of these systems for a while. Most everyday numbers were specified using the system of “parts.” On the other hand, astronomy and navigation required more precision, so the sexagesimal system was used in those fields. This included measuring time and angles. The fact that we still divide an hour into sixty minutes and a minute into sixty seconds goes back, via the Greek astronomers, to the Babylonian sexagesimal fractions; almost four thousand years later, we are still influenced by the Babylonian scribes.

2 Lengths Are Not Numbers

Things get more complicated with the mathematics of classical Greek and Hellenistic civilizations. The Greeks, of course, are famous for coming up with the first mathematical proofs. They were the first to attempt to do mathematics in a rigorously deductive way, using clear initial assumptions and careful statements. This, perhaps, is what led them to be very careful about numbers and their relations to other magnitudes.

Sometime before the fourth century B.C.E., the Greeks made the fundamental discovery of “incommensurable magnitudes.” That is, they discovered that it is not always possible to express two given lengths as (integer) multiples of a third length. It is not just that lengths and numbers are conceptually distinct things (though this was important too). The Greeks had found a proof that one cannot use numbers to represent lengths.

Suppose, they argued, you have two line segments. If their lengths are both given by numbers, then those numbers will at worst involve some fractions. By changing the unit of length, then, we can make sure that both of the lengths correspond to whole numbers. In other words, it must be possible to choose a unit length so that each of our segments consists of a whole number multiple of the unit. The two segments, then, could be “measured together,” i.e., would be “commensurable.”

Now here’s the catch: the Greeks could prove that this was not always the case. Their standard example had to do with the side and the diagonal of a square. We do not know exactly how they first established that these two segments are not commensurable, but it might have been something like this: if you subtract the side from the diagonal, you will get a segment shorter than either of them; if both side and diagonal are measured by a common unit, then so is the difference. Now repeat the argument: take the remainder and subtract it from the side until we get a second remainder smaller than the first (it can be subtracted twice, in fact). The second remainder will also be measured by the common unit. It turns out to be quite easy to show that this process will never terminate; instead, it will produce smaller and smaller remainder segments. Eventually, the remainder segment will be smaller than the unit that supposedly measures it a whole number of times. That is impossible (no whole number is smaller than 1, after all), and hence we can conclude that the common unit does not, in fact, exist.

Of course, the diagonal does in fact have a length. Today, we would say that if the length of the side is one unit, then the length of the diagonal is units, and we would interpret this argument as showing that the number is not a fraction. The Greeks did not quite see in what sense could be a number. Instead, it was a length, or, even better, the ratio between the length of the diagonal and the length of the side. Similar arguments could be applied to other lengths; for example, they knew that the side of a square of area 1 and a square of area 10 are incommensurable.

The conclusion, then, is that lengths are not numbers: instead, they are some other kind of magnitude. But now we are faced with a proliferation of magnitudes: numbers, lengths, areas, angles, volumes, etc. Each of these must be taken as a different kind of quantity, not comparable with the others.

This is a problem for geometry, particularly if we want to measure things. The Greeks solved this problem by relying heavily on the notion of a ratio. Two quantities of the same type have a ratio, and this ratio was allowed to be equal to the ratio of two quantities of another type: equality of two ratios was defined using Eudoxus’s theory of proportion, the latter being one of the most important and deep ideas of Greek geometry. So, for example, rather than talking about a number called π, which to them would not be a number at all, they would say that “the ratio of the circle to the square on its radius is the same as the ratio of the circumference to the diameter.” Notice that one of the two ratios is between two areas, the other between two lengths. The number π itself had no name in Greek mathematics, but the Greeks did compare it with ratios between numbers: ARCHIMEDES [VI.3] showed that it was just a little bit less than the ratio of 22 to 7 and just a little bit more than the ratio of 223 to 71.

Doing things this way seems ungainly to us, but it worked very well. Furthermore, it is philosophically satisfying to conceive of a great variety of magnitudes organized into various kinds (segments, angles, surfaces, etc.). Magnitudes of the same kind can be related to one another by ratios, and ratios can be compared with each other because they are relations perceived by our minds. In fact, the word for ratio, both in Greek and in Latin, is the same as the word for “reason” or “explanation” (logos in Greek, ratio in Latin). From the beginning, “irrational” (alogos in Greek) could mean both “without a ratio” and “unreasonable.”

Inevitably, this austere theoretical system was somewhat disconnected from the everyday needs of people who needed to measure things such as lengths and angles. Astronomers kept right on using sexagesima1 approximations, as did mapmakers and other scientists. There was some “leakage” of course: in the first century C.E., Heron of Alexandria wrote a book that reads like an attempt to apply the theoreticians discoveries to practical measurement. It is to him, for example, that we owe the recommendation to use as an approximation for π. (Presumably, he chose Archimedes’ upper bound because it was the simpler number.) In theoretical mathematics, however, the distinction between numbers and other kinds of magnitudes remained firm.

The history of numbers in the West over the fifteen hundred years that followed the classical Greek period can be seen as having two main themes: first, the Greek compartmentalization between different kinds of quantities was slowly demolished; second, in order to do this the notion of number had to be generalized over and over again.

3 Decimal Place Value

Our system for representing whole numbers goes back, ultimately, to the mathematicians of the Indian subcontinent. Sometime before (probably well before) the fifth century C.E., they created nine symbols to designate the numbers from one to nine and used the position of these symbols to indicate their actual value. So a 3 in the units position meant three, and a 3 in the tens position meant three tens, i.e., thirty. This, of course, is what we still do; the symbols themselves have changed, but not the principle. At about the same time, a place marker was developed to indicate an unoccupied space; this eventually evolved into our zero.

Indian astronomy made extensive use of sines, which are almost never whole numbers. To represent these, a Babylonian-style sexagesimal system was used, with each “sexagesimal unit” being represented using the decimal system. So “thirty-three and a quarter” might be represented as 33 15', i.e., 33 units and 15 “minutes” (sixtieths).

Decimal place-value numeration was passed on from India to the Islamic world fairly early. In the ninth century C.E. in Baghdad, the recently established capital of the caliphate, one finds AL-KHWRIZM [VI.5] writing a treatise on numeration in the Indian style, “using nine symbols.” Several centuries later, al-Khwrizm’s treatise was translated into Latin. It was so popular and ínfluentía1 in late-medieval Europe that decimal numeration was often referred to as “algorism.”

It is worth noting that in al-Khwrizm’s writing zero still had a special status: it was a place holder, not a number. But once we have a symbol, and we start doing arithmetic using these symbols, the distinction quickly disappears. We have to know how to add and multiply numbers by zero in order to multiply multidigit numbers. In this way, “nothing” slowly became a number.

4 What People Want Is a Number

As Greek culture was displaced by other influences, the practical tradition became more important. One can see this in al-Khwrizm’s other famous book, whose title gave us the word “algebra.” The book is actually a compendium of many different kinds of practical or semi-practical mathematics problems. Al-Khwrizm opens the book with a declaration that tells us at once that we are no longer in the Greek mathematical world: “When I considered what people generally want in calculating, I found that it is always a number.”

The first portion of al-Khwrizm’s book deals with quadratic equations and with the algebraic manipulations (done entirely in words, with no symbols whatsoever) needed to deal with them. His procedure is exactly the quadratic formula we still use, which of course requires extracting a square root. But in every example the number whose square root we need to find turns out to be a square, so that the square root is easily found—and al-Khwrizm does get a number!

At other points in the book, however, we can see that al-Khwrizm is beginning to think of irrational square roots as number-like entities. He teaches the reader how to manipulate symbols with square roots in them, and gives (in words, of course) examples such as (20 - + ( - 10) = 10. In the second part of the book, which deals with geometry and measurement, one even sees an approximation to a square root: “The product is one thousand eight hundred and seventy-five; take its root, it is the area; it is forty-three and a little.”

The mathematicians of medieval Islam were influenced not only by the practical tradition represented by al-Khwrizm, but also by the Greek tradition, especially EUCLID’S [VI.2] Elements. One finds in their writing a mixture of Greek precision and a more practical approach to measurement. In Omar Khayyam’s Algebra, for example, one sees both theorems in the Greek style and the desire for numerical solutions. In his discussion of cubic equations Khayyam manages to find solutions by means of geometric constructions but laments his inability to find numerical values.

Slowly, however, the realm of “number” began to grow. The Greeks might have insisted that was not a number, but rather a name for a line segment, the side of a square whose area is 10, or a name for a ratio. Among the medieval mathematicians, both in Islam and in Europe, started to behave more and more like a number, entering into operations and even appearing as the solution of certain problems.

5 Giving Equal Status to All Numbers

The idea of extending the decimal place-value system to include fractions was discovered by several mathematicians independently. The most influential of these was STEVIN [VI.10], a Flemish mathematician and engineer who popularized the system in a booklet called De Thiende (“ The tenth”), which was first published in 1585. By extending place value to tenths, hundredths, and so on, Stevin created the system we still use today. More importantly, he explained how it simplified calculations that involved fractions, and gave many practical applications. The cover page, in fact, announces that the book is for “astrologers, surveyors, measurers of tapestries.”

Stevin was certainly aware of some of the issues created by his move. He knew, for example, that the decimal expansion for was infinitely long; his discussion simply says that while it might be more correct to say that the full infinite expansion was the correct representation, in practice it made little difference if we truncated it.

Stevin was also aware that his system provided a way to attach a “number” (meaning a decimal expansion) to every single length. He saw little difference between 1.1764705882 (the beginning of the decimal expansion of ) and 1.4142135623 (the beginning of the decimal expansion of ). In his Arithmetic he boldly declared that all (positive) numbers were squares, cubes, fourth powers, etc., and that roots were just numbers. He also says that “there are no absurd, irrational, irregular, inexplicable, or surd numbers.” Those were all terms used for irrational numbers, i.e., numbers that are not fractions.

What Stevin was proposing, then, was to flatten the incredible diversity of “quantities” or “magnitudes” into one expansive notion of number, defined by decimal expansions. He was aware that these numbers could be represented as lengths along a line. This amounted to a fairly clear notion of what we now call the positive real numbers.

Stevin’s proposal was made immensely more influential by the invention of logarithms. Like the sine and the cosine, these were practical computational tools. In order to be used, they needed to be tabulated, and the tables were given in decimal form. Very soon, everyone was using decimal representation.

It was only much later that it came to be understood what a bold leap this move represented. The positive real numbers are not just a larger number system; they are an immensely larger number system, whose internal complexity we still do not fully understand (see SET THEORY [IV.22]).

6 Real, False, Imaginary

Even as Stevin was writing, the next steps were being taken: under the pressure of the theory of equations, negative numbers and complex numbers began to be useful. Stevin himself was already aware of negative numbers, though he was clearly not quite comfortable with them. For example, he explained that the fact that -3 is a root of x² + x - 6 really means that 3 is a root of the associated polynomial x² - x - 6, obtained by replacing x by - x everywhere.

This was an easy dodge, but cubic equations created more difficult problems. The work of several Italian mathematicians of the sixteenth century led to a method for solving cubic equations. As a crucial step, this method involved extracting a square root. The problem was that the number whose root was needed sometimes came out negative.

Up until then, it had always turned out that when an algebraic problem led to the extraction of the square root of a negative number, the problem simply had no solution. But the equation x³ = 15x + 4 clearly did have a solution—indeed, x = 4 is one—it was just that applying the cubic formula required computing, .

It was BOMBELLI [VI.8], also a mathematician and engineer, who decided to bite the bullet and just see what happened. In his Algebra, published in 1572, he went ahead and computed with this “new kind of radical” and showed that he could find the solution of the cubic in this way. This showed that the cubic formula did indeed work in this case; more importantly, it showed that these strange new numbers could be useful.

It took a while for people to become comfortable with these new quantities. About fifty years later, we find both Albert Girard and DESCARTES [VI.11] saying that equations can have three sorts of roots: true (meaning positive), false (negative), and imaginary. It is not completely clear that they understood that these imaginary roots would be what we now call complex numbers; Descartes, at least, sometimes seems to be saying that an equation of degree n must have n roots, and that the ones that are neither “true” nor “false” must simply be imagined.

Slowly, however, complex numbers began to be used. They came up in the theory of equations, in debates about the logarithms of negative numbers, and in connection to trigonometry. Their connection with the sine and cosine functions (via the exponential) was turned into a powerful tool by EULER [VI.19] in the eighteenth century. By the middle of the eighteenth century, it was well-known that every polynomial had a complete set of roots in the complex numbers. This result became known as THE FUNDAMENTAL THEOREM OF ALGEBRA [V.13]; it was finally proved to everyone’s satisfaction by GAUSS [VI.26]. Thus, the theory of equations did not seem to require any further extension of the notion of number.

7 Number Systems, Old and New

Since complex numbers are clearly different from real numbers, their presence stimulated people to begin classifying numbers into different kinds. Stevin’s egalitarianism had its impact, but it could not quite erase the fact that whole numbers are nicer than decimals, and that fractions are generally easier to grasp than irrational numbers.

In the nineteenth century, all sorts of new ideas created the need for a more careful look at this classification. In number theory, Gauss and KUMMER [VI.40] started looking at subsets of the complex numbers that behaved in a way analogous to the integers, such as the set of all numbers a + b with a and b both integers. In the theory of equations, GALOIS [VI.41] pointed out that in order to do a careful analysis of the solvability of an equation one must start by agreeing on what numbers count as “rational.” So, for example, he pointed out that in ABEL’S [VI.33] theorem on the unsolvability of the quintic, “rational” meant “expressible as a quotient of polynomials in the symbols used as the coefficients of the equation,” and he noted that the set of all such expressions obeyed the usual rules of arithmetic.

In the eighteenth century, Johann Lambert had established that e and π were irrational, and conjectured that in fact they were transcendental, that is, that they were not roots of any polynomial equation. Even the existence of transcendental numbers was not known at the time; LIOUVILLE [VI.39] proved that such numbers exist in 1844. Within a few decades, it was proved that both e and π were transcendental, and later in the century CANTOR [VI.54] showed that in fact the vast majority of real numbers were transcendental. Cantor’s discovery highlighted, for the first time, that the system Stevin had popularized contained unexpected depths.

Perhaps the most important change in the concept of number, however, came after HAMILTON’S [VI.37] discovery, in 1843, of a completely new number system. Hamilton had noticed that coordinatizing the plane using complex numbers (rather than simply using pairs of real numbers) vastly simplified plane geometry. He set out to find a similar way to parametrize three-dimensional space. This turned out to be impossible, but led Hamilton to a four-dimensional system, which he called the QUATERNIONS [III.76]. These behaved much like numbers, with one crucial difference: multiplication was not commutative, that is, if q and q’ are quaternions, qq' and q'q are usually not the same.

The quaternions were the first system of “hypercomplex numbers,” and their appearance generated lots of new questions. Were there other such systems? What counts as a number system? If certain “numbers” can fail to satisfy the commutative law, can we make numbers that break other rules?

In the long run, this intellectual ferment led mathematicians to let go of the vague notion of “number” or “quantity” and to hold on, instead, to the more formal notion of an algebraic structure. Each of the number systems, in the end, is simply a set of entities on which we can do operations. What makes them interesting is that we can use them to parametrize, or coordinatize, systems that interest us. The whole numbers (or integers, to give them their 1atinized formal name), for example, formalize the notion of counting, while the real numbers parametrize the line and serve as the basis for geometry.

By the beginning of the twentieth century, there were many well-known number systems. The integers had pride of place, followed by a nested hierarchy consisting of the rational numbers (i.e., the fractions), the real numbers (Stevin’s decimals, now carefully formalized), and the complex numbers. Still more general than the complex numbers were the quaternions. But these were by no means the only systems around. Number theorists worked with several different fields of algebraic numbers, subsets of the complex numbers that could be understood as autonomous systems. Galois had introduced finite systems that obeyed the usual rules of arithmetic, which we now call finite fields. Function theorists worked with fields of functions; they certainly did not think of these as numbers, but their analogy to number systems was known and exploited.

Early in the twentieth century, Kurt Hensel introduced the p-adic numbers [III.51], which were built from the rational numbers by giving a special role to a prime number p. (Since p can be chosen at will, Hensel in fact created infinitely many new number systems.) These too “obeyed the usual rules of arithmetic,” in the sense that addition and multiplication behaved as expected; in modern language, they were fields. The p-adics provided the first system of things that were recognizably numbers but that had no visible relation to the real or complex numbers—apart from the fact that both systems contained the rational numbers. As a result, they led Ernst Steinitz to create an abstract theory of fields.

The move to abstraction that appears in Steinitz’s work had also occurred in other parts of mathematics, most notably the theory of groups and their representations and the theory of algebraic numbers. All of these theories were brought together into conceptual unity by NOETHER [VI.76], whose program came to be known as “abstract algebra.” This left numbers behind completely, focusing instead on the abstract structure of sets with operations.

Today, it is no longer that easy to decide what counts as a “number.” The objects from the original sequence of “integer, rational, real, and complex” are certainly numbers, but so are the p-adics. The quaternions are rarely referred to as “numbers,” on the other hand, though they can be used to coordinatize certain mathematical notions. In fact, even stranger systems can show up as coordinates, such as Cayley’s OCTONIONS [III.76]. In the end, whatever serves to parametrize or coordinatize the problem at hand is what we use. If the requisite system turns out not to exist yet, well, one just has to invent it.

Further Reading

Berlinghoff, W. P., and F. Q. Gouvêa. 2004. Math through the Ages: A Gentle History for Teachers and Others, expanded edn. Farmington, ME/Washington, DC: Oxton House/The Mathematical Association of America.

Ebbingaus, H.-D., et al. 1991. Numbers. New York: Springer.

Fauvel, J., and J. J. Gray, eds. 1987. The History of Mathematics: A Reader. Basingstoke: Macmillan.

Fowler, D. 1985. 400 years of decimal fractions. Mathematics Teaching 110:20–21.

———. 1999. The Mathematics of Plato’s Academy, 2nd edn. Oxford: Oxford University Press.

Gouvêa, F. Q. 2003. p-adic Numbers: An Introduction, 2nd edn. New York: Springer.

Katz, V. J. 1998. A History of Mathematics, 2nd edn. Reading, MA: Addison-Wesley.

———, ed. 2007. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton, NJ: Princeton University Press.

Mazur, B. 2002. Imagining Numbers (Particularly the Square Root of Minus Fifteen). New York: Farrar, Straus, and Giroux.

Menninger, K. 1992. Number Words and Number Symbols: A Cultural History of Numbers. New York: Dover. (Translated by P. Broneer from the revised German edition of 1957/58: Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl. Göttingen: Vandenhoeck und Ruprecht.)

Reid, C. 2006. From Zero to Infinity: What Makes Numbers Interesting. Natick, MA: A. K. Peters.