📚 Math in Society
⇩ Download ▾
Coming soon

14.3 The Hindu-Arabic Number System

The Evolution of a System

Our own number system, composed of the ten symbols {0,1,2,3,4,5,6,7,8,9} is called the Hindu-Arabic system. This is a base-ten (decimal) system since place values increase by powers of ten. Furthermore, this system is positional, which means that the position of a symbol has bearing on the value of that symbol within the number. For example, the position of the symbol 3 in the number 435,681 gives it a value much greater than the value of the symbol 8 in that same number. We’ll explore base systems more thoroughly later. The development of these ten symbols and their use in a positional system comes to us primarily from India.[i]

NUMBER = 5783216         # EDIT this and re-run

names = {0: "ones", 1: "tens", 2: "hundreds", 3: "thousands", 4: "ten thousands",
         5: "hundred thousands", 6: "millions", 7: "ten millions"}
digits = str(NUMBER)
print(f"{NUMBER:,} taken apart by position\n")
print(f"{'digit':>6}{'x place value':>18}{'as a power':>14}{'contributes':>16}   place name")
total = 0
for i, ch in enumerate(digits):
    p = len(digits) - 1 - i
    value = int(ch) * 10 ** p
    total += value
    print(f"{ch:>6}{'x ' + format(10 ** p, ','):>18}{'= ' + str(int(ch)) + ' x 10^' + str(p):>14}"
          f"{value:>16,}   {names.get(p, '10^' + str(p))}")
print(f"{'':>6}{'':>18}{'total':>14}{total:>16,}    match? {total == NUMBER}")

print("\nthe book's point, made with 435,681:")
demo = "435681"
for target in ("3", "8"):
    i = demo.index(target)
    p = len(demo) - 1 - i
    print(f"   the symbol {target} sits in the {names[p]} place, so it is worth {int(target) * 10 ** p:,}")
a = int("3") * 10 ** (len(demo) - 1 - demo.index("3"))
b = int("8") * 10 ** (len(demo) - 1 - demo.index("8"))
print(f"   the 3 is worth {a // b} times as much as the 8, even though 3 < 8")
print("\nA non-positional system (tally marks, Roman numerals) cannot do this:")
print("   the symbol carries its whole value with it, so long numbers need new symbols.")

Small portrait of the writer al-Biruni, a bearded man in a large white turban, shown from the shoulders up facing the viewer.It was not until the 15th century that the symbols that we are familiar with today first took form in Europe. However, the history of these numbers and their development goes back hundreds of years. One important source of information on this topic is the writer al-Biruni, whose picture is shown here.[ii] Al-Biruni, who was born in modern day Uzbekistan, had visited India on several occasions and made comments on the Indian number system. When we look at the origins of the numbers that al-Biruni encountered, we have to go back to the third century B.C.E. to explore their origins. It is then that the Brahmi numerals were being used.

The Brahmi numerals were more complicated than those used in our own modern system. They had separate symbols for the numbers 1 through 9, as well as distinct symbols for 10, 100, 1000,…, also for 20, 30, 40,…, and others for 200, 300, 400, …, 900. The Brahmi symbols for 1, 2, and 3 are shown below.[iii]

Table of the three earliest Brahmi numerals, captioned Brahmi one, two, three: one is a single horizontal stroke, two is two stacked horizontal strokes, three is three stacked horizontal strokes.

These numerals were used all the way up to the 4th century C.E., with variations through time and geographic location. For example, in the first century C.E., one particular set of Brahmi numerals took on the following form[iv]:

Table of first century Brahmi numerals for 1 through 9, each modern digit above its Brahmi form: one, two and three are one, two and three stacked horizontal strokes; four is a plus sign; five resembles a lower case h; six is a small looped squiggle; seven resembles a question mark hook; eight is a flattened S lying on its side; nine is a hooked stroke like a comma with a tall stem.

From the 4th century on, you can actually trace several different paths that the Brahmi numerals took to get to different points and incarnations. One of those paths led to our current numeral system, and went through what are called the Gupta numerals. The Gupta numerals were prominent during a time ruled by the Gupta dynasty and were spread throughout that empire as they conquered lands during the 4th through 6th centuries. They have the following form[v]:

Table of the fourth to sixth century Gupta numerals for 1 through 9. One, two and three are still one, two and three horizontal strokes, but the strokes are joined by cursive ties; four through nine are curved pen forms, a looped y for four, an upright with a side hook for five, a large double loop for six, a hooked eta shape for seven, an S curve for eight and a squared backwards three for nine.

How the numbers got to their Gupta form is open to considerable debate. Many possible hypotheses have been offered, most of which boil down to two basic types[vi]. The first type of hypothesis states that the numerals came from the initial letters of the names of the numbers. This is not uncommon…the Greek numerals developed in this manner. The second type of hypothesis states that they were derived from some earlier number system. However, there are other hypotheses that are offered, one of which is by the researcher Ifrah. His theory is that there were originally nine numerals, each represented by a corresponding number of vertical lines. One possibility is this:[vii]

Table illustrating Ifrah's hypothesis that the nine numerals began as groups of vertical strokes: one, two and three are a single row of one, two and three strokes; four is two rows of two; five is a row of three above a row of two; six is two rows of three; seven is a row of four above a row of three; eight is two rows of four; and nine is three rows of three.

Because these symbols would have taken a lot of time to write, they eventually evolved into cursive symbols that could be written more quickly. If we compare these to the Gupta numerals above, we can try to see how that evolutionary process might have taken place, but our imagination would be just about all we would have to depend upon since we do not know exactly how the process unfolded.

The Gupta numerals eventually evolved into another form of numerals called the Nagari numerals, and these continued to evolve until the 11th century, at which time they looked like this:[viii]

Table of eleventh century Nagari numerals for 1 through 9 and, for the first time in this sequence, 0. One resembles a lower case q, two and three are close to the modern symbols written with a curled tail, four resembles a figure eight, five resembles a modern 4, six is a looped form with a descender, seven resembles a filled comma or 9, eight resembles a small flat-topped bracket, nine resembles a lower case e, and zero is a small filled ring.

Note that by this time, the symbol for 0 has appeared! The Mayans in the Americas had a symbol for zero long before this, however, as we shall see later in the chapter.

These numerals were adopted by the Arabs, most likely in the eighth century during Islamic incursions into the northern part of India.[ix] It is believed that the Arabs were instrumental in spreading them to other parts of the world, including Spain (see below).

Other examples of variations up to the eleventh century include:

Devangari, eighth century[x]:

Row of nine eighth century Devanagari numerals for 1 through 9, handwritten in ink with no labels: a hooked stroke for one, then forms close to a modern 2 and 3 but with looped tails, a crossed loop like a figure eight for four, a modern-looking 4 for five, an S shape for six, a modern-looking 7, a shape like a less-than sign for eight and a curled epsilon for nine.

West Arab Gobar, tenth century[xi]:

Row of nine tenth century West Arab Gobar numerals for 1 through 9, handwritten with no labels: a plain vertical stroke for one, a modern-looking 2, a crossed z for three, a low flourish for four, a form like a 9 with a straight tail for five, then shapes very close to the modern 6, 7, 8 and 9.

Spain, 976 C.E.[xii]:

Row of nine numerals for 1 through 9 as written in Spain in 976 C.E., handwritten with no labels: a plain vertical stroke for one, a looped 2, a doubled loop for three, a crossed flourish for four, a form like a modern 4 for five, a squared hook for six, then shapes close to the modern 7, 8 and 9.

Finally, one more graphic[xiii] shows various forms of these numerals as they developed and eventually converged to the 15th century in Europe.

Flow chart of the descent of our numerals, each stage shown as a boxed row of nine or ten handwritten symbols split into a group of five and a group of four or five. Brahmi numerals at the top lead down to Indian (Gvalior), which leads down to Sanskrit-Devanagari (Indian) and also branches left to West Arabic (gubar) and right to East Arabic (still used in Turkey). Below those sit two more boxed rows, 15th century on the left and 16th century (Durer) on the right, whose symbols are recognisably our own 1 to 9 and 0.

The Positional System

More important than the form of the number symbols is the development of the place value system. Although it is in slight dispute, the earliest known document in which the Indian system displays a positional system dates back to 346 C.E. However, some evidence suggests that they may have actually developed a positional system as far back as the first century C.E.

The Indians were not the first to use a positional system. The Babylonians used a positional system with 60 as their base. However, there is not much evidence that the Babylonian system had much impact on later numeral systems, except with the Greeks. Also, the Chinese had a base-10 system, probably derived from the use of a counting board[xiv]. Some believe that the positional system used in India was derived from the Chinese system.

Wherever it may have originated, it appears that around 600 C.E., the Indians abandoned the use of symbols for numbers higher than nine and began to use our familiar system where the position of the symbol determines its overall value.[xv] Numerous documents from the seventh century demonstrate the use of this positional system.

Interestingly, the earliest dated inscriptions using the system with a symbol for zero come from Cambodia. In 683, the 605th year of the Saka era is written with three digits and a dot in the middle. The 608th year uses three digits with a modern 0 in the middle.[xvi] The dot as a symbol for zero also appears in a Chinese work (Chiu-chih li). The author of this document gives a strikingly clear description of how the Indian system works:

Using the [Indian] numerals, multiplication and division are carried out. Each numeral is written in one stroke. When a number is counted to ten, it is advanced into the higher place. In each vacant place a dot is always put. Thus the numeral is always denoted in each place. Accordingly there can be no error in determining the place. With the numerals, calculations is easy…” [xvii]

Transmission to Europe

It is not completely known how the system got transmitted to Europe. Traders and travelers of the Mediterranean coast may have carried it there. It is found in a tenth-century Spanish manuscript and may have been introduced to Spain by the Arabs, who invaded the region in 711 C.E. and were there until 1492.

In many societies, a division formed between those who used numbers and calculation for practical, every day business and those who used them for ritualistic purposes or for state business.[xviii] The former might often use older systems while the latter were inclined to use the newer, more elite written numbers. Competition between the two groups arose and continued for quite some time.

Woodcut of the contest between the abacist and the algorist. Two men sit at counting tables under a Gothic arch with the robed figure of Arithmetic standing between them, numerals running down her gown and a banner overhead. The man on the left works with a pen, Hindu-Arabic numerals written across his tabletop; the man on the right pushes round counters along the ruled horizontal lines of a counting board.In a 14th century manuscript of Boethius’ The Consolations of Philosophy, there appears a well-known drawing of two mathematicians. One is a merchant and is using an abacus (the “abacist”). The other is a Pythagorean philosopher (the “algorist”) using his “sacred” numbers. They are in a competition that is being judged by the goddess of number. By 1500 C.E., however, the newer symbols and system had won out and has persevered until today. The Seattle Times recently reported that the Hindu-Arabic numeral system has been included in the book The Greatest Inventions of the Past 2000 Years.[xix]

One question to answer is why the Indians would develop such a positional notation. Unfortunately, an answer to that question is not currently known. Some suggest that the system has its origins with the Chinese counting boards. These boards were portable and it is thought that Chinese travelers who passed through India took their boards with them and ignited an idea in Indian mathematics.[xx] Others, such as G. G. Joseph propose that it is the Indian fascination with very large numbers that drove them to develop a system whereby these kinds of big numbers could easily be written down. In this theory, the system developed entirely within the Indian mathematical framework without considerable influence from other civilizations.

[i] www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_numerals.html

[ii] www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Al-Biruni.html

[iii] www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_numerals.html

[iv] www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_numerals.html

[v] Ibid

[vi] Ibid

[vii] Ibid

[viii] Ibid

[ix] Katz, page 230

[x] Burton, David M., History of Mathematics, An Introduction, p. 254-255

[xi] Ibid

[xii] Ibid

[xiii] Katz, page 231.

[xiv] Ibid, page 230

[xv] Ibid, page 231.

[xvi] Ibid, page 232.

[xvii] Ibid, page 232.

[xviii] McLeish, p. 18

[xix] seattletimes.nwsource.com/news/health-science/html98/invs_20000201.html, Seattle Times, Feb. 1, 2000

[xx] Ibid, page 232.

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.

These eBooks are a prerelease and are not yet certified conformant with WCAG 2.1 AA or ADA Title II. Every page is built against an automated accessibility gate, and the published editions will meet ADA Title II requirements when they release in late September 2026. If something is unusable, please tell us.