Hindu-Arabic Numerals

The Hindu-Arabic numeral system is the way of writing numbers with the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, in which the value of a digit depends on its place. It was developed in India and reached Europe through Arabic-speaking scholars, which is why it is called the Hindu-Arabic or Indo-Arabic numeral system (both names mean the same) and, in everyday speech, simply “Arabic numerals.” Today it is the most widely used number system in the world.

The page explains the system and its place value, shows how to read Roman numerals as Hindu-Arabic numbers, follows the numerals from the first Indian inscriptions to Fibonacci’s Liber abaci, tells the history of zero and compares the international system of grouping digits (million, billion) with the Indian one (lakh, crore).

What are Hindu-Arabic numerals? The ten digits and place value

With ten symbols you can write every number, however large. The trick is place value: each place is worth ten times the place to its right. The number 3,205 means 3 thousands, 2 hundreds, 0 tens and 5 ones. The 0 is a placeholder that keeps the tens place empty, so that 3,205 is not mistaken for 325. On the number systems page you can see the same idea with other bases such as 2, 8 and 16.

Compare this with Roman numerals, where X always means 10, wherever it stands, or with Egyptian numerals, which have a separate picture for each power of ten. Those systems need many symbols for large numbers and are clumsy for written arithmetic. In the Hindu-Arabic system you add, subtract, multiply and divide on paper with the same few rules for any size of number. That is the main reason it won.

The digits in other scripts

The ten digits exist in several scripts. The ones we use in Europe and America are called Western Arabic numerals. In many Arabic-speaking countries in the east, people write the Eastern Arabic-Indic digits, and in India the Devanagari digits are in everyday use. All three are shapes of the same system.

Roman numerals to Hindu-Arabic numerals

School tasks such as “write the Hindu-Arabic numerals” simply ask you to turn Roman numerals into ordinary numbers. Read from left to right, subtract a smaller symbol that stands before a larger one, and add everything else. XXIX is 10 + 10 + (10 − 1) = 29, and MCMXCIX is 1,000 + 900 + 90 + 9 = 1,999. The full rules and a chart from 1 to 100 are on the Roman numerals page.

History of the Hindu-Arabic numerals

1. about 250 BC The Brahmi numerals appear on inscriptions of the emperor Ashoka. They are the ancestors of our digits, but they are not yet a place value system and have no zero.
2. first centuries AD In India a place value system with ten digits develops. The Bakhshali manuscript uses a dot for an empty place. Its age is disputed: radiocarbon dating at Oxford in 2017 gave the 3rd or 4th century for one leaf and later dates for others, so it is a collection of different layers.
3. 594 A donation charter of king Dadda III is the oldest dated Indian document in place value notation, although some historians think the date was added later.
4. 628 Brahmagupta writes the first rules for calculating with zero as a number.
5. about 825 Al-Khwarizmi in Baghdad describes calculating with the Indian digits. The word “algorithm” comes from his name.
6. 876 An inscription in Gwalior shows a round zero in the numbers 270 and 50. It is the first dated Indian place value inscription with zero that nobody disputes.
7. 976 The Codex Vigilanus, written in northern Spain, is one of the earliest European manuscripts with the new digits, still without zero.
8. 1202 Leonardo of Pisa (Fibonacci) publishes the Liber abaci, which opens with the nine Indian figures and the sign for zero.
9. 1518 and 1522 Adam Ries publishes arithmetic books in German that teach calculating with the digits to a wide public.

The details of the oldest dates are debated, so treat the earliest ones as ranges. What is solid is the sequence: Brahmi digits first, then place value in India, then transfer to the Arabic-speaking world, then to Europe. The scholar al-Biruni, who visited India in the 1020s, wrote that the Arabs used the most regular of the Indian figures.

The history of zero

Zero has two jobs, and they were not discovered at the same time. First it was a placeholder, a mark that says “nothing in this place.” Later it became a number with rules: 3 + 0 = 3 and 3 · 0 = 0.

• Babylon: For over a thousand years the scribes left a gap. From about 400 BC a special sign marked an empty place, usually in the middle of a number.
• Maya: A shell-shaped sign for zero appears in inscriptions and books, used in the calendar and as a placeholder. See Mayan numbers.
• India: The placeholder became a number. Brahmagupta gave the rules in 628, though he made one error: he set 0 divided by 0 equal to 0.
• Europe: Fibonacci introduced the sign in 1202 but treated it as a “sign,” not as a number like the others. The Arabic word sifr became both “cipher” and, by way of Italian, “zero.” It took until the 17th century for zero to be fully accepted as a number.

Roman numerals never had a zero, as the section on the Roman numeral for zero explains.

How the numerals reached Europe

The road ran through Baghdad, where scholars translated Indian and Greek works in the House of Wisdom, and through the Arabic-speaking parts of Spain and North Africa. On the way the shapes of the digits changed; our 2, 3, 4 and 5 look quite different from the Brahmi signs. The ancestors of the European digits were used in North Africa and Spain.

Fibonacci spent his youth in North Africa, where his father worked as a merchant, and learned the Indian method there. His Liber abaci of 1202 taught merchants to calculate with it, with many practical problems about money and goods. One of them, about rabbits, produced the Fibonacci sequence.

Acceptance was slow. In 1299 the guild of money changers in Florence forbade the use of the new numerals in official account books. The reason given was fraud, since a 0 is easy to turn into a 6 or a 9. According to historians the ban applied to published records, not to the calculating itself. Until about 1550, accounts in many parts of Europe were still kept in Roman numerals. Printed arithmetic books, such as those of Adam Ries in the 16th century, finally made the digits common property.

International and Indian number system

Large numbers are named and grouped differently around the world. In the international system, used in the United States, the United Kingdom and most of the world, digits are grouped in threes from the right: 123,456,789 is read “one hundred twenty-three million, four hundred fifty-six thousand, seven hundred eighty-nine.” The names are thousand, million, billion and trillion, where a billion is a thousand million.

In the Indian system, used in India, Pakistan, Bangladesh, Nepal and Sri Lanka, the last three digits form one group and all further digits are grouped in twos: the same number is written 12,34,56,789. The units are lakh (100,000) and crore (10,000,000), and 100 lakh make one crore. So 25,000,000 is 2,50,00,000, read “two crore fifty lakh.”

The Indian names continue every two powers of ten: lakh (10⁵), crore (10⁷), arab (10⁹) and kharab (10¹¹). Only the grouping and the names differ, not the digits and not the place value.

For other number words, see how counting habits are hidden in languages in the history of number systems: French quatre-vingts for 80 and Danish halvtreds for 50 are leftovers of counting in twenties.

Number systems compared

Practice problems with answers

Frequently asked questions

Sources and further reading

• MacTutor History of Mathematics: Indian numerals
• MacTutor History of Mathematics: A history of Zero
• Encyclopaedia Britannica: Hindu-Arabic numerals and Liber abaci
• Bodleian Libraries: Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol ‘zero’ (14 September 2017)
• Norman Biggs: Mathematics of currency and exchange: arithmetic at the end of the thirteenth century, BSHM Bulletin 24 (2009), pp. 67–77, and the Gresham College lecture of 2008.
• Wikipedia: Indian numbering system and Wikipedia: Billion
• Georges Ifrah: The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley, New York 2000.