Zero: How an Indian Idea Became the World’s Most Powerful Number

Zero feels so obvious that it is hard to see how strange it was. How can nothing be a number? Why did great civilisations with advanced mathematics manage without it? This guide follows the idea from empty spaces on Babylonian clay tablets to Brahmagupta’s rules in India, across the Arab world to Fibonacci’s Italy, and finally into the binary code inside every computer.

Why zero is hard

Counting begins with things you can see: three goats, five stones. Nobody counts “zero goats” except to say that there are none. So early number systems had no need for a symbol for nothing. The difficulty only appears when you want to write large numbers compactly, and need to show that a place is empty.

Roman numerals are a good example. XXIV, MMXXVI and CCC are built by adding symbols together, and a place never needs to be marked empty. This works for recording numbers but is terrible for calculating: try multiplying CLXVII by XXIII on paper. Positional systems, where the position of a digit tells you its value, make arithmetic easy, but they need a way to mark an empty place. That is what zero first did.

Try it: the place-value explorer

Type any whole number up to nine digits. The explorer breaks it into place values, shows what happens to the number if the zeros are simply dropped, and prints it in binary, the base that computers use.

Try 3007, then 307, then 37. The digits 3 and 7 are the same each time, but the number is completely different. The zeros do the work of saying “nothing in the hundreds place” and “nothing in the tens place”.

Placeholders before the number

The Babylonians wrote numbers in base 60 on clay tablets from about 2000 BCE, with a gap to show an empty place. A gap is ambiguous, so later scribes began to use a mark of two small wedges as a placeholder, from around 300 BCE. The Maya of Central America independently developed a shell-shaped zero for their calendar system. In both cases zero marked an empty position, but it was not treated as a number you could add, subtract or multiply.

India: zero becomes a number

In India the placeholder took a different path. A dot appears in the Bakhshali manuscript, a collection of mathematical text on birch bark, found in 1881 in a village near Peshawar, in what is now Pakistan. Radiocarbon dating done at the University of Oxford in 2017 showed that the manuscript’s 70 leaves come from at least three different periods, with the earliest material dating to as early as the 3rd or 4th century CE, centuries earlier than scholars had thought. In this manuscript the dot marks orders of magnitude such as tens, hundreds and thousands, a placeholder, not yet a number.

The leap to treating zero as a number is credited to Indian mathematicians. In 628 CE the astronomer-mathematician Brahmagupta, working at Bhinmal in Rajasthan, wrote the Brahmasphutasiddhanta. In it he defined zero as the result of subtracting a number from itself and set out rules: adding zero to a number leaves it unchanged, and multiplying by zero gives zero. He also dealt with negative numbers as debts. Division by zero defeated him, as it would others for centuries.

The oldest zero carved in stone is at the Chaturbhuj temple in Gwalior, in an inscription dated 876 CE, where a small circle appears as part of the number 270. Placed in a temple inscription for all to see, it shows that the symbol was in everyday use by then.

From shunya to zero

The Sanskrit word for empty or void is shunya. Arabic scholars translated it as sifr, meaning empty, which gave us both “cipher” and, through Italian zefiro and then zero, our word zero. When the Italian merchant and mathematician Leonardo of Pisa, known as Fibonacci, published Liber Abaci in 1202, he told European readers that the nine Indian figures together with the sign 0 could write any number. Adoption was slow. Some cities, including Florence in 1299, restricted the new digits in commercial records, suspecting that a 0 could be altered into a 6 or 9. But the efficiency of written arithmetic won out.

Place value in Python

n = 2026
digits = list(str(n))
for i, d in enumerate(reversed(digits)):
    print(f"10^{i}: digit {d} -> {int(d) * 10**i}")
print(bin(n))

Output:

10^0: digit 6 -> 6
10^1: digit 2 -> 20
10^2: digit 0 -> 0
10^3: digit 2 -> 2000
0b11111101010

The zero in the hundreds place contributes nothing but keeps the 2 in the thousands place. Without it, you would read 226. The last line shows the same number in binary, where every digit is a 0 or a 1.

The trouble with dividing by zero

Brahmagupta’s difficulty with division by zero is not a flaw of his era but a real puzzle. Division asks “how many times does this fit in that”. Divide 1 by smaller and smaller numbers and the answer gets bigger and bigger. Slide the divisor toward zero and watch.

There is no number you can name for “1 divided by 0”, so mathematicians leave it undefined. Calculus later made the idea of “approaching zero” precise with the idea of limits.

Zero inside every computer

Digital computers store everything as strings of 0s and 1s: numbers, text, pictures and programs. The binary system that Leibniz described in 1703 is a place-value system just like decimal, only with two digits instead of ten, so it depends on zero as much as our everyday numbers do. If you want to follow that story, read our guide to the transistor, the tiny switch that stores a 0 or a 1.

Mind map: zero at a glance

Common myths

• Myth: Zero was invented by one person on one day. It grew through several cultures and centuries.
• Myth: The Maya and the Babylonians had zero as a number. They used it as a placeholder; treating it as a number came in India.
• Myth: Roman numerals had a zero. They did not need one, as they are not a place-value system.
• Myth: Zero is neither odd nor even. Zero is an even number, because it is divisible by two.

Frequently asked questions

Keep exploring

Sources

• Zero (mathematics), Encyclopaedia Britannica
• Indian mathematics, Encyclopaedia Britannica
• Brahma-sphuta-siddhanta, Encyclopaedia Britannica
• Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol zero, University of Oxford, Bodleian Libraries
• Chaturbhuj Temple, Gwalior, Wikipedia

Featured image: Image: Augustus Hoernle (1841-1918), via Wikimedia Commons, Public domain. Dates for early events are approximate.