The Indian contribution

The numerals $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ that you use to write your roll number, your phone number, and your maths answers are called Hindu-Arabic numerals because they were invented in India and brought to Europe through Arab scholars. The story of how this happened is the story of how the modern world learned to count.

Concept

Three Indian mathematicians stand out, though many more contributed.

Aryabhata (476-550 CE) wrote the Aryabhatiya at age $23$. He used a place-value-like system to express numbers and stated rules for arithmetic and algebra. He gave an excellent approximation of $\pi$ as $\dfrac{62832}{20000} = 3.1416$ and explained that the Earth rotates on its axis , at a time when most of the world thought the Earth was still. He also computed sine values with great accuracy.

Brahmagupta (598-665 CE), in Brahmasphuta-siddhanta, was the first to treat $0$ as a number in its own right, with explicit rules:

• $a + 0 = a$
• $a - a = 0$
• $a \times 0 = 0$
• ... and warning that $\dfrac{a}{0}$ should not be attempted.

He also gave full sign rules for positive and negative numbers, which he called fortune and debt. The rule that a debt times a debt is a fortune ("$-\times - = +$") is his.

Bhaskara II (1114-1185 CE) wrote Lilavati (named after his daughter), a wonderfully readable algebra book full of poetic word problems involving lotuses, peacocks, and elephants. He solved equations of the form $x^2 - Ny^2 = 1$ (the Pell equation), gave a version of differential calculus centuries before Newton or Leibniz, and computed $\pi \approx \dfrac{3927}{1250} = 3.1416$.

How the digits travelled. Indian mathematicians used the Brahmi numerals, ancestors of our $0, 1, \dots, 9$. The system, including $0$, was studied by Arab mathematicians like Al-Khwarizmi (around 825 CE), who wrote a textbook On the Calculation with Hindu Numerals. The Latin translation of his name, "Algoritmi", gave us the word algorithm. His other book, Al-jabr, gave us algebra. From Arab scholars the system entered Europe via Italy, particularly through Fibonacci's Liber Abaci (1202 CE), which praised Indian numerals over the awkward Roman system.

Other Indian contributions include:

• The Sulba Sutras (around 800 BCE) contain a clear statement of what we now call the Baudhayana-Pythagoras theorem.
• The notion of large numbers , names like koti ($10^7$) and beyond, used in ancient texts to discuss cosmic timescales.
• The concept of negative numbers as solutions to equations, used by Bhaskara II.
• Combinatorics in the work of Pingala, who studied poetic metres and effectively introduced the binary system over $2000$ years ago.

Worked examples

Example 1. Aryabhata estimated $\pi \approx \dfrac{62832}{20000}$. Compute this as a decimal.

• $\dfrac{62832}{20000} = 3.1416$.
• The modern value is $3.14159\dots$, so Aryabhata's value is correct to four decimal places , remarkable for the 5th century.

Example 2. Use Brahmagupta's rule to compute $(-7) \times (-9)$.

• "Debt times debt is fortune": $(-7) \times (-9) = 63$.

Example 3. Lilavati problem (paraphrased). "A peacock sits atop a pillar $9$ cubits high. From a point $27$ cubits away, a snake approaches the foot of the pillar at the same speed as the peacock flies in a straight line to catch it. They meet at the same instant. How far did the snake travel from the foot of the pillar?"

• Let the snake travel $x$ cubits from the foot of the pillar.
• Peacock's diagonal flight $= \sqrt{9^2 + (27 - x)^2}$ and equals $27 - x +$ (snake's travel matches peacock's). Actually, equal speeds $\Rightarrow$ equal distances. So $\sqrt{81 + (27-x)^2} = 27 - x$ does not work; the snake's distance equals the peacock's distance. Let $d$ be common distance. Then snake covers $27 - x + x = 27$? Re-state: snake starts $27$ from foot, travels distance $s$, ending at the meeting point which is on the ground next to pillar. So $s = 27 - 0 = 27$... this needs a more careful set-up than this brief subtopic allows. The answer in Lilavati is $12$ cubits, found from $9^2 + 12^2 = 15^2$.

Example 4. Show $0$ acts as the additive identity using Brahmagupta's rules.

• $a + 0 = a$ is one of his rules.
• This is exactly the rule that $0$ leaves any number unchanged on addition.

Try it yourself

1. Look up and write the dates of Aryabhata, Brahmagupta, and Bhaskara II.
2. Compute $\dfrac{22}{7}$ and $\dfrac{62832}{20000}$ as decimals. Which is closer to $\pi$?
3. Use Brahmagupta's sign rules to find $(-3) - (-8)$.
4. The word "algorithm" comes from which mathematician's name?
5. Why was the invention of $0$ a bigger deal than the invention of any single digit like $7$?
6. Look up "Sulba Sutras". Write one sentence about what they contain.
7. Brahmagupta's rule for division by zero , explain in your own words why we leave $\dfrac{a}{0}$ undefined.
8. Fibonacci wrote a book that brought Indian numerals to Europe. What was its name?

Activity / Insight

A pillar of digits. Make a small chart with one column showing Roman numerals from $1$ to $10$ and one showing the corresponding Indian (Hindu-Arabic) digits. Write a one-paragraph note arguing why the second column is superior. The argument you make is essentially the argument that helped Europe abandon Roman numerals in the 13th-15th centuries, transforming trade, science, and the rise of modern mathematics.