The ten digits you use every day, 0 to 9, and the idea that the position of a digit determines its value, came from India. Between roughly the 5th and 12th centuries CE, Indian mathematicians not only perfected the decimal place-value system but also treated zero as a number, worked confidently with negatives and solved equations that stumped Europe for another thousand years.
Early Roots
Mathematics in South Asia is very old. The Sulba Sutras, composed roughly between 800 and 200 BCE, gave rules for building fire altars of precise shapes and sizes. They contain a clear statement of what we call Pythagoras' theorem and a remarkably accurate approximation for \( \sqrt{2} \):
$$ \sqrt{2} \approx 1 + \frac{1}{3} + \frac{1}{3 \cdot 4} - \frac{1}{3 \cdot 4 \cdot 34} = \frac{577}{408} \approx 1.414216 $$
The Brahmi numerals, used from around the 3rd century BCE, had separate symbols for 1 to 9. Over the following centuries these evolved into the digits we use now, and by the early centuries CE a place-value system was developing.
The Bakhshali manuscript, found in 1881 near Peshawar in present-day Pakistan, uses a dot as a placeholder for zero. Radiocarbon tests published in 2017 gave a range of dates for its birch-bark leaves, from around the 3rd to the 10th century CE; how to interpret these dates is still debated by historians.
The Decimal Place-Value System with Zero
Earlier systems, such as the Babylonian, had place value but no consistent zero; others, such as Roman numerals, had neither. The Indian system combined three ideas:
- Only ten symbols, 0 to 9.
- Place value in base 10: in 305, the 3 means three hundreds.
- Zero as a placeholder, so 305 cannot be confused with 35.
With these, any whole number can be written compactly, and arithmetic becomes a matter of simple column methods. A dated inscription from 876 CE at a temple in Gwalior, central India, shows the numbers 270 and 50 written with a round zero, one of the earliest clearly dated examples.
Aryabhata (476–around 550 CE)
Aryabhata worked at Kusumapura, near Pataliputra (modern Patna). In 499 CE, aged just 23, he completed the Aryabhatiya, a compact work in 121 verses covering arithmetic, algebra, trigonometry and astronomy. Among its achievements:
- He stated that the Earth rotates on its axis, explaining the daily motion of the stars, an idea many later astronomers rejected.
- He gave a table of sine differences, a forerunner of trigonometric tables. Our word "sine" descends, through Arabic and Latin mistranslations, from the Sanskrit jya, meaning "bowstring".
- He gave methods for solving linear equations in whole numbers, called kuttaka ("the pulveriser").
Worked Example 1: Aryabhata's value of pi
Aryabhata wrote, in verse: "Add 4 to 100, multiply by 8, and add 62,000. This is approximately the circumference of a circle whose diameter is 20,000."
Step 1: Add 4 to 100:
$$ 100 + 4 = 104 $$
Step 2: Multiply by 8:
$$ 104 \times 8 = 832 $$
Step 3: Add 62,000:
$$ 832 + 62000 = 62832 $$
Step 4: Since circumference \( = \pi \times \) diameter,
$$ \pi \approx \frac{62832}{20000} = 3.1416 $$
This is correct to four decimal places (the true value is 3.14159...). Notice that Aryabhata called it approximate (asanna), suggesting he knew pi could not be expressed exactly.
Brahmagupta (598–around 668 CE)
Brahmagupta lived in Bhillamala (modern Bhinmal, Rajasthan) and in 628 CE wrote the Brahmasphutasiddhanta. It contains the earliest known rules for arithmetic with zero and negative numbers, which he called "fortunes" (positives) and "debts" (negatives):
- A debt minus zero is a debt; a fortune minus zero is a fortune.
- Zero minus zero is zero.
- The product of a debt and a fortune is a debt.
- The product of two debts is a fortune.
- A number multiplied by zero is zero.
His one mistake was claiming that \( 0 \div 0 = 0 \); today we leave division by zero undefined. Even so, treating zero as a number in its own right was a giant conceptual step.
Brahmagupta also gave the general solution of the quadratic equation in words, and studied equations of the form \( Nx^2 + 1 = y^2 \), now (misleadingly) called Pell's equation. He posed the case \( 92x^2 + 1 = y^2 \), reportedly remarking that anyone who solved it within a year was a true mathematician. Its smallest solution is \( x = 120, y = 1151 \). Check: \( 92 \times 14400 + 1 = 1324801 = 1151^2 \).
Worked Example 2: Brahmagupta's formula for a cyclic quadrilateral
Brahmagupta discovered that a quadrilateral whose vertices lie on a circle (a cyclic quadrilateral) with sides \( a, b, c, d \) has area
$$ A = \sqrt{(s-a)(s-b)(s-c)(s-d)}, \qquad s = \frac{a+b+c+d}{2} $$
Find the area of a cyclic quadrilateral with sides 7, 24, 20 and 15.
Step 1: Semi-perimeter:
$$ s = \frac{7 + 24 + 20 + 15}{2} = \frac{66}{2} = 33 $$
Step 2: The four factors:
$$ s - 7 = 26, \quad s - 24 = 9, \quad s - 20 = 13, \quad s - 15 = 18 $$
Step 3: Multiply:
$$ 26 \times 9 \times 13 \times 18 = 234 \times 234 = 54756 $$
Step 4: Square root:
$$ A = \sqrt{54756} = 234 $$
Check: This quadrilateral can be split along a diameter of length 25 into two right triangles, 7-24-25 and 15-20-25 (since \( 7^2 + 24^2 = 625 = 15^2 + 20^2 \)). Their areas are \( \tfrac{1}{2}(7)(24) = 84 \) and \( \tfrac{1}{2}(15)(20) = 150 \), totalling 234. Brahmagupta's formula agrees perfectly.
Later Giants: Bhaskara II and Kerala
Bhaskara II (1114–around 1185) wrote the Lilavati, a beloved arithmetic textbook written as problems addressed to a girl named Lilavati, and gave a general method, the chakravala, for Pell's equation. From the 14th century, Madhava of Sangamagrama and the Kerala school discovered infinite series for sine, cosine and pi, such as
$$ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots $$
about 250 years before these series appeared in Europe.
How the System Reached the World
Indian astronomical texts were translated into Arabic in Baghdad in the 8th century. Around 825 CE, al-Khwarizmi wrote a book on calculating with Hindu numerals, later translated into Latin. In 1202, Fibonacci of Pisa promoted the "nine Indian figures" and zero in his Liber Abaci. That is why we call them Hindu-Arabic numerals.
Did You Know?
- The word "zero" comes from the Sanskrit shunya ("empty"), translated into Arabic as sifr, which also gave us the word "cipher".
- Indian scholars named very large powers of ten; the Buddhist text Lalitavistara lists number names reaching \( 10^{53} \).
- Aryabhata's name was given to India's first satellite, launched in 1975.
Key Takeaways
- India perfected the decimal place-value system with zero, the basis of all modern arithmetic.
- Aryabhata (499 CE) gave \( \pi \approx 3.1416 \) and early sine tables.
- Brahmagupta (628 CE) stated the first rules for zero and negative numbers.
- Brahmagupta's formula gives the area of a cyclic quadrilateral, and he studied Pell-type equations.
- The Kerala school discovered infinite series for pi and trigonometric functions centuries before Europe.
Whether it is quadratic equations, series or geometry, Amathify's subject experts and PhD scholars offer step-by-step solutions and assignment help, so contact us on WhatsApp or through the Ask a Question form.
Stuck on a problem or assignment?
Our experts and PhD scholars solve maths, physics and chemistry — step by step.