A pinch of history

Forming expressions with symbols and solving equations was an important part of mathematical work in ancient India. The area was called Bījagaṇita, now known as algebra. The word bīja means SEED: just as a tree is hidden inside a seed, the answer is hidden inside an unknown number, and solving is like helping the tree grow.

In Chapter 18 of his Brāhmasphuṭasiddhānta (628 CE), Brahmagupta explained how to add, subtract and multiply unknown numbers using letters — much as we use x or y. It was one of the earliest known works in algebra in history. The mathematician and Fields Medalist David Mumford said, 'Brahmagupta is the key person in the creation of algebra as we know it'.

In the 8th century Indian mathematical ideas were translated into Arabic and influenced Al-Khwarizmi, who lived in present-day Iraq. Around 825 CE he wrote Hisab al-jabr wal-muqabala, meaning 'calculation by restoring and balancing'. By the 12th century it was translated into Latin and reached Europe, and the word AL-JABR from its title gave us the word ALGEBRA.

The old notation is worth seeing. Instead of letters, mathematicians of Brahmagupta's time used symbols such as yā, kā, nī, pī and lo. The symbol yā was short for yāvat-tāvat, 'as much as needed'; kā and nī came from colour names, kālaka (black) and nīlaka (blue). Known quantities had a fixed form, rūpa, and were written rū.

So 2x + 1 was written 'yā 2 rū 1', with the indication of unknown or known coming FIRST in each term. A dot over a number meant it was negative, so 2x − 8 was 'yā 2 rū 8' with a dot. And an equation like 3x + 4 = 2x + 8 was written as two lines, one below the other, rather than with an equal sign.

Example 16, from Bhāskarāchārya's Bījagaṇita (1150 CE): one man has ₹300 and 6 horses; another has 10 horses and a debt of ₹100; they are equally rich and every horse costs the same. Let a horse cost ₹x. Then 300 + 6x = 10x − 100, so 400 + 6x = 10x, so 400 = 4x, and x = 100. Each horse costs ₹100.

Finally, a FORMULA. Aryabhata (499 CE) first proposed a systematic way to solve problems with a single unknown, and Brahmagupta set it out. For any equation of the form Ax + B = Cx + D, the solution is x = (D − B) ÷ (A − C). So 650m + 4000 = 500m + 5050 is solved simply by m = (5050 − 4000) ÷ (650 − 500). Ancient Indian mathematicians were excellent at turning complex ideas into simple procedures that everyone could use.