Bijaganita: algebra in ancient India

What does bijaganita mean?

Writing numbers with symbols and solving equations was a big part of mathematics in ancient India. This branch was called bijaganita, which we now call algebra. Bija means seed. A seed holds a whole tree inside it, and an unknown number holds the answer to a problem; solving the problem is like letting the tree grow, one step at a time.

Brahmagupta and the unknowns

In Chapter 18 of his book Brahmasphutasiddhanta (628 CE), Brahmagupta showed how unknown numbers written as symbols can be added, subtracted and multiplied, much as we use x and y today. It is one of the earliest known works on algebra anywhere. The mathematician David Mumford, a winner of the Fields Medal, called Brahmagupta the central figure in creating algebra as we know it.

How the ideas travelled

During the 8th century, works of Indian mathematics were put into Arabic, and they shaped the thinking of Al-Khwarizmi, who lived in what is now Iraq. Around 825 CE he wrote Hisab al-jabr wal-muqabala, meaning “the calculation of restoring and balancing”. In the 12th century his book was translated into Latin and became popular in Europe, and al-jabr turned into the word algebra that we still use.

The old Indian notation

From Brahmagupta’s time, Indian mathematicians used short syllables for unknowns. ya was short for yavat-tavat, “as much as”, and was used for the first unknown. ka and ni came from the colour names kalaka (black) and nilaka (blue), and pi and lo were used too. (Extra: these two are usually explained as pitaka, yellow, and lohita, red.) A known number was marked ru, from rupa, form.

The sign came first in each term, so 2x + 1 was written ya 2 ru 1. A dot over a number showed it was negative, so 2x − 8 was ya 2 ru 8 with a dot over the 8. In an equation, the two sides were written one below the other instead of with an equals sign.

Brahmagupta’s rule

A systematic way of solving problems with one unknown was first given by Aryabhata (499 CE), and Brahmagupta set it out in 628 CE. For any equation of the form Ax + B = Cx + D, the answer is

x = (D − B) ÷ (A − C).

Why? Take Cx and B away from both sides: Ax − Cx = D − B, that is (A − C)x = D − B, and divide by A − C. For 5x + 4 = 3x + 8 the rule gives (8 − 4) ÷ (5 − 3) = 2. For the savings equation 650m + 4000 = 500m + 5050 it gives (5050 − 4000) ÷ (650 − 500) = 7 straight away.

Bhaskaracharya’s horse problem

Bhaskaracharya wrote the book Bijaganita in 1150 CE. One of its problems: a man has ₹300 and 6 horses; another has 10 horses but owes ₹100. If they are equally rich and every horse costs the same, what is the price of a horse? Let a horse cost ₹x. Then 300 + 6x = 10x − 100. Adding 100 to both sides gives 400 + 6x = 10x, and taking 6x from both sides gives 400 = 4x, so x = 100. A horse costs ₹100.

A sharing problem from the Bakhshali Manuscript

The Bakhshali Manuscript, which the chapter dates to 300 CE, has a sharing puzzle. The first person gets an unknown amount, the second twice as much, the third three times the second and the fourth four times the third, and 132 is shared in all. With a for the first share, the shares are a, 2a, 6a and 24a, so 33a = 132 and a = 4.

Why it matters

Indian mathematicians turned hard ideas into simple procedures that anyone could follow, and algebra still lets us describe patterns, prove claims such as “odd + odd is even” and solve problems of many kinds. Practise the methods in solving simple equations and word problems on equations. Answers to the textbook questions are in the NCERT solutions for Part 2 Chapter 7; see all of Part 2 Chapter 7 notes.