A Pinch of History

Like general fractions, general integers (including zero and the negative numbers) were first conceived of and used in Asia, thousands of years ago, before they eventually spread across the world in more modern times.

The first known instances of the use of negative numbers occurred in the context of accounting. In one of China's most important mathematical works, The Nine Chapters on Mathematical Art (Jiuzhang Suanshu)—which was completed by the first or second century CE—positive and negative numbers were represented using red and black rods, much like the way we represented them using green and red tokens!

There was a strong culture of accountancy also in India in ancient times. The concept of credit and debit was written about extensively by Kautilya in his Arthaśhāstra (c. 300 BCE), including the recognition that an account balance could be negative. The explicit use of negative numbers in the context of accounting is seen in a number of ancient Indian works, including in the Bakśhālī manuscript from around the year 300 CE, where a negative number was written using a special symbol placed after the number (rather than before the number as we do today).

The first general treatment of positive numbers, negative numbers, and zero—all on an equal footing as equally-valid numbers on which one can perform the basic operations of addition, subtraction, multiplication, and even division—was given by Brahmagupta in his Brāhma-sphuṭa-siddhānta in the year 628 CE. Brahmagupta gave clear and explicit rules for operations on all numbers—positive, negative, and zero—that essentially formed the modern way of understanding these numbers that we still use today!

Some of Brahmagupta's key rules for addition and subtraction of positive numbers, negative numbers and zero are given below:

Brahmagupta's Rules for Addition (Brāhma-sphuṭa-siddhānta 18.30, 628 CE)

1. The sum of two positives is (e.g., 2 + 3 = 5).

2. The sum of two negatives is . To add two negatives, add the numbers (without the signs), and then place a sign to obtain the result (e.g., (– 2) + (– 3) = – 5).

3. To add a positive number and a negative number, subtract the number (without the sign) from the number (without the sign), and place the sign of the number to obtain the result (e.g., – 5 + 3 = – 2, 2 + (– 3) = – 1 and – 3 + 5 = 2).

4. The sum of a number and its inverse is (e.g., 2 + (– 2) = 0).

5. The sum of any number and zero is the (e.g., – 2 + 0 = – 2 and 0 + 0 = 0).

Brahmagupta's Rules for Subtraction (Brāhma-sphuṭa-siddhānta 18.31-18.32)

1. If a smaller positive is subtracted from a larger positive, the result is (e.g., 3 – 2 = ).

2. If a larger positive is subtracted from a smaller positive, the result is (e.g., 2– 3 = ).

3. Subtracting a negative number is the same as the corresponding positive number (e.g., 2 – (–3) = 2 + 3).

4. Subtracting a number from itself gives (e.g., 2 – 2 = 0 and – 2 – (– 2) = ).

5. Subtracting zero from a number gives the (e.g., – 2 – 0 = and 0 – 0 = ). Subtracting a number from zero gives the number's (e.g., 0 – (– 2) = ).

Once you understand Brahmagupta's rules, you can do addition and subtraction with any numbers whatsoever — positive, negative, and zero!

Brahmagupta was the first to describe zero as a number on an equal footing with positive numbers as well as with negative numbers, and the first to give explicit rules for performing arithmetic operations on all such numbers, positive, negative, and zero—forming what is now called a . It would change the way the world does mathematics.

However, it took many centuries for the rest of the world to adopt zero and negative numbers as numbers. These numbers were transmitted to, accepted by, and further studied by the Arab world by the 9th century, before making their way to Europe by the 13th century.

Surprisingly, negative numbers were still not accepted by many European mathematicians even in the 18th century. Lazare Carnot, a French mathematician in the 18th century, called negative numbers 'absurd'. But over time, zero as well as negative numbers proved to be indispensable in global mathematics and science, and are now considered to be critical numbers on an equal footing with and as important as positive numbers—just as Brahmagupta had recommended and explicitly described way back in the year 628 CE!

This abstraction of arithmetic rules on all numbers paved the way for the modern development of algebra, which we will learn about in future classes.