Brahmagupta's rules

The Indian mathematician Brahmagupta, in his book the Brahma-sphuta-siddhanta, written in the year 628 CE, was the first to treat positive numbers, negative numbers and zero all as equally good numbers. He gave clear rules for working with them, which we still use today.

His rules for addition:

• (1) The sum of two positives is positive: 2 + 3 = 5.
• (2) The sum of two negatives is negative: add the numbers without their signs, then put a minus sign in front: (-2) + (-3) = -5.
• (3) To add a positive number and a negative number, subtract the smaller number (without its sign) from the greater number (without its sign), and give the answer the sign of the greater number: -5 + 3 = -2, 2 + (-3) = -1 and -3 + 5 = 2.
• (4) The sum of a number and its inverse is zero: 2 + (-2) = 0.
• (5) A number plus zero is the same number: -2 + 0 = -2.

A short way to remember rules 2 and 3: same signs, add and keep the sign; different signs, subtract and keep the sign of the bigger one.

His rules for subtraction:

• A smaller positive taken from a larger positive gives a positive (3 - 2 = 1); a larger positive taken from a smaller positive gives a negative (2 - 3 = -1).
• Subtracting a negative number is the same as adding the corresponding positive number: 2 - (-3) = 2 + 3 = 5.
• A number minus itself is zero (2 - 2 = 0, -2 - (-2) = 0).
• Subtracting zero changes nothing (-2 - 0 = -2), and subtracting a number from zero gives its inverse (0 - (-2) = 2).

With these rules you can say what kind of answer to expect. (positive) - (negative) is always positive, because subtracting a negative is adding a positive, and positive + positive is positive. (negative) + (negative) is always negative, and so is (negative) - (positive), which is (negative) + (negative). But (positive) + (negative) can be positive or negative (or zero): it depends on which number is greater without its sign.