Page 278 - Ganit Kaushal

10.9 HISTORICAL GLIMPSE (Brahmagupta’s Rules of Addition & Subtraction)

The first general treatment of positive numbers, negative numbers, and zero, all on 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

A1. The sum of two positives is positive. [e.g., 2 + 3 = 5, 9 + 7 = 16]
A2. The sum of two negatives is negative. To add two negatives, add the numbers (without the signs), and
then place a minus sign to obtain the result.
[e.g., (–2) + (–3) = –5, (–7) + (–6) = –13, (–12) + (–13) = –25]
A3. To add a positive number and a negative number, subtract the smaller number (without the sign) from
the greater number (without the sign), and place the sign of the greater number to obtain the result.
[e.g., – 5 + 3 = – 2, 2 + (– 3) = – 1 and – 3 + 5 = +2]
A4. The sum of a number and its inverse is zero. [e.g., 2 + (–2) = 0, (–3) + 3 = 0]
A5. The sum of any number and zero is the same number. [e.g., – 2 + 0 = – 2 and 0 + 3 = 3]

Brahmagupta’s Rules for Subtraction

S1. If a smaller positive is subtracted from a larger positive, the result is positive.
[e.g., 3 – 2 = 1, 13 – 5 = 8, 33 – 15 = 18]

S2. If a larger positive is subtracted from a smaller Remark
positive, the result is negative. To do so, subtract the Once you understand Brahmagupta’s
smaller from the larger and put the negative sign. rules, you can do addition and subtraction
[e.g., 2 – 3 = –1, 6 – 11 = −5, 21 – 33 = −12] with any numbers whatsoever – positive,

S3. Subtracting a negative number is the same as adding negative, and zero! If the numbers used
the corresponding positive number. in addition and subtraction are large,
then it is always better to do it by using
[e.g., 2 – (– 3) = 2 + 3 = 5, 6 – (−11) = 6 + 11 = 17] Brahmagupta’s rules instead of doing it
S4. Subtracting a number from itself gives zero. using the number line.
[e.g., 2 – 2 = 0 and – 3 – (– 3) = 0]
S5. Subtracting zero from a number gives the same number. [e.g., – 2 – 0 = – 2 and 0 – 0 = 0]
S6. Subtracting a number from zero gives the number’s inverse. [e.g., 0 – (–2) = 2, 0 − (3) = −3]

Ex 1. Use Brahmagupta’s rules to solve the following.
(a) (−7) + (−8) + (−90) (b) 50 − (−40) – (−2)
(c) 280 + (− 130) – 96 (d) –10025 + 149
(e) 2884 + (−2884) (f) −522 + (−160)
Sol. (a) (−7) + (−8) + (−90) = − 7 – 8 − 90 = −15 − 90 [ −7 − 8 = −15, Rule A2]

= −105 [By rule A2]

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