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Knowledge Desk
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 Brahma-sphuta-siddhanta in the year 628 CE.
Brahmagupta’s Rules for Addition (Brahma-sphuta-siddhanta 18.30, 628 CE):
1. The sum of two positives is positive.
2. 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.
3. 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.
4. The sum of a number and its inverse is zero.
5. The sum of any number and zero is the same number.

Example 8: Add the following:
(a) (+59) + (+ 30) (b) 86 + (–10)
(c) (–32) + 32 (d) (–200) + 101 + (–20) + 210

Solution: (a) (+59) + (+ 30) = 59 + 30 = 89 (b) (86) + (–10) = 86 – 10 = 76
(c) (–32) + 32 = –32 + 32 = 0
(d) (–200) + 101 + (–20) + 210 = (–220) + 311 = 91

Properties of Addition of Integers
(a) Closure Property: Sum of two integers is always an integer.

(i) 13 + 72 = 85 is an integer (ii) 17 + (–9) = 8 is an integer
(iii) (–7) + (–3) = –10 is an integer
(b) Commutative Property: For any two integers, the addition of integers is commutative.

(i) 4 + 3 = 3 + 4 = 7 (ii) (–5) + 8 = 3 = 8 + (–5)
(iii) (–23) + (–17) = (–40) = (–17) + (–23)
(c) Associative Property: For any three integers, the addition of integers is associative.

(i) 3 + {(–4) + (–16)} = {3 + (–4)} + (–16) = –17 (ii) (–18) + {12 + 16} = {(–18) + 12} + 16 = 10
(iii) (–9) + {(–5) + (–8)} = {(–9) + (–5)} + (–8) = –22
(d) Additive Identity: ‘0’ is an additive identity. Adding ‘0’ to an integer does not affect the integer.

(i) 8 + 0 = 8 or 0 + 8 = 8 (ii) (–29) + 0 = (–29) or 0 + (–29) = (–29)
(e) Additive Inverse: For any integer, if there exists another integer with the opposite sign such that
the sum of those two integers is zero. Then the two integers are additive inverse to each other.
(i) 23 + (–23) = 0 ⇒ 23 is the additive inverse of (–23) and vice versa.
(ii) 238 + (–238) = 0 ⇒ 238 is the additive inverse of (–238) and vice versa.

Quick Check
Identify the property in the following.

1. [12 + (–19)] + (–34) = 12 + [(–19) + (–34)] 2. (–123) + 341 = 341 + (–123)
3. (–123) + 0 = (–123) 4. (–2) + 2 = 0 = 2 + (–2)

289 The Other Side of Zero