Page 24 - Start Up Mathematics_7

(d) Existence of additive identity: Since a + 0 = a = 0 + a for every integer a, ∴ 0 is the
additive identity.
(e) Existence of additive inverse: If a + (–a) = 0 = (–a) + a, then –a is called the additive
inverse of a.
4. Subtraction: If a and b are any integers, we define a – b = n iff a = b + n, also a – b = a + (–b)
Properties
Closure: a – b is again an integer.
Commutative and associative properties are not true for subtraction in integers.
5. Multiplication: For any integers a and b, we define (–a)(–b) = ab and (–a)(b) = (a)(–b) = –(ab)
Properties
(a) Closure: a × b is again an integer. (b) Commutative: a × b = b × a

(c) Associative: (a × b) × c = a × (b × c) (d) Distributive: a × (b + c) = (a × b) + (a × c)
(e) Existence of multiplicative identity: Since a × 1 = a = 1 × a for every integer a, ∴ 1 is
the multiplicative identity.
6. Division: If a and b ≠ 0 are any integers, then a is divisible by b, i.e., a ÷ b, if there exists a
unique integer c such that a = bc.
(a) For any integer a, 0 ÷ a = 0 and a ÷ 0 is not defined.
(b) The properties, i.e., closure, commutative and associative are not true for integers.
7. Order of operations: When division, multiplication, addition and subtraction appear in
an expression without brackets, multiplication and division are done first in order of their
appearance from left to right followed by addition and subtraction in order of their appearance
from left to right. Any calculation in parenthesis is done first or one can follow the order given
in BODMAS.

Integrating Fun and Facts

The Barrel Operators
Given below are the mathematical barrels which are capable of performing the fundamental
operations marked on them. When a number is put from the top as input, the operation marked
on the label is performed and the answer comes out of the outlet as output. Series (a) is solved
as an illustration. Complete series (b) and (c).
18

(a) +12 ÷(–3) ×5 +53 –2

30 –10 –50 3 1

2

(b) ×1 +2 –2 ×(–2) ÷(–2)

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