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PROPERTIES OF ADDITION ON INTEGERS EXAMPLE 14: (i) The successor of 19 is (19 + 1) = 20.
(ii) The successor of -18 is (-18 + 1) = -17.
(iii) The predecessor of 10 is (10 - 1) = 9.
(i) CLOSURE PROPERTY OF ADDITION The sum of two integers is always an integer. (iv) The predecessor of -20 is (-20 - 1) = - 21.

EXAMPLE 9: (i) 3 + 5 = 8, and 8 is an integer. EXAMPLE 15: Find the sum of -8, 23, -32, -17 and -63.
(ii) 3 + (-8) = - 5, and -5 is an integer.
(iii) (-3) + (-9) = -12, and -12 is an integer. SOLUTION: (-8) + 23 + (-32) + (-17) + (-63)
(iv) 16 + (-7) = 9, and 9 is an integer. = [(-8) + 23] + (-32) + [(-17) + (-63)]
= [15 + (-32) ] + (-80) = (-17) + (-80) = - 97.

(ii) COMMUTATIVE LAW OF ADDITION if a and b are any two integers then a + b = b + a. EXAMPLE 16: Find an integer a such that
(i) 2 + a = 0 (ii) a + (-6) = 0
EXAMPLE 10 : (i) (-3) + 8 = 5, and 8 + (-3) = 5. SOLUTION: (i) 2 + a = 0 => (-2) + [2 +a] = (-2) + 0 [adding (-2) on both sides]

:. (-3)+ 8=8+(-3). => [(-2) + 2] + a= - 2 [by associatlve law of additlon and property of 0]
=> 0 + a= - 21·: (-2) + 2 = 0]
(ii) (-4) + (-6) = -10, and (-6) + (-4) = -10. => a= -2.
Hence, a = - 2.
(iii) ASSOCIATIVE LAW OF ADDITION if a, b, care any three integers then(a + b) + c = a+ (b + c).
(ii) a + (-6) = 0
EXAMPLE 11: Consider the integers -5, -7 and 3. => [a + (-6) + 6 = 0 + 6 [adding 6 on both sides]
=> a + [(-6) + 6] = 6 [by associative law of addition and property of O l
We have: [(-5) + (-7) ) + 3 = (-12) + 3 = -9. => a + 0 = 6 [·: (-6) + 6 = O]
And, (-5) + 1(-7) + 31 = (-5) + (-4) = - 9. => a = 6.
:. [(-5) + (-7)) + 3 = (-5) + [(-7) + 3).
Hence, a = 6.
(iv) if a is any integer then a + 0 = a and O + a = a. SUBTRACTION OF INTEGERS

EXAMPLE 12: (i) 8 + 0 = 8 (ii) (-3) + 0 = -3 (iii) 0 + (-5) = -5 We have learnt how to subtract two whole numbers.

Remark 0 is called the additive identity. We defined subtraction as an inverse process of addition. For example, to subtract 4 from 9 is the same as to find
a number which when added to 4 gtves 9.
(v) The sum of an integer and its opposite ts 0.
Clearly, the answer is 5.
Thus, if a ts an integer then a+ (-a)= 0. Thus, 9 - 4 = 5.
a and -a are called opposites or negatives or additive inverses of each other.
We extend the same idea to subtraction of integers. Suppose we want to subtract (-4) from 6.
EXAMPLE 13: 3 + (-3) = 0 and (-3) + 3 = 0. Clearly, we want a number which when added to (-4) gtves 6.
Now, on the number line, find out how many steps should be taken from -4 to reach 6.
Thus, the additive inverse of 3 is -3.
And, the additive inverse of -3 is 3.
Remark Clearly, the additive inverse of 0 is 0. -5 -4 -3 -2 -1 0 1 2 3 4 5 6

SUCCESSOR AND PREDECESSOR OF AN INTEGER We see that the number of steps taken is 10.
Let a be an integer. :. 6 - (-4) = 10.
Then, (a+ 1) is called the successor of a. Also, we know that 6 + 4 = 10.
And, (a - 1) is called the predecessor of a. Thus, 6 - (-4) = 6 + 4 = 10.