Category : 7th Class
Integers
N= {1, 2.3, 4, …….}
W= {0, 1, 2, 3 …...}
1, 2, 3, 4, .... etc., are called positive integers and are denoted by\[{{Z}^{+}}\].
\[\therefore \]\[{{Z}^{+}}\]= {1, 2, 3, 4, .......}
-1, - 2, -3, - 4, ...... etc., are called negative integers and are denoted by\[{{Z}^{-}}\].
\[\therefore \]\[{{Z}^{-}}\]= {..............-3,-2,-1}
Note: 1.Usually negative numbers are placed in brackets to avoid confusion arising due to two signs I evaluations. e.g., 3+ (-5) =-2
(i) To represent quantities like profit, income, increase, rise, high, north, east, above, depositing, climbing and so on, positive numbers are used.
(ii) To represent quantities like loss, expenditure, decrease, fall, low, south, west, below, withdrawing, sliding and so on, negative numbers are used.
Note: 1. 0 is neither positive nor negative.
3.\[\frac{1}{2}\] and 0.3 are not integers as they are not whole numbers.
On a number line when we
(a) Add a positive integer, we move to the right.
(b) Add a negative integer, we move to the left.
(c) Subtract a positive integer, we move to the left.
(d) Subtract a negative integer, we move to the right.
(i) Closure property: Closure property is satisfied with respect to addition, subtraction and multiplication in the set of integers.
For a, b e Z, a + b e Z, a - b \[\in \] Z and a b\[\in \] Z.
(ii) Commutative property: Commutative property is satisfied with respect to addition and multiplication in the set of integers.
If a, b \[\in \]Z, then a + b = b + a and a \[\times \] b = b \[\times \] a.
If a, b, c \[\in \] Z, then
a + (b + c) = (a + b) + c = c + (b + a) and a \[\times \] (b \[\times \] c) = (a \[\times \] b) \[\times \] c = c \[\times \] (b \[\times \] a).
For a, b and c e Z, a (b + c) = ab + ac and a (b - c) = ab - ac.
For a e Z, a+0=a=0+a and a\[\times \]1 = a = 1 \[\times \] a.
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