Following are the properties of addition :
(i) Commutative property
(ii) Associative property
(iii) Identity property
(iv) Property of additive inverse
(v) Distributive property
(vi) Property of equality
Two real numbers can be added in any order.
In other words, addition is commutative for real numbers.
We have 8 + (- 3) = 5 and (- 3) + 8 = 5
So,
8 + (- 3) = (- 3) + 8
In general, for any two real numbers a and b we can say,
a + b = b + a
Therefore addition of real numbers is commutative.
Observe the following example :
Consider the real numbers 5, – 4 and 7.
Look at
5 + [(– 4) + 7] = 5 + 3 = 8
and
[5 + (– 4)] + 7 = 1 + 7 = 8
Therefore,
5 + [(– 4) + 7] = [5 + (– 4)] + 7
In general, for any real numbers a, b and c, we can say,
a + (b + c) = (a + b) + c.
Therefore addition of real numbers is associative.
When we add zero to any real number, we get the same real number.
Observe the examples :
5 + 0 = 5
0 + 7 = 7
-2 + 0 = -2
0 + (-3) = -3
In general, for any real number a,
a + 0 = a
0 + a = a
Therefore, zero is the additive identity for real numbers.
The sum of a real number and its additive inverse is zero.
3 + (-3) = (-3) + 3 = 0
Note :
(i) If a number is positive, then its additive inverse is the same number with negative sign.
(ii) If a number is negative, then its additive inverse is the same number with positive sign.
Examples :
-5 is the additive inverse of 5
7 is the additive inverse of -7
Consider the real numbers 12, 9, 7.
Look at
12 x (9 + 7) = 12 x 16 = 192
12 x (9 + 7) = 12 x 9 + 12 x 7 = 108 + 84 = 192
Thus
12 x (9 + 7) = (12 x 9) + (12 x 7)
In general, for any real numbers a, b, c.
a x (b + c) = (a x b) + (a x c)
Therefore, multiplication is distributive over addition.
Two sides of an equation remain equal, if the same number is added to each side.
For example, if x = y, then
x + z = y + z
Example :
3 = 3
Add 5 to each side.
3 + 5 = 3 + 5
8 = 8
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