Properties of Real Numbers
Commutative Properties
(1) Commutative Property of Addition
The commutative property of addition states that numbers may be added in any order without affecting the sum.
If a and b are two real numbers, then
a + b = b + a
Consider an example with real numbers.
(2) Commutative Property of Multiplication
The commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.
If a and b are two real numbers, then
a ⋅ b = b ⋅ a
Consider an example with real numbers.
Note : Neither subtraction nor division is commutative.
Example : 17 – 4 ≠ 4 – 17
Similarly, 20 ÷ 4 ≠ 20 ÷ 4
Associative Properties
(1) Associative Property of Addition
The associative property of addition states that numbers may be grouped differently without affecting the sum.
If a, b and c are three real numbers, then
a + (b + c) = (a + b) + c
Consider an example with real numbers.
(2) Associative Property of Multiplication
The associative property of multiplication states that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.
If a, b and c are three real numbers, then
a (b c) = (a b) c
Consider an example with real numbers.
Note : Neither subtraction nor division is associative.
Example : 8 – (5 – 10) ≠ (8 – 5) – 10
Similarly, 20 ÷ (4 ÷ 2) ≠ (20 ÷ 4) ÷ 2
Distributive Property
The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.
In other words, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.
If a, b and c are three real numbers, then
a ⋅ (b + c) = a ⋅ b + a ⋅ c
Also, (a + b) ⋅ c = a ⋅ b + b ⋅ c
In the above example, 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.
When describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.
Identity Properties
(1) Identity Property of Addition
The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.
If a be any real numbers, then
a + 0 = a = 0 + a
For example, we have (-5) + 0 = -5 = 0 + (-5)
(2) Identity Property of Multiplication
The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.
If a be any real numbers, then
a · 1 = a = 1 · a
For example, we have (-5) · 1 = -5 = 1 · (-5)
Inverse Properties
(1) Inverse Property of Addition
The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted by −a, that, when added to the original number, results in the additive identity, 0.
If a be any real numbers, then
a + (-a) = 0 = (-a) + a
For example, if a = -5, the additive inverse is 5, since (-5) + 5 = 0.
(2) Inverse Property of Multiplication
The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal), denoted by (1/a), that, when multiplied by the original number, results in the multiplicative identity, 1.
If a be any real numbers, then
