Basic Number Properties

The basic properties of real numbers are based on a straightforward concept. You could even call it “common sense” math because no complicated analysis is required. Real numbers have four (4) fundamental properties: commutative, associative, identity, and distributive. These properties are only applicable to addition and multiplication operations.

I. Commutative Property

For Addition

The sum of two or more real numbers is always the same regardless of the order in which they are added. In other words, real numbers can be added in any order because the sum remains the same.

Examples:

a) [latex]a + b = b + a[/latex]

b) [latex]5 + 7 = 7 + 5[/latex]

c) [latex]{}^ – 4 + 3 = 3 + {}^ – 4[/latex]

d) [latex]1 + 2 + 3 = 3 + 2 + 1[/latex]

For Multiplication

The product of two or more real numbers is not affected by the order in which they are being multiplied. In other words, real numbers can be multiplied in any order because the product remains the same.

Examples:

a) [latex]a \times b = b \times a[/latex]

b) [latex]9 \times 2 = 2 \times 9[/latex]

c) [latex]\left( { – 1} \right)\left( 5 \right) = \left( 5 \right)\left( { – 1} \right)[/latex]

d) [latex]m \times {}^ – 7 = {}^ – 7 \times m[/latex]

II. Associative Property

For Addition

The sum of two or more real numbers is always the same regardless of how you group them. When you add real numbers, any change in their grouping does not affect the sum.

Examples:

the quantity a plus b added to c is equal to a added to the quantity b plus c

For Multiplication

The product of two or more real numbers is always the same regardless of how you group them. When you multiply real numbers, any change in their grouping does not affect the product.

Examples:

the quantity a times b multiplied to c is equal to a times to the quantity b times c

III. Identity Property

For Addition

Any real number added to zero (0) is equal to the number itself. Zero is the additive identity since [latex]a + 0 = a[/latex] or [latex]0 + a = a[/latex]. You must show that it works both ways!

Examples:

fifteen plus zero is equal to fifteen and negative one plus zero is negative one

For Multiplication

Any real number multiplied by one (1) is equal to the number itself. The number one is the multiplicative identity since [latex]a \times 1 = a[/latex] or [latex]1 \times a = 1[/latex]. You must show that it works both ways!

Examples:

seven times one is seven while negative five times one is negative five

IV. Distributive Property of Multiplication over Addition

Multiplication distributes over Addition

Adding two or more real numbers and multiplying the sum to an outside number, is the same as multiplying the outside number to every number inside the parenthesis, then adding their products.

Examples:

a)   

a times the quantity b plus c is equal to the quantity a times b plus the quantity a times c which can also be written as (a)(b+c) = +.

b)   

(3)(4+5) = + --> (3)(9)=(12)+(15) --> 27=27

c)   

(-2){} = + -->(-2)(4) = 2+(-10) --> -8 = -8

The following is the summary of the properties of real numbers as discussed above:

This is a table showing all the Basic Properties of Numbers. Firstly, the Commutative Property of both addition and multiplication, respectively: a+b=b+a and (a)(b)=(b)(a). Secondly, the Associative Property of both addition and multiplication, respectively: (a+b)+c=a+(b+c) and (a*b)c=a(b*c). Identity Property of addition and multiplication: a+0=a and (a)(1)=a. Finally, the Distributive Property of Multiplication of Addition which is a(b+c)= (a*b)+(a*c).