Real numbers have unique properties which make them particularly useful in everyday life.

First , Real numbers are an ordered set of numbers. This means real numbers are sequential. The numerical value of every real number fits between the numerical values two other real numbers.

Everyone is familiar with this idea since all measurements (weight, the purchasing power of money, the speed of a car, etc.) depend upon the fact that some numbers have a higher value than other numbers. Ten is greater than five, and five is greater than four . . . and so on.

Second , we never run out of real numbers. The quantity of real numbers available is not fixed. There are an infinite number of values available. The availability of numbers expands without end. Real numbers are not simply a finite "row of separate points" on a number line. There is always another real number whose value falls between any two real numbers (this is called the "density" property).

Third , when real numbers are added or multiplied, the result is always another real number (this is called the "closure" property). [This is not the case with all arithmetic operations. For example, the square root of a -1 yields an imaginary number.]

With these three points in mind, the question is: How can we use real numbers in practical calculations? What rules apply?

The following properties of real numbers answers these types of questions. The property characteristics which follow show how much latitude you have to change the mechanics of calculations which use real numbers without changing the results.

• Associative Property
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• Commutative Property
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• Distributive Property
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• Identity Property
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• Inverse Property

Distributive Property . . .

5 + 6

This problem would ordinarily be written as expression which contains parentheses:

3 * ( 5 + 6 )

To evaluate the expression, you could add the 5 and the 6 together first, and then multiply by 3

5 + 6 = 11

3 * 11 = 33

However, you can also multiply each of the two numbers by 3 separately, and then add the results together.

3 * 5 = 15

3 * 6 = 18

15 + 18 = 33

The results are the same either way.

Simply put, the Distributive Property states that it makes no difference whether you:

- multiply 3 times each number individually, then add the result

or

- add the two numbers together, then multiply the result by 3

The Distributive Law states that the following expressions are equivalent to one another:

3 * ( 5 + 6 ) = 3 * 5 + 3 * 6 = 33

Mental Math - the Distributive Law simplifies calculations

Knowing that the two expressions shown above are equivalent is very useful when multiplying numbers using mental math.

For example, multiply the following two numbers using mental math:

5 * 723

Using the Distributive Law, this number 723 can be arbitrarily spit into parts, and re-written as shown below. This is called Decomposing the Multiplier .

5 * 723 = 5 * ( 700 + 20 + 3 )

The multiplication process can be completed as follows:

5 * ( 700 + 20 + 3 ) = 5 * 700 + 5 * 20 + 5 * 3

= 3500 + 100 + 15

= 3615

Algebra - the Distributive Law helps simplify algebraic expressions

The Distributive Law is even more important in Algebra. The values of the variables used in Algebra are unknown. The Distributive Law allows you to simplify an algebraic expression without knowing the actual values represented by the variables.

The Distributive Law can be stated using algebraic notation as follows:

a * ( b + c ) = a * b + a * c

If more than three variables are used, the Distributive Law can be stated:

a * ( b + c + d + e + ... + z ) = a * b + a * c + a * d + a * e + ... + a * z

Example Problems in Algebra using the Distributive Law :

Example Problems Using the Horizontal Format :

Example 1:

3 ( x + 9 ) = 3 * ( x + 9 )

= 3 * x + 3 * 9

= 3x + 27

Example 2:

r ( x + 9 ) = r * ( x + 9 )

= r * x + r * 9

= rx + 9r

Example Problems Using a Multiplication Table :

Example 1:

3 ( x + 9 )

(+3) ( + x + 9 )

Multiplication Table

(+3) ( + x + 9 )

= [ (+3) * (+x) ] + [ (+3) * (+9) ]

= 3x + 27

Example 2:

( x + 2 ) ( x - 5 )

( + x + 2 ) ( + x - 5 )

Multiplication Table

= [ x * x ] - [ x * 5 ] + [ 2 * x ] - [ 2 * 5 ]

= x² - 5x + 2x - 10

= x² + [- 5x + 2x] - 10

= x² - 3x - 10

Example Problems Using the Vertical Format :

Example 1:

( x + 2 ) ( x - 5 )

( + x + 2 ) ( + x - 5 )

Example 2:

( x + 2 ) ( x² + x - 5 )

( + x + 2 ) ( + x² +  x  −  5 )