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The Associative Property
of
Addition and Multiplication




Associative Property - Properties of Real Numbers : 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?
  • How should numbers be added, subtracted, multiplied, and divided? What latitude do we have?


  • Does it matter what we do first? second? third? . . .


  • Can we add a series of numbers together in any order? Will the final answer be the same regardless of the order we choose?


  • Can we multiply a series of numbers together in any order? Will the final answer be the same regardless of the order we choose?

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


  • Commutative Property


  • Distributive Property


  • Identity Property


  • Inverse Property

The Seven Fundamental Properties of Real Numbers - click description Return to Math Properties Page Return to Math Properties Page Return to Math Properties Page To See All "Math Properties" click here




Math - Associative Property
Math - Commutative Property
Math - Distributive Property
Math - Identity Property
Math - Inverse Property
Math - Closure Property & Density Property
Return to Top of Page Return To "Top of Page" click here Associative Property - Addition and Multiplication
(Note that the Associative Property only applies to addition and
multiplication. It does not apply to other operations in arithmetic.)

Associative Property of Addition:

Any series of addends can be grouped in any way and added in any order without changing the results. This does apply to all real numbers (including: fractions, decimals, and negative numbers).

Example 1: (positive integers) 1 + (5 + 9) = (1 + 5) + 9 = (1 + 9) + 5 Example 2: (negative numbers) 1 + ([-5] + 9) = (1 + [-5]) + 9 = (1 + 9) + [-5] Example 3: (fractions) ¾ + (½ + ) = (¾ + ½) + = (¾ + ) + ½ Example 4: (decimals) 1.1 + (0.0055 + 0.09) = (1.1 + 0.0055) + 9 = (1.1 + 0.09) + 0.0055 Example 5: (algebraic notation) a + (b + c) = (a + b) + c = (a + c) + b

Associative Property of Multiplication:

Any series of factors can be grouped in any way and multiplied in any order without changing the results. This does apply to all real numbers (including: fractions, decimals, and negative numbers).

Example 1: (positive integers) 2 * (5 * 9) = (2 * 5) * 9 = (2 * 9) * 5 Example 2: (negative numbers) 2 * ([-5] * 9) = (2 * [-5]) * 9 = (2 * 9) * [-5] Example 3: (fractions) ¾ * (½ * ) = (¾ * ½) * = (¾ * ) * ½ Example 4: (decimals) 1.1 * (0.0055 * 0.09) = (1.1 * 0.0055) * 9 = (1.1 * 0.09) * 0.0055 Example 5: (algebraic notation) a * (b * c) = (a * b) * c = (a * c) * b



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