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The Identity Property
of
Addition and Multiplication
. . .
keys to solving equations and adding fractions

Identity 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
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Identity Property



The concept of individual identity can be applied in a variety of interesting ways. People working in different fields of study have developed their own specialized systems to identify and catalog what is important to them. Examples of these differences can be seen in how individual identity is used in biological identification (DNA), music, psychology, archaeology, architecture, etc.

However, the examples used here will focus exclusively on numbers .



Identity: - an example is shown below


A personal driver's license number is one form of identity.

Your driver's license number allows you to be recognized as a separate driver who is different from all other drivers. Your driver's license number is a distinct and unique number which is yours alone.

However, your diver's license number is also part of a large group of license numbers. It is only one number within a large group of numbers (called a set).




Identity Property: - an example is shown below


An identity property only applies to a group of numbers as a whole (a set). It does not apply to individual numbers within the set.


The identity property can be illustrated using the "clock" analogy as follows:

Math - Example of Identity Property

The clock (shown above) cannot display all real numbers. The clock can only display hours, minutes, and seconds up to and including 12 hours.

The set of numbers used by the clock is: all times up to 12 hours.



Identity

The time displayed by the clock has an identity. It is 2:15. 2:15 is distinct and unique from all the other times which could be displayed by the clock (am and pm are not shown the a 12 hour clock used in this example).

Changes to Identity

The time on the clock will change.

One hour later, the time on the clock will say 3:15. 3:15 has a completely different identity than 2:15.



Identity Property (The Identity Property is a property which applies to the entire set of numbers which can be displayed by the clock. The identity property does not apply to individual numbers by themselves.)

Of all the times (numbers) that can be displayed by the clock, the number 12 hours has a unique quality. If 12 hours is added to any time shown on the clock, the clock continues to show the original time. (The original time does not loose its identity.)

The time currently displayed on the clock is 2:15. However, if 12 hours is added to 2:15, the clock will continue to show 2:15.

The number 12 hours is called an identity element (also called a neutral element). Every number on the clock remains unchanged whenever 12 hours is added to it.

Because the number 12 hours has this unique quality, the set of numbers used by the clock possesses an Identity Property of addition.



Identity Properties for the set of all Real Numbers -


An additive identity is a number that can be added to any number without changing the value of that other number.

The additive identity for the set of all real numbers is 0 (zero). The number 0 can be added to any real number without changing its value.


Example 1: (positive integers) 2 + 0 = 2


Example 2: (negative numbers) -2 + 0 = -2


Example 3: (fractions) ¾ + 0 = ¾


Example 4: (decimals) 1.1 + 0 = 1.1


Example 5: (algebraic notation) a + 0 = a



A multiplicative identity is a number that can be multiplied by any number without changing the value of that other number.

The multiplicative identity for the set of all real numbers is 1 (one). Any real number can be multiplied by the number 1 without changing its value.


Example 1: (positive integers) 2 * 1 = 2


Example 2: (negative numbers) -2 * 1 = -2


Example 3: (fractions) ¾ * 1 = ¾


Example 4: (decimals) 1.1 * 1 = 1.1


Example 5: (algebraic notation) a * 1 = a




The Multiplicative Identity of 1 is the key to adding fractions with different denominators:

Example 1: add the following two fractions

Math - Fraction + Math - Fraction


(1) Using the Multiplicative Identity of 1, multiply each fraction by 1.


Math - Adding Fractions using the Multiplicative Identity
Note:
Multiplying a fraction by 1
does not change the value
of the fraction:

Math - fraction multiplied by 1 is still Math - fraction
Math - fraction multiplied by 1 is still Math - fraction
(2) Replace the Multiplicative Identity of 1, with a fraction. Select the fraction as follows:

  • The fraction must equal 1. The numerator and
    denominator of the fraction must be the same.

    Examples of the type of fractions which can be
    used are:

    • Math - Fraction


  • Choose fractions which will produce common
    denominators     when multiplied with the original
    fractions.

    For this example those fractions can be:

    • Math - Adding Fractions using the Multiplicative Identity



Math - Adding Fractions using the Multiplicative Identity
Note:

Math - Fraction multiplied by Math - Fraction is still Math - Fraction


Math - Fraction multiplied by Math - Fraction is still Math - Fraction

(3) Complete the Multiplication of the two groups of fractions. Note that after the multiplication is complete, the denominators of both of the remaining fractions are the same:
Math - Adding Fractions using the Multiplicative Identity
Note:

The denominators of both of the remaining fractions are the same.
(4) Add the Fractions. The two remaining fractions can now be
added since their denominators are the same:
Math - Adding Fractions using the Multiplicative Identity
(5) The Final Answer:
Math - Adding Fractions using the Multiplicative Identity

Example 2: add the following two fractions

Math - Adding Fractions using the Multiplicative Identity + Math - Adding Fractions using the Multiplicative Identity


(1) Using the Multiplicative Identity of 1, multiply each fraction by 1.


Math - Adding Fractions using the Multiplicative Identity
Note:

Multiplying a fraction by 1
does not change the value
of the fraction:

Math - Fraction multiplied by 1 is still Math - Fraction
Math - Fraction multiplied by 1 is still Math - Fraction
(2) Replace the Multiplicative Identity of 1, with a fraction. Select the fraction as follows:

  • The fraction must equal 1. The numerator and
    denominator of the fraction must be the same.

    Examples of the type of fractions which can be
    used are:

    • Math - Adding Fractions using the Multiplicative Identity


  • Choose fractions which will produce common
    denominators     when multiplied with the original
    fractions.

    For this example those fractions can be:

    • Math - Adding Fractions using the Multiplicative Identity



Math - Adding Fractions using the Multiplicative Identity
Note:

Math - Fraction multiplied by Math - Fraction is still Math - Fraction

Math - Fraction multiplied by Math - Fraction is still Math - Fraction

(3) Complete the Multiplication of the two groups of fractions. Note that after the multiplication is complete, the denominators of both of the remaining fractions are the same:
Math - Adding Fractions using the Multiplicative Identity
Note:

The denominators of both of the remaining fractions are the same.
(4) Add the Fractions. The two remaining fractions can now be
added since their denominators are the same:
Math - Adding Fractions using the Multiplicative Identity
(5) The Final Answer:
Math - Adding Fractions using the Multiplicative Identity


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