Number Properties

Date: 01/26/98 at 21:30:47
From: Leslie Seagle
Subject: Number Properties

My daughter is trying to learn about number properties, and is having 
an extremely difficult time understanding the definitions of: closure 
for addition, commutative for addition, associative for addition, 
identity for addition, and so on all the way through multiplication.  

What we are looking for are very distinct definitions for these terms, 
in order for her to create her own examples based on the definitions.  
Could you please help with this? Anything you could provide would be 
great. 

Thanks a million, 
Leslie and Amanda

Date: 01/27/98 at 09:18:11
From: Doctor Anthony
Subject: Re: Number Properties

ADDITION
--------

(1) Closure for addition of integers

    This means that if you add two integers (whole numbers), you get 
    another integer. So, as long as you start with two integers you 
    will always end with an integer. You don't move outside integers 
    into fractions or square roots or whatever. You are 'enclosed' in 
    a universe of integers.  

(2) Commutative property for addition

    This means that the order you write down the two numbers does not 
    affect the answer. So  3 + 7 = 7 + 3  Both give the same answer: 
    10.

(3) Associative property for addition

    If we have three numbers to add, say 3 + 9 + 4, we can proceed in 
    two ways.

      (3 + 9) + 4 =  12 + 4 = 16    or

      3 + (9 + 4) =   3 + 13 = 16

    In the first situation we first 'associated' the 3 and the 9. In 
    the second situation we first 'associated' the 9 and the 4. 

(4) Identity element for addition

    The identity element leaves any other element unchanged if added.

    Clearly 0 is the identity element for addition:  5 + 0 = 5


MULTIPLICATION
--------------

(1) Closure for multiplication of integers

    Yes, if you multiply two integers you get an integer.

(2) Commutative property for multiplication 

    Yes:  3 x 4 = 4 x 3   Both = 12

(3) Associative property for multiplication

    Yes:  (3 x 4) x 5 = 3 x (4 x 5)
               12 x 5 = 3 x 20
                   60 = 60

(4) Identity element for multiplication  

    Clearly the identity element for multiplication is 1:

         5 x 1 = 5   and so on.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/