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Basic Real Number Properties


Date: 07/31/97 at 06:21:26
From: Tuesday
Subject: Basic Real Number Properties

Hi! Dr. Math,

I'm a grade seven student here in the Philippines, and we have this 
research paper and I really need more material about Basic Real Number 
Properties: associative, commutative, closure, identity, inverse, 
distributive.

Thank you,

Tuesday


Date: 08/01/97 at 18:41:29
From: Doctor Tom
Subject: Re: Basic Real Number Properties

Hello Tuesday,

To talk about these properties, you need to think not only about the
numbers themselves, but the operations you perform on them.  Let's
just start with + (addition) and x (multiplication).

The operations + and x are associative.  That means that if you
want to figure out what 3+4+5 is, you can start by adding 3 and 4
and then add 5 to that, or you can add 4 and 5 first, and then
add 3 to that.  It's usually written like this:

    (3+4)+5 = 3+(4+5)

The parentheses tell you what to do first.  This is true of all
real numbers - there is nothing special about 3, 4, and 5, so you
often see it written:

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

where a, b, and c are any real numbers.

If you replace "+" by "x", exactly the same thing is true; you
can multiply in any order.

The other way to see why it is important to recognize that this is
a property is to look for operations that are NOT associative.

Is - (subtraction) associative?  Let's check:

is (10 - 9) - 1 = 10 - (9 - 1) ?

No.  On the left, we get 0 and on the right, we get 2.

Is division associative?  Check some examples.

Think about the other properties in the same way.  + and x
are commutative. That means a+b = b+a and a*b = b*a. Subtraction
and division are not commutative.

Closure means that if you add any two real numbers you'll get a real
number. Same if you multiply two numbers. Subtraction is closed, but
division is not. You cannot divide by zero.

The identity for addition is a number that can be added to any
other number and not change the other number.  So zero is the
additive identity.  Add zero to anything and it doesn't change.
Similarly, 1 is the multiplicative identity.  Multiply anything
by 1 and it doesn't change.

The additive inverse of a number is something you can add to
the original number to get the additive identity.  Additive
inverses always exist.  The inverse of 4 is -4 since if you
add 4 and -4, you get 0, the additive identity.

All numbers but 0 have a multiplicative inverse.  The inverse
of 7 is 1/7; the inverse of 92 is 1/92.  But 0 has no inverse
because you can't multiply anything by zero and get 1 - you
always get zero.

Finally, the distributive law shows the interaction between
addition and multiplication.  It states that:

    ax(b+c) = axb + axc.

In other words, you can either add the b and c first, before
multiplying, or you can multiply a by each of b and c, and
add the results, and the final answer will be the same.  The
distributive law holds for any three real numbers a, b, and c.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Addition
Elementary Definitions
Elementary Division
Elementary Multiplication
Elementary Subtraction
Middle School Definitions
Middle School Division

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