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### Closure Axiom

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Date: 06/12/97 at 19:01:11
From: Stacy Stojakovic
Subject: Closure Axiom - What does it really mean?

In my algebra book, the closure axiom states:

The set of real numbers is closed under addition and multiplication.

That is,

1. x+y is a unique real number
2. xy is a unique real number

I have re-examined this axiom many times and I am still confused.

Could you please explain it to me?
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Date: 06/13/97 at 08:29:49
From: Doctor Anthony
Subject: Re: Closure Axiom - What does it really mean?

The idea of 'closure' is actually very simple. If you add together
two whole numbers, you will always get another whole number. If you
multiply two whole numbers, you will get a whole number as a result.
So we say that whole numbers (integers) are 'closed' under the

What about division?  Well 12 divided by 2 is 6, which is a whole
number, so in this case we get a whole number result. But 12 divided
by 5 = 2 and 2/5, so now we have moved out of the field of whole
numbers. If we divide two whole numbers we cannot guarantee that the
result will still be a whole number. So the set of whole numbers is
not closed under the operation of division.

Positive whole numbers are closed under addition - you always get a
positive whole number in the result. But they are not closed under
subtraction, since, for example,  4 - 9 = -5  and -5 is not a positive
whole number.

To decide whether a set of numbers is closed under some operation or
other, look for cases where the result is no longer in the set you
started with.

In the case of real numbers, which include positive, negative,
fractional, and irrational (like sqrt(2)) numbers, the operations of
addition, multiplication, division and subtraction are all closed (apart
from division by zero which is not defined). But taking square roots is
not closed because if, for example, we try sqrt(-5), we no longer get a
real number as a result. In fact, we have moved into the realm of
complex numbers.

The idea of 'closure' does not apply only to operations on sets of
numbers. We can have operations on vectors and matrices, for example,
which might or might not be closed.

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Basic Algebra

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