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Is Zero a Real Number?


Date: 09/26/2001 at 15:27:49
From: Patrick McNamara
Subject: "0" and "-0"

Dr. Math, 

My first question is, "Is 0 a real number?"
I will, based upon your answers, be asking you more questions.


Date: 09/26/2001 at 23:42:09
From: Doctor Peterson
Subject: Re: "0" and "-0"

Hi, Patrick.

Yes, since zero is an integer, and all integers are real numbers, zero 
is a real number.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   


Date: 09/27/2001 at 15:09:40
From: (anonymous)
Subject: "0"

My friend Sean and I think we have disproved that 0 is a real number.  
0 doesn't follow several of the properties, such as the inverse 
property of multiplication, and several division properties. In the 
definition of opposites, "Any 2 points on a number line that are 
equidistant from the origin (0)." Also, "any real number multiplied 
by -1 will give you the opposite of that number." So, in essence, 
0 * -1 =0. However, 0 could not possibly be the opposite of 0 
according to the definition of opposites, because, even though 0 is 
equidistant from zero in coordinates to 0, they are not two separate 
points, thus proving that 0 doesn't follow all real number properties.


Date: 09/27/2001 at 22:17:40
From: Doctor Peterson
Subject: Re: "0"

Hi, Patrick and Sean.

You have to be very careful when you talk about definitions. The 
concept of real number starts with integers, which include zero, as I 
said, so their properties are determined based on a definition that 
includes zero as a real number! To then use those properties to prove 
that zero is not one of the numbers they refer to suggests that you 
have misunderstood them. And in fact, it is all too common to state 
such properties too broadly, without giving all the conditions.

First, nobody says that all real numbers have an inverse; rather, we 
say that all real numbers EXCEPT ZERO have an inverse. Yes, that means 
that zero is unique among the real numbers; but it doesn't mean it is 
not one of them. Zero is in fact the starting point of the whole 
system (you use it as part of your definition of "opposite"), and it's 
really not too surprising that the foundation should have some unique 
properties.

Second, when we say that every real number has an opposite, that does 
not mean that the opposite has to be a DIFFERENT real number. 
Mathematicians use words very carefully, and when we mean "different" 
or "distinct" or "unique" we say so explicitly. Zero is its own 
opposite; there is no other number the same distance (zero) from the 
origin.

Your definition of "opposite" is a poor one, and not one on which we 
would base any careful reasoning. It's really just a description, to 
help you visualize the concept. Since it sounds as if you enjoy deeper 
mathematical thinking, why don't you look in a good library for a 
solid text on algebra or number theory, one that you can follow but 
that you find challenging, and start going through it. Once you get a 
feel for how really solid definitions and theorems are made, you'll be 
able to take off on your own and start really discovering good ideas.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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