Proof Question: Prove integer + 1/2 is not an integer

In summary, there is a contradiction in the statement that 0.5 plus an integer equals an integer. It is not acceptable to say that by definition of integers, 0.5 plus an integer is not an integer. Instead, one must prove it by showing that dividing 0.5 by 2 always results in a fraction, not a whole number. Additionally, it is possible to prove that 1/2 is not an integer by using its definition and the well-ordering principle of positive integers.
  • #1
kamui8899
15
0
I was in the middle of proving something when I reached a contradiction, that .5 + an integer = an integer. However, this cannot be true, and I'm curious if its acceptable to just say that by definition of integers .5 + an integer is not an integer, or do I have to prove it?
Furthermore, if I have to prove it, how would I go about this? I would say let x and y be integers, so x + .5 = y, right?
Since x and y are integers then x = x/1 and y = y/1, so x/1 + 1/2 = y/1.
2x/2 + 1/2 = y/1
so
(2x + 1/2)/2 = y/1
and then... If I said that 2x +1/2 was not a whole number so dividing it by two must give a fraction, and thus it can't be reduced to a whole number over 1... That doesn't sound like it works though becuase its just restating what I was trying to prove... Not to mention I'm not sure I can even say that a fraction divided by two doesn't give a whole number... Any ideas? Thanks.
 
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  • #2
Are you saying that

[tex]\frac{(2x+\frac{1}{2})}{2}=\frac{y}{1}[/tex]

shows that [itex]\frac{1}{2}+n1=n2[/itex]?

n is an arbitrary integer.

(there's a problem with this post. unwanted spacing)
 
  • #3
There is a MUCH quicker way.
supose 0.5+N=M where N and M are integers. Then 0.5=M-N. But if M and N are integers, M-N is an integer. But this implies 0.5 is an integer. This is a cointradiction. Done.
 
  • #4
How do you figure Setting x=x/1 forces x to be an integer?
 
  • #5
To show 1/2 is not an integer, use its definition:

1/2 is the number that satisfies the equation 2x=1.

Now, for any integer n, 2n is never 1 (why? because 2n is always even and 1 is odd, and no integer is both even and odd.) Hence, 1/2 cannot be an integer.

Of course, to argue that no integer is both even and odd uses the quotient-remainder theorem, which in turn relies on the well-ordering principle of the positive integers.
 

What does it mean for an integer + 1/2 to not be an integer?

When we say that an integer + 1/2 is not an integer, it means that the resulting number is not a whole number. In other words, it has a decimal component.

Can you provide an example of an integer + 1/2 that is not an integer?

Yes, for example, if we take the integer 3 and add 1/2 to it, we get 3.5. This number is not a whole number and therefore not an integer.

Why is it impossible for an integer + 1/2 to be an integer?

This is because integers are defined as whole numbers, without any fractions or decimals. When we add 1/2 to an integer, we are essentially introducing a fractional component, making it impossible for the result to be a whole number and thus not an integer.

How can we prove that an integer + 1/2 is not an integer?

We can prove this using basic algebraic principles. We can take any integer, add 1/2 to it, and show that the resulting number is not a whole number. This will serve as a proof that an integer + 1/2 is not an integer.

Is there a real-life application for proving that an integer + 1/2 is not an integer?

Yes, this concept is important in many fields such as mathematics, physics, and computer science. For example, in computer programming, understanding the difference between integers and non-integers is crucial for writing accurate and efficient code.

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