Why is -1/(0^2) = -Infinity? Exploring the Difference from 1/0

In summary, the difference between -1/(0^2) and 1/0 on a calculator is due to how the calculator calculates the limit of these functions as they approach 0. While -1/(0^2) approaches negative infinity, 1/0 is undefined. This is because -1/(0^2) has a continuous extension to x=0 in the extended real numbers, while 1/0 does not. It is important to use critical thinking and not solely rely on a calculator for mathematical subtleties.
  • #1
cscott
782
1
Why does my calculator tell me -1/(0^2) = -infinity. How is this different from 1/0?
 
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  • #2
1/0 isn't anything, since 1/x approaches infinity if x --> 0 from the right (x>0) and 1/x approaches -infinity if x --> 0 from the left (x<0).
By contrast, -1/x^2 is always negative, so it approaches negative infinity as x --> 0 no matter which direction you come from.
Bear in mind, I'm not saying that -1/0^2 equals - infinity. It's not really defined actually.
What kind of calculator are you using anyway? Mine always says "error - divide by 0" if I put in 1/0.
 
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  • #3
I think his question is why his calculator (TI-89) says 1/0 is undefined but 1/02 says infinity.

It just has to do with how the calculator calculates things I guess.

Edit: I tried it for other powers and it seems like a/0n is given as infinity for even n and "undef" for odd n if n is positive.

If n is negative it gives a/0n as 0. If n is zero it gives it as a, but writes a warning message saying that 00 was replaced by 1.Edit: I think latex is broken...
 
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  • #4
Alright, thanks.

...and I agree, Latex is broken.
 
  • #5
Never trust your calculator for mathematical 'subtleties' like this, use your mind :smile:
 
  • #6
Incidnetally, the function

-1/x²

does have a continuous extension to x=0 in the extended real numbers. Whereas

-1/x

does not.
 
  • #7
Hurkyl said:
Incidnetally, the function

-1/x²

does have a continuous extension to x=0 in the extended real numbers. Whereas

-1/x

does not.
That makes sense. The calculator is probably computing the limit of those functions as they go to 0, which is infinity for even functions and undefined for odd.
 

1. Why is -1/(0^2) equal to -Infinity?

The mathematical concept of division by zero is undefined, meaning that it has no definite value. In this case, -1 divided by 0 squared results in a number that is infinitely small, but not equal to zero. This is why it is represented as -Infinity.

2. Why is the result different from 1/0?

1 divided by 0 is also undefined, but it is represented as Infinity because it is a number that is infinitely large. This is because as the denominator approaches 0, the quotient approaches infinity.

3. Can -1/(0^2) be simplified to a different value?

No, it cannot be simplified to a different value. As mentioned before, division by zero is undefined, and the result of -Infinity is the most accurate representation.

4. How is this concept relevant in mathematics?

The concept of division by zero is relevant in mathematical proofs and equations, as it can lead to contradictions and inconsistencies. It is also used in fields such as calculus and complex analysis to understand the behavior of functions near points where the denominator is equal to zero.

5. Can -1/(0^2) ever have a finite value?

No, it cannot. The result will always be -Infinity, regardless of the value of -1. This is because any number divided by a number that is infinitely small will result in a number that is infinitely large.

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