My brothers was to teach me the basics that chathamtownfc.netematics and we had actually some confusion about the optimistic and an adverse behavior the Zero. After reading a few post ~ above this we involved know that it depends on the paper definition of its use.

Why do we take it 1/0 as confident infinity quite than negative infinity (we come close come zero from an unfavorable axis)?




You are watching: What is -1^0

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The various other comments space correct: $frac10$ is undefined. Similarly, the limit of $frac1x$ together $x$ ideologies $0$ is additionally undefined. However, if you take it the border of $frac1x$ together $x$ viewpoints zero indigenous the left or native the right, friend get negative and confident infinity respectively.


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$1/x$ does tend to $-infty$ as you technique zero from the left, and also $infty$ as you approach from the right:

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That these boundaries are not equal is why $1/0$ is undefined.




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The place where you frequently see $1 colorred/ 0 = infty$ is when doing arithmetic in the projective line. (I"ve added color come $colorred/$ to much better distinguish it from the ordinary department operation top top the genuine numbers) The binary operation $colorred/$ is defined for every pair the projective real numbers except $(0,0)$ and also $(infty, infty)$:

$ x colorred/ y = x/y$ once $y eq 0$$ x colorred/ 0 = infty$$ x colorred/ infty = 0$$infty colorred/ x = infty$

where $x,y$ represent ordinary real numbers. (one can define the various other arithmetic to work too)

The projective line has actually only one unlimited element. In the projective line, the same number $infty$ is at both "ends" that the plain line. Over there is another common number mechanism -- the prolonged real number -- that has two limitless elements: $+infty$ and $-infty$. Make details note that $1 colorcyan/ 0$ is undefined because that the arithmetic of extended real numbers. (where again I"ve added color come distinguish)

Unfortunately, people often usage $infty$ instead of $+infty$. So, when someone writes $infty$, it have the right to be unclear whether or no they room doing arithmetic in the projective genuine line, or in the expanded real line.