My brother was teaching me the basics that usmam.orgematics and we had actually some confusion around the optimistic and negative behavior the Zero. After analysis a few post on this we pertained to know the it depends on the paper definition of the use.

Why execute we take it 1/0 as hopeful infinity quite than an adverse infinity (we come close come zero from negative axis)?




You are watching: 1 = 0 is infinity or not defined

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The other comments are correct: $\frac10$ is undefined. Similarly, the limit of $\frac1x$ as $x$ approaches $0$ is additionally undefined. However, if you take the limit of $\frac1x$ as $x$ philosophies zero native the left or from the right, girlfriend get negative and optimistic infinity respectively.


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

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That these boundaries are no equal is why $1/0$ is undefined.




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The ar where you generally see $1 \colorred/ 0 = \infty$ is once doing arithmetic in the projective line. (I"ve added color to $\colorred/$ to much better distinguish it from the ordinary division operation on the actual numbers) The binary procedure $\colorred/$ is identified for every pair that projective real numbers other than $(0,0)$ and $(\infty, \infty)$:

$ x \colorred/ y = x/y$ once $y \neq 0$$ x \colorred/ 0 = \infty$$ x \colorred/ \infty = 0$$\infty \colorred/ x = \infty$

where $x,y$ signify ordinary genuine numbers. (one can define the various other arithmetic operations too)

The projective line has actually only one unlimited element. In the projective line, the exact same number $\infty$ is in ~ both "ends" the the simple line. Over there is an additional common number mechanism -- the prolonged real number -- that has two boundless elements: $+\infty$ and $-\infty$. Make specific note that $1 \colorcyan/ 0$ is undefined for the arithmetic of extended real numbers. (where again I"ve included color to distinguish)

Unfortunately, human being often usage $\infty$ rather of $+\infty$. So, once someone writes $\infty$, it deserve to be unclear whether or not they room doing arithmetic in the projective real line, or in the expanded real line.