The answers to 1^(1/2) and 1/0

[robly18]

Not sure if it's wrong of me to bump this... but...

Why do people consider 1/0 to be undefined anyways?

Also, is it just +∞ or ±∞?

I think it's ±∞ because... well, if you look at a reciprocal function, you'll notice the 2 branches move in opposite directions at the graphs vertical asymptote(s), thus they seem to reach either positive or negative ∞.

Well, it is... Sorta.

The thing is, infinity isn't a number. In the real number system anyway. I recommend you take a look at the thread they're talking about.

Long story short, infinity isn't a number in the real number system, so we say it's undefined, like the square root of negative one.

However, there are number systems in which it exists, like the complex numbers. However, by using those systems you lose some things. Specifically, you lose the ability to order things.

See, if 1/0 = ±∞, then you must ask yourself: is 1 inferior or superior to it?

Well, 1 < +∞, but 1 > -∞. Since both of those are equal to 1/0, then 1/0 < 1 < 1/0, making 1/0 < 1/0. We get a contradiction.

As a result, we call the extended real number line, as it's called, not ordered. things like 1 < 2 become meaningless, because the whole concept is nullified.

So basically 1/0 is undefined in the reals but defined in the extended number line the same way sqrt(-1) is undefined in reals but defined in the complexes.

[wtg62]

Oh, alright.

[exfret]

Robly expressed the more official view of 1/0. This is my viewpoint:

I still see no loss by adding infinity to the number system. Operators like > and < will have to be redefined, sure, but that doesn't lose you anything. Also, infinity is more like a place you reach when you go so far. All infinities kind of exist there in the same place. Infinity/infinity is exactly like 0/0. Also, I disagree with people who say it's only a concept (not talking about you robly, you acknowledged it's possible number-ness). Simply define it as the for which + 1 = . It's also useful to use it as a number in a variety of other contexts as a number. There are many things that I think fit better together if 1/0 = infinity, but for some reason, we don't let that be.